A pathwise regularization by noise phenomenon for the evolutionary p-Laplace equation

Abstract

We study an evolutionary p-Laplace problem whose potential is subject to a translation in time. Provided the trajectory along which the potential is translated admits a sufficiently regular local time, we establish existence of solutions to the problem for singular potentials for which a priori bounds in classical approaches break down, thereby establishing a pathwise regularization by noise phenomena for this nonlinear problem.

1 Introduction

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb {R}^d$, $T>0$ and $N \in {\mathbb {N}}$. We are interested in the evolutionary p-Laplace system

$$\begin{aligned} \begin{aligned} \partial _t u - \textrm{div}\,S(\nabla u)&= b(u){} & {} \text { on } \Omega \times [0,T], \\ u|_\Omega&=0{} & {} \text { on } \partial \Omega \times [0,T],\\ u(0, \cdot )&=u_0{} & {} \text { on } \Omega , \end{aligned} \end{aligned}$$

(1.1)

where $S(\xi ) = \left|\xi \right|^{p-2} \xi \in \mathbb {R}^{d\times N}$, $p\in (1,\infty )$ and $b:{\mathbb {R}}^N \rightarrow {\mathbb {R}}^N$.

The p-Laplace operator $\textrm{div}\, S(\nabla u)$ is a prominent example of a maximal monotone operator. The famous theory of monotone operators traces back to the early works of Minty [32] and Browder [5]. It inspired many mathematicians to study well-posedness of monotone evolution equations and perturbations of it, see, e.g., [1, 8, 28, 29, 31, 34, 35]. However, if the potential b(u) cannot be treated as a compact perturbation, well-posedness breaks down and solutions may blow up in finite time, see, e.g., [12, 38].

In the following, we intend to study the effect of translations of the potential b along so-called regularizing paths w, that is, we are interested in the problem

$$\begin{aligned} \begin{aligned} \partial _t u - \textrm{div}S(\nabla u)&= b(u-w){} & {} \text { on } \Omega \times [0,T], \\ u|_\Omega&=0{} & {} \text { on } \partial \Omega \times [0,T],\\ u(0, \cdot )&=u_0{} & {} \text { on } \Omega . \end{aligned} \end{aligned}$$

(1.2)

From a physical point of view, translating the potential in time can be interpreted as uncertainty as to the location of the origin of the potential. By a “regularizing path”, we will understand a continuous path w that admits a sufficiently regular local time¹ such that techniques from pathwise regularization by noise à la [7, 16, 27] become applicable. Let us briefly sketch some main ideas of these approaches, in particular, in the spirit of [27], which we will be able to employ also in the study of (1.2).

1.1 Pathwise regularization by noise in a nutshell

The starting point for investigations of this type usually consists in the study of averaging operators along a path w, given by

$$\begin{aligned} (T^{-w}_tb)(u):= \int _0^tb(u-w_s)ds. \end{aligned}$$

As was recognized already in [7], paths enjoying so-called $\rho$ irregularity [7, Definition 1.3], i.e., paths whose Fourier transform of the occupation measure decreases sufficiently rapidly, lead to a regularization effect in that the function $u\mapsto (T^{-w}_tb)(u)$ will enjoy higher regularity than b. The reason for this gain of regularity can be made clear as follows: Assuming w to also admit a local time L, we can rewrite the averaging operator thanks to the occupation times formula as

$$\begin{aligned} \int _0^tb(u-w_s)ds=\int _{\mathbb {R}^N}b(u-z)L_t(z)dz=(b*L_t)(u). \end{aligned}$$

If the Fourier transform of the occupation measure is decreasing rapidly in a certain quantifiable sense, the local time L will enjoy some high quantifiable spatial regularity. By Young’s inequality (see, for example, (2.1) or [30] for a generalization to the Besov space setting), this implies that the regularity of $T^{-w}_tb$ will essentially increase by the regularity of $L_t$ with respect to the regularity of b. While the canonical paths for which this gain in regularity can be quantified are realizations of fractional Brownian motion [7, 14, 16], $\rho$ irregularity is in fact a typical property among Hölder continuous paths in the sense of prevalence [15].

Taking this observation of increased regularity of the averaging operator as a starting point, one can exploit this local gain of regularity to further study Riemann sum-type expressions of the form

$$\begin{aligned} {\mathcal {I}}^n_{T}=\sum _{[s,t]\in {\mathcal {P}}^n([0,T])}(T^{-w}_{s,t}b)(u_s) \end{aligned}$$

for partitions ${\mathcal {P}}^n([0,T])$ of [0, T] and a continuous function u. The tool that then ensures the convergence of such Riemann sums in the limit $|{\mathcal {P}}^n|\rightarrow 0$ is Gubinelli’s Sewing Lemma [11, 24], also cited in the Appendix as Lemma 6.3. Let us stress at this point already that since the Sewing Lemma is formulated in a “Hölder-space setting”, convergence of $({\mathcal {I}}^n)_n$ will, however, always require at least some Hölder regularity of the function u. Provided this is available, i.e., u is sufficiently Hölder regular and the averaging operator $(T^{-w}b)$ sufficiently regular in time and space, the Sewing Lemma ensures the convergence of $({\mathcal {I}}^n)_n$ as $|{\mathcal {P}}^n|\rightarrow 0$. Note that as this convergence eventually only requires information on the regularity of the averaging operator and not the regularity of b, this construction (which can be shown to coincide with the classical Lebesgue integral for regular b) naturally extends the definition of the Lebesgue integral to irregular and potentially even distributional b, i.e., one can define

$$\begin{aligned} {\mathcal {I}}_{T}=\lim _{|{\mathcal {P}}^n|\rightarrow 0}\sum _{[s,t]\in {\mathcal {P}}^n([0,T])}(T^{-w}_{s,t}b)(u_s)=:\int _0^Tb(u_s-w_s)ds \end{aligned}$$

Moreover, even in the case of a regular nonlinearity b where the above can be defined as a classical Lebesgue integral, this definition provides alternative a priori bounds through the Sewing Lemma of which we will crucially make use in our work.

1.2 Application to the p-Laplace equation with shifted potential

A first approach in studying (1.2) might consist in investigating the weak formulation

$$\begin{aligned} \langle u_t-u_s, \varphi \rangle +\int _s^t \langle S(\nabla u_r), \nabla \varphi \rangle dr=\int _s^t\langle b(u_r-w_r), \varphi \rangle dr, \end{aligned}$$

where the right-hand side should be interpreted as

$$\begin{aligned} \int _s^t\langle b(u_r-w_r), \varphi \rangle dr:=\lim _{|{\mathcal {P}}^n|\rightarrow 0}\sum _{[s',t']\in {\mathcal {P}}^n([s,t])}\langle (T^{-w}_{s',t'}b)(u_{s'}), \varphi \rangle . \end{aligned}$$

Note, however, that the classical monotone operator approach to this problem, working on the Gelfand triple $\big (W^{1,p}_0(\Omega )\cap L^2(\Omega ), L^2(\Omega ), (W^{1,p}_0(\Omega )\cap L^2(\Omega ))'\big )$, only yields a priori bounds that permit to conclude $u\in C([0,T], L^2(\Omega ))$. In particular, no additional Hölder regularity in time on this spatial regularity scale is obtained, meaning the Sewing argument can’t be closed making the right-hand side ill defined. To circumvent this problem, we employ a strong formulation to the problem, i.e., we strive for solutions u that satisfy

$$\begin{aligned} u_t-u_s-\int _s^t \text{ div } S(\nabla u_r)dr=\int _s^t b(u_r-w_r)dr \end{aligned}$$

(1.3)

understood as an equality in $L^2(\Omega )$ and where for singular b the right-hand side is understood in the sense

$$\begin{aligned} \int _s^t b(u_r-w_r):=\lim _{|{\mathcal {P}}^n|\rightarrow 0}\sum _{[s',t']\in {\mathcal {P}}^n([s,t])} (T^{-w}_{s',t'}b)(u_{s'}), \end{aligned}$$

(1.4)

where the convergence on the right-hand side holds in $L^2(\Omega )$, uniformly in time on [0, T]. Note that while such strong solutions naturally require higher regularity of the initial condition, namely $u_0\in W^{1,p}_0(\Omega )$, they allow for further a priori bounds in $C^{0,1/2}\big ([0,T];L^2(\Omega )\big )$ provided b is sufficiently regular (refer to Sect. 3). Having for regular b such a priori bounds at our disposal, we can then harness the regularizing effect of the averaging operator $T^{-w}$ as discussed above to obtain a priori bounds that are robust even when considering singular potentials b (refer to Sect. 4). The so obtained new a priori bounds for singular b can then be used to obtain solutions to (1.3) using classical monotonicity arguments (refer to Sect. 5). In summary, this allows us to prove our main theorem:

Theorem 1.1

(Existence of robustified solution) Let $d, N\in {\mathbb {N}}$, $\Omega$ a bounded Lipschitz domain in ${\mathbb {R}}^d$, $p > \frac{2d}{d+2}$ and $u_0 \in W^{1,p}_0(\Omega )$. For $r\in [1,\infty )$ and $q\in [r, \infty )$ let $b:\mathbb {R}^N\rightarrow \mathbb {R}^N$ satisfy $b\in L^{2q}({\mathbb {R}}^N)$. Suppose that $w:[0, T]\rightarrow \mathbb {R}^N$ is continuous and admits a local time L which satisfies $L\in C^{0,\gamma }\big ([0,T]; W^{1, r'}({\mathbb {R}}^N)\big )$ for some $\gamma \in (1/2, 1)$.

Then there exists a robustified solution

$$\begin{aligned} u \in \left\{ v \in L^\infty \big (0,T; L^2(\Omega ) \cap W^{1,p}_0(\Omega ) \big )\big | \, \partial _t v, \textrm{div}S(\nabla v) \in L^2([0,T] \times \Omega ) \right\} \end{aligned}$$

to (1.2) in the sense of Definition 2.2. Moreover, the following a priori bound is valid

$$\begin{aligned} \begin{aligned}&\left\Vert u\right\Vert _{L^\infty (0,T; L^2(\Omega ))}^2 + \left\Vert \nabla u\right\Vert _{L^\infty (0,T; L^p(\Omega ))}^p + \left\Vert \partial _t u\right\Vert _{L^2([0,T]\times \Omega )}^2 + \left\Vert \textrm{div}S(\nabla u)\right\Vert _{L^2([0,T]\times \Omega )}^2 \\&\hspace{3em} \lesssim \left\Vert u_0\right\Vert _{L^2(\Omega )}^2+\left\Vert \nabla u_0\right\Vert _{L^p(\Omega )}^p + \left\Vert b\right\Vert _{L^{2q}({\mathbb {R}}^N)}^4\left\Vert L\right\Vert _{C^{0,\gamma }([0,T];W^{1, r'}({\mathbb {R}}^N))}^2. \end{aligned} \end{aligned}$$

(1.5)

Suppose moreover that b satisfies the monotonicity condition, for all $u,v \in {\mathbb {R}}^N$,

$$\begin{aligned} (b(u)-b(v))\cdot ( u-v) \;\leqslant \;0, \end{aligned}$$

(1.6)

then robustified solutions to (1.2) in the sense of Definition 2.2 are unique.

Let us illustrate this established regularization effect by means of a concrete example. A detailed verification of the claims of the example can be found in “Appendix 6”.

Example 1.2

Let $K > 0$ and define the potential

$$\begin{aligned} b(u) := - |u|^{\eta -1}u {{\textbf {1}}}_{\{|u|\;\leqslant \;K\}}. \end{aligned}$$

(1.7)

Let $d\in {\mathbb {N}}$ and $p> \frac{2d}{d+2}$. Then the following statements are true:

• (1) If $\eta \;\leqslant \;-1$, then (1.1), i.e., the problem without regularizing path does not have a weak solution for $u_0=0$.

• (2) Let $\eta \in (-N/2,0)$ and

  $$\begin{aligned} H <\frac{1}{(2-4\eta ) \vee N} \end{aligned}$$

  (1.8)

  and $w^H$ be a N-dimensional fractional Brownian motion with Hurst parameter H. Then for almost any realization $w^H(\omega ):[0,T]\rightarrow \mathbb {R}^N$ (1.2) has a robustified solution for any $u_0\in W^{1, p}_0(\Omega )$.

Essentially, (1) shows that in the unperturbed setting the origin is a singular state, which is due to the singularity of b in zero. In contrast to this, the presence of the highly oscillating path $w^H(\omega )$ ensures that the solution u does not spend too much time in the singularity of b. Condition (1.8) ensures that this effect is quantitatively sufficiently strong for the singularity not to obstruct existence theory, as formulated in our main theorem above. Overall, we can thus observe a regularization phenomenon in dimension $N\;\geqslant \;3$. In particular, for $N=3$ and $\eta =-1$, there is no weak solution to (1.1) for $u_0=0$, while (1.2) with $u_0=0$ admits a robustified solution for almost every realization $w^H(\omega )$ of a fractional Brownian motion, provided $H<1/6$.

Remark 1.3

Let us stress that we are employing a pathwise regularization by noise argument in the spirit of [27] relying on the study of local times and their regularities and not in the spirit of [7, 16] based on the study of averaging operators. While results on the regularizing effect of averaging operators are slightly less optimal for realizations of fractional Brownian motion for example, they allow for a completely pathwise, i.e., analytical treatment. In contrast, arguing in the spirit of [7, 16] would require to employ a tightness argument. Moreover, if the perturbing path is the realization of a stochastic process w, note that as our proof employs a compactness argument, we lose measurability of our solution with respect to the probability space on which w is defined.

1.3 Remarks on the literature

Starting with the seminal work [7], taken up again by [16, 17, 27, 36] pathwise regularization by noise has seen considerable developments in recent years. Areas in which such techniques have been successfully implemented include particle systems [26], distribution dependent SDEs [21, 22], multiplicative SDEs [3, 6, 10, 20, 33] and perturbed SDEs [23]. An extension of pathwise regularization techniques to the two parameter setting with applications to regularization by noise for the stochastic wave equation was established in [4]. Pathwise regularization by noise for the multiplicative stochastic heat equations with spatial white noise was treated in [2, 9].

1.4 Outline of the paper

In Sect. 2 we introduce basic notation and the notion of solution. Section 3 addresses quantified but implicit a priori bounds for strong solutions. In Sect. 4 we close the a priori bounds from Sect. 3. Lastly, Sect. 5 copes with the identification of the limit and a discussion on uniqueness. In Appendix 6 we verify Example 1.2, present results related to occupation measures, local times and sewing techniques and give details on the identification of limits for monotone equations.

2 Mathematical setup

Let $\Omega \subset \mathbb {R}^d$ for $d \;\geqslant \;1$ be a bounded Lipschitz domain. For some given $T>0$ we denote by $I:= [0,T]$ the time interval and write $\Omega _T:= I \times \Omega$ for the time space cylinder. We write $f \lesssim g$ for two nonnegative quantities f and g if f is bounded by g up to a multiplicative constant. Accordingly, we define $\gtrsim$ and $\eqsim$. Moreover, we denote by c a generic constant which can change its value from line to line. For $r\in [1, \infty ]$, we denote by $r' = r/(r-1)$ its Hölder conjugate. We do not distinguish between scalar, vector and matrix-valued functions.

2.1 Function spaces

As usual, for $q \in [1,\infty ]$, let $L^q(\Omega )$ denote the Lebesgue space and $W^{1,q}(\Omega )$ the Sobolev space on the domain $\Omega$, respectively. Furthermore, we denote by $W^{1,q}_0(\Omega )$ the Sobolev space with zero boundary values. For $q < \infty$ it is the closure of $C^\infty _c(\Omega )$ (smooth functions with compact support) in the $W^{1,q}(\Omega )$-norm. Additionally, we denote by $W^{-1,q'}(\Omega )$ the dual of $W^{1,q}_0(\Omega )$. We abbreviate function spaces on the domain by $L^p_x:= L^p(\Omega )$ and on the full space by $L^p:=L^p(\mathbb {R}^N)$ with suitable modifications for Sobolev norms. The inner product in $L^2_x$ is denoted by $(\cdot , \cdot )$, and duality pairings are written as $\langle \cdot , \cdot \rangle$.

For a Banach space $\left( X, \left\Vert \cdot \right\Vert _X \right)$ let $L^q(I;X)$ be the Bochner space of Bochner-measurable functions $u: I \rightarrow X$ satisfying $t \mapsto \left\Vert u(t)\right\Vert _X \in L^q(I)$. Moreover, C(I; X) is the space of continuous functions with respect to the norm-topology. We also use $C^{0,\alpha }(I;X)$, $\alpha \in (0,1]$, for the space of $\alpha$-Hölder continuous functions. For $u\in C^{0,\alpha }(I;X)$, we denote by $\left[ u\right] _{C^{0, \alpha }(I; X)}:=\sup _{s\ne t\in I}\frac{\left\Vert u_t-u_s\right\Vert _X}{|t-s|^\alpha }$ the corresponding semi-norm and by $\left\Vert u\right\Vert =\sup _{t\in I}\left\Vert u_t\right\Vert _x+\left[ u\right] _{C^{0, \alpha }(I; X)}$ the corresponding norm. We abbreviate the notation $L^q_t X:= L^q(I;X)$ and $C_t X = C(I;X)$. If a Banach space $\left( X, \left\Vert \cdot \right\Vert _X \right)$ embeds continuously into another Banach space $\left( Y, \left\Vert \cdot \right\Vert _Y \right)$, we write $X\hookrightarrow Y$. If the embedding is moreover compact, we write $X\hookrightarrow \hookrightarrow Y$. If a sequence $(u_n)_n\subset X$ converges to $u\in X$ weakly, respectively weakly star, in a Banach space $\left( X, \left\Vert \cdot \right\Vert _X \right)$, we write $u\rightharpoonup u$, respectively, $u\overset{*}{\rightharpoonup } u$.

For $s\in \mathbb {R}^+$ we further note by $H^{s}$ the space of Bessel potentials,

$$\begin{aligned} H^{s}:=\{ f\in {\mathcal {S}}'|\ \left\Vert f\right\Vert _{H^{s}}:=\left\Vert {\mathcal {F}}^{-1} (1+|\xi |^2)^{s/2}{\mathcal {F}}f\right\Vert _{L^2}<\infty \} \end{aligned}$$

Let us also recall a particular instance of Young’s convolution inequality adapted to our setting, i.e.,

$$\begin{aligned} \left\Vert f*g\right\Vert _{C^{0,1}}\lesssim \left\Vert f\right\Vert _{L^r}\left\Vert g\right\Vert _{W^{1,r'}}, \end{aligned}$$

(2.1)

which is a consequence of $D_x (f*g)=f*(D_xg)$ and Young’s convolution inequality in the classical setting.

2.2 Solution concepts

In the following, let us discuss different notions of solutions to (1.2). We begin with the classical notions of weak and strong solutions before passing on to so-called robustified solutions that exploit the gain in regularity due to the regularizing path w as discussed in the introduction.

Definition 2.1

(Classical) A function u is called weak solution to (1.2) if

• (1) (Regularity) $u \in C_{t} L^2_x \cap L^{p}_t W^{1,p}_{0,x}$, $b(u_\cdot - w_\cdot ) \in L^{p'}_t W^{-1,p'}_x$ and

• (2) (Tested equation) for all $t \in I$ and $\xi \in C^\infty _{c,x}$

  $$\begin{aligned}&\int _\Omega (u_t - u_0) \cdot \xi \,\text {d}x + \int _0^t \int _\Omega S(\nabla u) : \nabla \xi \,\text {d}x \,\text {d}s \nonumber \\&\quad = \int _0^t \left\langle b(u - w), \xi \right\rangle _{W^{-1,p'}\times W^{1,p}_{0,x}} \,\text {d}s. \end{aligned}$$

  (2.2)

A weak solution u is called strong solution to (1.2) if additionally

• (1) (Regularity) $\partial _t u, \, \textrm{div}S(\nabla u), \, b(u_\cdot - w_\cdot ) \in L^{2}_t L^2_x$ and

• (2) (Point-wise equation) for almost all $(t,x) \in \Omega _T$

  $$\begin{aligned} \begin{aligned} \partial _t u - \textrm{div}S(\nabla u) = b(u- w). \end{aligned} \end{aligned}$$

  (2.3)

Definition 2.2

(Robustified) We call u a solution to (1.2) in the robustified sense if

$$\begin{aligned} u \in \left\{ v \in C^{0, 1/2}_tL^2_x \cap L^\infty _tW^{1,p}_{0,x} \big | \, \partial _t v, \textrm{div}S(\nabla v) \in L^2_t L^2_x \right\} , \end{aligned}$$

(2.4)

and for any $t\in I$ we have

$$\begin{aligned} u_t-u_0-\int _0^t \textrm{div}S(\nabla u_r) \,\text {d}r=({\mathcal {I}}A^u)_{0,t} \end{aligned}$$

(2.5)

understood as an equality in $L^2_x$ where ${\mathcal {I}}A^u$ denotes the sewing² of the germ

$$\begin{aligned} A^u_{s,t}=(b*L_{s,t})(u_s). \end{aligned}$$

In this case, we have, in particular, also $\partial _t{\mathcal {I}}A^u\in L^2_tL^2_x$.

The next two lemmata verify that the concept of robustified solutions coincides with classically defined strong solutions in the smooth setting.

Lemma 2.3

Let b be smooth and bounded and assume that $w:[0, T]\rightarrow \mathbb {R}^N$ is measurable and admits a local time $L\in C^{0, 1/2+\epsilon }_tL^2_x$ for some $\epsilon >0$. Then any strong solution $u\in C^{0,1/2}_tL^2_x$ to (1.2) is a solution in the robustified sense.

Proof

Since u is a strong solution, it suffices to show that for any $s<t\in [0,T]$ we have

$$\begin{aligned} ({\mathcal {I}}A^u)_{s,t}=\int _s^t b(u_r-w_r) dr \end{aligned}$$

Remark first that $A^u$ does admit a sewing with values in $L^2_x$. Indeed,

$$\begin{aligned} \left\Vert (\delta A^u)_{svt}\right\Vert _{L^2_x}&=\left\Vert (b*L_{v,t})(u_s)-(b*L_{v,t})(u_v)\right\Vert _{L^2_x}\\&\lesssim \left\Vert b*L_{v,t}\right\Vert _{C^{0,1}}\left\Vert u_s-u_v\right\Vert _{L^2_x}\\&\lesssim \left\Vert b\right\Vert _{H^1}\left\Vert L\right\Vert _{C^{0,1/2+\epsilon }_t L^2}|t-s|^{1/2+\epsilon }\left\Vert u\right\Vert _{C^{0,1/2}_t L^2_x}|t-s|^{1/2} \end{aligned}$$

for some $\epsilon >0$, meaning we may apply the Sewing Lemma 6.3. Moreover, we have

$$\begin{aligned}&\left\Vert A^u_{s,t}-\int _s^tb(u_r-w_r)dr\right\Vert _{L^2_x}=\left\Vert \int _s^t b(u_s-w_r)-b(u_r-w_r)dr\right\Vert _{L^2_x}\\&\quad \;\leqslant \;\left\Vert b\right\Vert _{C^{0,1}}\left\Vert u\right\Vert _{C^{0,1/2}_tL^2_x}\int _s^t|r-s|^{1/2}dr\lesssim \left\Vert b\right\Vert _{C^{0,1}}\left\Vert u\right\Vert _{C^{0,1/2}_tL^2_x}|t-s|^{3/2}, \end{aligned}$$

from which we conclude that

$$\begin{aligned}&\left\Vert ({\mathcal {I}}A^u)_{s,t}-\int _s^t b(u_r-w_r) dr\right\Vert _{L^2_x}\\&\quad \;\leqslant \;\left\Vert A^u_{s,t}-({\mathcal {I}}A^u)_{s,t}\right\Vert _{L^2_x}+\left\Vert A^u_{s,t}-\int _s^tb(u_r-w_r)dr\right\Vert _{L^2_x}= O(|t-s|^{1+\epsilon }). \end{aligned}$$

Hence, the function

$$\begin{aligned} t\rightarrow ({\mathcal {I}}A^u)_{0,t}-\int _0^tb(u_r-w_r)dr \end{aligned}$$

is constant. As it starts in zero, this concludes the claim. $\square$

Lemma 2.4

Let b be smooth and bounded and let $w:[0, T]\rightarrow \mathbb {R}^N$ be continuous with local time $L \in C^{0,1/2+\epsilon }_t L^2_x$ for some $\epsilon > 0$. Assume $u\in C^{0, 1/2}_tL^2_x$, then

• (1) the sewing is weakly differentiable in time, i.e.,

  $$\begin{aligned} t\mapsto ({\mathcal {I}}A^u)_{t} \in W^{1,2}(0,T; L^2_x), \end{aligned}$$

  and moreover, we find

  $$\begin{aligned} \partial _t \big [ ({\mathcal {I}}A^u)\big ] \big |_t&:= \lim _{h\rightarrow 0} h^{-1}\big ( ({\mathcal {I}}A^u)_{t+h} - ({\mathcal {I}}A^u)_{t} \big ) = b(u_t - w_t), \end{aligned}$$

  where the convergence holds strongly in $L^2_x$.

• (2) If u is moreover a robustified solution, then it is also a strong solution, that is for almost all $t \in I$

  $$\begin{aligned} \partial _t u - \textrm{div}S(\nabla u) = b(u - w) \end{aligned}$$

  as an equation in $L^2_x$.

Proof

First, we will address (1).

Recall that $({\mathcal {I}}A^u)$ is constructed as the germ

$$\begin{aligned} A_{s,t} = \int _s^t b(u_s - w_\tau ) \,\text {d}\tau = \big ( b * L_{s,t} \big ) (u_s). \end{aligned}$$

Additionally, we have as in the previous lemma

$$\begin{aligned} \left\Vert ({\mathcal {I}}A^u)_{s,t} - A_{s,t}\right\Vert _{L^2_x} \lesssim \left\Vert b\right\Vert _{H^1} \left\Vert L\right\Vert _{C^{0,1/2+\epsilon }_t L^2} \left\Vert u\right\Vert _{C^{0,1/2}_t L^2_x} \left|t-s\right|^{1+\epsilon }. \end{aligned}$$

As w is uniformly continuous due to the compactness of [0, T], we find for any $\epsilon >0$ a $\delta >0$ such that for all $s, t\in [0, T]$ such that $|t-s|<\delta$ we have $|w_s-w_t|<\epsilon$. Thus, choosing $h<\delta$, we have

$$\begin{aligned} \left\Vert \frac{A_{t,t+h}}{h} -b(u_{t} - w_t)\right\Vert _{L^2_x}&= \left\Vert h^{-1} \int _{t}^{t+h} b(u_{t} - w_\tau ) - b(u_t - w_t) \,\text {d}\tau \right\Vert _{L^2_x} \\&\;\leqslant \;\left\Vert h^{-1} \int _{t}^{t+h} \left[ b\right] _{C^{0,1}}\left|w_\tau - w_t\right|\,\text {d}\tau \right\Vert _{L^2_x}\lesssim \epsilon \left[ b\right] _{C^{0,1}} \end{aligned}$$

Overall, this implies

$$\begin{aligned} \lim _{h\rightarrow 0}\left\Vert \frac{({\mathcal {I}}A^u)_{t+h} - ({\mathcal {I}}A^u)_{t} }{h} -b(u_t-w_t)\right\Vert _{L^2_x}= 0, \end{aligned}$$

establishing the first claim. Next, we take a look at (2). By part (1) the right-hand side of (2.5) is weakly differentiable in $L^2_x$ and due to the regularity assumptions $\partial _t u \in L^2_tL^2_x$ and $\textrm{div}S(\nabla u) \in L^2_t L^2_x$ also the left-hand side is differentiable. Therefore, we may rescale (2.5) by $\left|t-s\right|^{-1}$ and pass to the limit $t \rightarrow s$ to obtain

$$\begin{aligned} \partial _t u - \textrm{div}S(\nabla u) = b(u-w). \end{aligned}$$

$\square$

Remark 2.5

Note that the key difference between strong and robustified solutions lies in the fact that the latter can exploit the regularization property of the local time L associated with w. In particular, provided this local time is sufficiently regular, the definition is meaningful even in instances in which b only enjoys distributional regularity as discussed in the introduction.

3 Classical a priori bounds for strong solutions

Global a priori bounds are generally inaccessible for singular potentials b. However, if we restrict ourselves to regularized approximations of b, i.e., we assume that there exists $(b_\varepsilon )_{\varepsilon \in (0,1)} \in C^{0,1}$ such that $b_\varepsilon \rightarrow b \in L^{2q}$, then for each $\varepsilon \in (0,1)$ the classical theory is applicable and existence of strong solutions to (1.2) for smooth b is a classical result. It can be found, e.g., in the books [18, 31, 39]. The objective of the present section is to trace the precise form of the a priori estimates that will be robustified in the next section.

Theorem 3.1

Let $p \in (1,\infty )$, $u_0 \in L^2_x \cap W^{1,p}_{0,x}$ and $b_\varepsilon \in C^{0,1}$. Then there exists a unique solution $u^\varepsilon$ solving 

$$\begin{aligned} \begin{aligned} \partial _t u^\epsilon -\textrm{div}S(\nabla u^\epsilon )&=b_\epsilon (u^\epsilon -w){} & {} \text { on } \Omega \times [0,T]\\ u^\epsilon |_\Omega&=0{} & {} \text { on } \partial \Omega \times [0,T]\\ u^\epsilon (0, \cdot )&=u_0{} & {} \text { on } \Omega . \end{aligned} \end{aligned}$$

(3.1)

in the strong sense of Definition 2.1. Moreover, the following a priori bounds are valid

$$\begin{aligned} \begin{aligned}&\sup _{t \in I} \left\Vert \nabla u_t^\varepsilon \right\Vert _{L^p_x}^p + \int _0^T \left\Vert \partial _t u_t^\varepsilon \right\Vert _{L^2_x}^2 + \left\Vert \textrm{div}S(\nabla u^\varepsilon _t)\right\Vert _{L^2_x}^2 \,\text {d}t \\&\quad \lesssim \left\Vert \nabla u_0\right\Vert _{L^p_x}^p + \int _0^T \int _\Omega \left|b_\varepsilon (u_t^{\varepsilon }(x) - w_t)\right|^2 \,\text {d}x \,\text {d}t, \end{aligned} \end{aligned}$$

(3.2)

and

$$\begin{aligned} \sup _{t \in I} \left\Vert u_t^{\varepsilon }\right\Vert _{L^2_x}^2 + \int _0^T \left\Vert \nabla u_t^{\varepsilon }\right\Vert _{L^p_x}^p \,\text {d}t \lesssim \left\Vert u_0\right\Vert _{L^2_x}^2 + \int _0^T \int _\Omega \left|b_\varepsilon (u_t^{\varepsilon }(x) - w_t)\right|^2 \,\text {d}x \,\text {d}t. \end{aligned}$$

(3.3)

Proof

The existence of a unique strong solution $u^\varepsilon$ to (3.1) is standard, see, e.g., [18, 31, 39], so let us just recall the main steps employed: In order to obtain existence, one first performs a Galerkin projection to the problem. The existence of solutions to the so obtained finite-dimensional problem is done through a fixed point theorem. Next, using monotonicity of the p-Laplace operator, one establishes a priori bounds uniformly along solutions to the projected problems. By Banach–Alaoglu, one extracts a weak-* convergent subsequence, whose limit one has to identify as a solution to the problem. Identifying the limit in the nonlinearity b is done thanks to the Aubin–Lions Lemma 6.6. Identifying the limit in the p-Laplace operator is done with Minty’s Lemma 6.5. Finally, uniqueness is obtained by monotonicity of the p-Laplace operator. We will argue on the weak (3.3) and strong (3.2) energy estimates separately.

The weak energy estimate (3.3) naturally occurs when multiplying (3.1) by $u^\varepsilon$. Integration in space and integration by parts imply

$$\begin{aligned} \partial _t \left( \frac{1}{2} \left\Vert u_t^\varepsilon \right\Vert _{L^2_x}^2 \right) + \left\Vert \nabla u_t^\varepsilon \right\Vert _{L^p_x}^p = \int _{\Omega } b_\varepsilon (u_t^\varepsilon (x) - w_t) \cdot u_t^\varepsilon (x) \,\text {d}x. \end{aligned}$$

Integration in time together with Hölder’s and Young’s inequalities results in

$$\begin{aligned}&\frac{1}{2} \left\Vert u_t^\varepsilon \right\Vert _{L^2_x}^2 + \int _0^t \left\Vert \nabla u_s^\varepsilon \right\Vert _{L^p_x}^p \,\text {d}s = \frac{1}{2} \left\Vert u_0\right\Vert _{L^2_x}^2 + \int _0^t \int _{\Omega } b_\varepsilon (u_s^\varepsilon (x) - w_s) \cdot u_s^\varepsilon (x) \,\text {d}x \,\text {d}s \nonumber \\&\quad \;\leqslant \;\frac{1}{2} \left\Vert u_0\right\Vert _{L^2_x}^2 + t\int _0^t \int _{\Omega } \left|b_\varepsilon (u_s^\varepsilon (x) - w_s)\right|^2 \,\text {d}x \,\text {d}s + \frac{1}{4t} \int _0^t \left\Vert u_s^\varepsilon \right\Vert _{L^2_x}^2 \,\text {d}s. \end{aligned}$$

(3.4)

Finally, take the supremum in t over (0, T) to conclude

$$\begin{aligned} \sup _{t \in (0,T)} \frac{1}{4} \left\Vert u_t^\varepsilon \right\Vert _{L^2_x}^2 \;\leqslant \;\frac{1}{2} \left\Vert u_0\right\Vert _{L^2_x}^2 + T\int _0^T \int _{\Omega } \left|b_\varepsilon (u_s^\varepsilon (x) - w_s)\right|^2 \,\text {d}x \,\text {d}s. \end{aligned}$$

This and (3.4) establish the inequality (3.3).

The strong energy estimate follows from squaring both sides of (3.1) and integration in space and time

$$\begin{aligned} \int _0^t \int _{\Omega } \left|\partial _t u_s^\varepsilon - \textrm{div}S(\nabla u_s^\varepsilon )\right|^2 \,\text {d}x \,\text {d}s = \int _0^t \int _{\Omega } \left|b_\varepsilon (u_s^\varepsilon (x) - w_s)\right|^2 \,\text {d}x \,\text {d}s. \end{aligned}$$

Note that, due to integration by parts,

$$\begin{aligned}&-2 \int _0^t \int _{\Omega } \partial _t u_s^\varepsilon \cdot \,\textrm{div}S(\nabla u_s^\varepsilon ) \,\text {d}x \,\text {d}s = 2 \int _0^t \int _{\Omega } \partial _t \nabla u_s^\varepsilon : S(\nabla u_s^\varepsilon ) \,\text {d}x \,\text {d}s \\&\hspace{2em} = \frac{2}{p} \int _0^t \partial _t \left\Vert \nabla u_s^\varepsilon \right\Vert _{L^p_x}^p \,\text {d}s = \frac{2}{p} \left( \left\Vert \nabla u_t^\varepsilon \right\Vert _{L^p_x}^p - \left\Vert \nabla u_0\right\Vert _{L^p_x}^p\right) . \end{aligned}$$

Therefore, we obtain

$$\begin{aligned}&\frac{2}{p} \left\Vert \nabla u_t^\varepsilon \right\Vert _{L^p_x}^p + \int _0^t \left\Vert \partial _t u_s^\varepsilon \right\Vert _{L^2_x}^2 +\left\Vert \textrm{div}S(\nabla u_s^\varepsilon )\right\Vert _{L^2_x}^2 \,\text {d}s \\&\hspace{2em} =\frac{2}{p} \left\Vert \nabla u_0\right\Vert _{L^p_x}^p + \int _0^t \int _{\Omega } \left|b_\varepsilon (u_s^\varepsilon (x) - w_s)\right|^2 \,\text {d}x \,\text {d}s. \end{aligned}$$

The claim (3.2) immediately follows after taking the supremum in t over (0, T). $\square$

We embed $W^{1,2}_t L^2_x \hookrightarrow C^{0,1/2}_t L^2_x$ as the Sewing Lemma is designed for the Hölder scale.

Corollary 3.2

Let the assumptions of Theorem 3.1 be satisfied. Then

$$\begin{aligned} \left\Vert u^\varepsilon \right\Vert _{C^{0,1/2}_t L^2_x}^2 \lesssim \left\Vert u_0\right\Vert _{L^2_x}^2 + \left\Vert \nabla u_0\right\Vert _{L^p_x}^p + \int _0^T \int _\Omega \left|b_\varepsilon (u_t^{\varepsilon }(x) - w_t)\right|^2 \,\text {d}x \,\text {d}t. \end{aligned}$$

(3.5)

Proof

Due to (3.3), it holds

$$\begin{aligned} \left\Vert u^\varepsilon \right\Vert _{L^\infty _t L^2_x}^2 \lesssim \left\Vert u_0\right\Vert _{L^2_x} + \int _0^T \int _\Omega \left|b_\varepsilon ( u_t^\varepsilon (x) - w_t)\right|^2 \,\text {d}x \,\text {d}t. \end{aligned}$$

(3.6)

The fundamental theorem of calculus and Hölder’s inequality reveal

$$\begin{aligned} \left\Vert u_t^\varepsilon - u_s^\varepsilon \right\Vert _{L^2_x}^2 = \left\Vert \int _s^t \partial _t u_r^\varepsilon \,\text {d}r\right\Vert _{L^2_x}^2 \;\leqslant \;\left|t-s\right| \int _s^t \left\Vert \partial _t u_r^\varepsilon \right\Vert _{L^2_x}^2 \,\text {d}r. \end{aligned}$$

The estimate (3.2) bounds

$$\begin{aligned} \left[ u^\varepsilon \right] _{C^{0,1/2}_t L^2_x}^2 \lesssim \left\Vert \nabla u_0\right\Vert _{L^p_x}^p + \int _0^T \int _\Omega \left|b_\varepsilon (u_t^{\varepsilon }(x) - w_t)\right|^2 \,\text {d}x \,\text {d}t. \end{aligned}$$

(3.7)

Adding (3.6) and (3.7) verifies the claim. $\square$

4 Robustified a priori bounds for the mollified problem

Given a singular potential $b\in L^{2q}$, we have seen in the previous section that for a mollification $b_\epsilon =b*\rho _\epsilon$, we obtain a unique strong solution $u^\varepsilon$ to

$$\begin{aligned} \begin{aligned} \partial _t u^\epsilon - \textrm{div}S(\nabla u^\epsilon ) = b_\epsilon (u^\epsilon -w). \end{aligned} \end{aligned}$$

(4.1)

In the following, we show that in harnessing the regularizing effect due to the perturbing path w, robust a priori bounds uniformly in $\epsilon >0$ can be obtained. Toward this end, we first show that the right-hand side of (3.2), (3.3) and (3.5) can be robustified in the sense of the following identification.

Lemma 4.1

Let $r\in [1,\infty )$ and $\gamma > 1/2$. Suppose $w:[0, T]\rightarrow \mathbb {R}^N$ is continuous and admits a local time L such that $L\in C^{0, \gamma }_tW^{1, r'}$ and $b\in L^{2q}$ for $q\in [r, \infty )$. Let $u^\epsilon$ be the unique strong solution to (3.1) of Theorem 3.1 associated to the mollification $b^\epsilon$ of b. Then for any $\epsilon >0$ fixed, we have

$$\begin{aligned} \int _0^T\int _\Omega |b_\epsilon (u^\epsilon _r-w_r)|^2 \,\text {d}x \,\text {d}r=({\mathcal {I}}A^\epsilon )_{0,T} \end{aligned}$$

(4.2)

where ${\mathcal {I}}$ denotes the sewing of the germ

$$\begin{aligned} A^\epsilon _{s,t}=\int _\Omega (b_\epsilon ^2*L_{s,t})(u^\epsilon _s) \,\text {d}x, \end{aligned}$$

and where we used the shorthand notation $b_\epsilon ^2(u):= |b_\epsilon (u)|^2$. Moreover, there holds the a priori bound

$$\begin{aligned} |({\mathcal {I}}A^\epsilon )_{s,t}|\lesssim \left\Vert b\right\Vert _{L^{2q}}^2\left\Vert L\right\Vert _{C^{0, \gamma }_tW^{1, r'}}(1+\left\Vert u^\epsilon \right\Vert _{C^{0, 1/2}_tL^2_x})|t-s|^\gamma . \end{aligned}$$

Proof

Recall that by Corollary 3.2, we have $u^\epsilon \in C^{0, 1/2}_tL^2_x$ for $\epsilon >0$ fixed. The first part of the claim is established similarly to Lemma 2.3, the main difference being that we exploit the regularity gained from the local time L in order to obtain the a priori bound in the second part of the claim. Let us start by remarking that for $\epsilon >0$ fixed, $A^\epsilon$ does indeed admit a sewing as

$$\begin{aligned} |\delta A^\epsilon _{s,r,t}|&\;\leqslant \;\int _\Omega | (b_\epsilon ^2*L_{r,t})(u^\epsilon _s)- (b_\epsilon ^2*L_{r,t})(u^\epsilon _r)| \,\text {d}x\\&\;\leqslant \;\left\Vert b_\epsilon ^2*L_{r,t}\right\Vert _{C^{0,1}}\left\Vert u^\epsilon _r-u^\epsilon _s\right\Vert _{L^1_x}\\&\lesssim \left\Vert b_\epsilon ^2\right\Vert _{L^q}\left\Vert L_{r,t}\right\Vert _{W^{1, q'}}\left\Vert u^\epsilon \right\Vert _{C^{0, 1/2}_tL^2_x}|r-s|^{1/2}\\&\lesssim \left\Vert b\right\Vert _{L^{2q}}^2\left\Vert L\right\Vert _{C^{0,\gamma }_tW^{1, r'}}\left\Vert u^\epsilon \right\Vert _{C^{0, 1/2}_tL^2_x}|r-s|^{1/2}|t-r|^{\gamma }, \end{aligned}$$

where we exploited that due to continuity of w, $(L_t)_t$ is of compact support uniformly in $t\in [0,T]$ and thus $\left\Vert L_t\right\Vert _{W^{1, q'}}\lesssim \left\Vert L_t\right\Vert _{W^{1, r'}}$. Moreover, note that for $\epsilon >0$ fixed, we have

$$\begin{aligned} \left| A^\epsilon _{s,t}-\int _s^t\int _\Omega b_\epsilon ^2(u^\epsilon _r-w_r) \,\text {d}x \,\text {d}r \right|&=\left| \int _s^t \int _\Omega b^2_\epsilon (u^\epsilon _s-w_r)-b^2_\epsilon (u^\epsilon _r-w_r) \,\text {d}x \,\text {d}r\right| \\&\lesssim \left[ b_\epsilon ^2\right] _{C^{0,1}}\left[ u^\epsilon \right] _{C^{0, 1/2}_tL^2_x}\int _s^t|r-s|^{1/2} \,\text {d}r\\&=\left[ b_\epsilon ^2\right] _{C^{0,1}}\left[ u^\epsilon \right] _{C^{0, 1/2}_tL^2_x}|t-s|^{3/2}. \end{aligned}$$

Similar to Lemma 2.3, we can thus conclude that indeed

$$\begin{aligned} \int _0^T\int _\Omega b_\epsilon ^2(u^\epsilon _r-w_r) \,\text {d}x \,\text {d}r=({\mathcal {I}}A^\epsilon )_{0,T} \end{aligned}$$

for any $\epsilon >0$ fixed. Moreover, exploiting the a priori bounds that come with the Sewing Lemma 6.3, we infer

$$\begin{aligned} |({\mathcal {I}}A^\epsilon )_{s,t}|\;\leqslant \;|A^\epsilon _{s,t}|+|({\mathcal {I}}A^\epsilon )_{s,t}-A^\epsilon _{s,t}|\lesssim \left\Vert b\right\Vert _{L^{2q}}^2\left\Vert L\right\Vert _{C^{0,\gamma }_tW^{1, r'}}(1+\left\Vert u^\epsilon \right\Vert _{C^{0, 1/2}_tL^2_x}). \end{aligned}$$

where we used that

$$\begin{aligned} |A^\epsilon _{s,t}|\lesssim \left\Vert b^2_\epsilon *L_{s,t}\right\Vert _{L^\infty }\lesssim \left\Vert b\right\Vert _{L^{2q}}^2\left\Vert L\right\Vert _{C^{0, \gamma }_tL^{r'}}|t-s|^\gamma , \end{aligned}$$

which completes the proof. $\square$

Remark 4.2

The sewing enables the local time to regularize the interplay of $b_\varepsilon$ and $u^\varepsilon$. In particular, the square $\left|b_\varepsilon (u^\varepsilon - w)\right|^2$ only acts on $b_\varepsilon$ and not on $u^\varepsilon$. Therefore, it is possible to work in the $L^1_x$ framework for $u^\varepsilon$. Indeed, a short inspection of the proof of Lemma 4.1 shows that Hölder regularity of $u^\varepsilon$ as a $L^1_x$-valued function is sufficient for the identification. Moreover, upon replacing $u^\epsilon$ by a generic function $v\in C^\alpha _tL^1_x$, the identification (4.2) holds provided $\alpha +\gamma >1$. However, as our main application is the closing of the a priori bound in Corollary 4.3, we decided to formulate the result immediately on the $L^2(\Omega )$ scale.

Corollary 4.3

Let $r\in [1,\infty )$ and $\gamma > 1/2$. Suppose $w:[0, T]\rightarrow \mathbb {R}^N$ is continuous and admits a local time L such that $L\in C^{0, \gamma }_tW^{1, r'}$ and $b\in L^{2q}$ for $q\in [r, \infty )$. Let $u^\epsilon$ be the unique solution to (3.1) of Theorem 3.1 associated to the mollification $b_\epsilon$ of b. Then we have the a priori bound

$$\begin{aligned} \begin{aligned}&\left\Vert u^\varepsilon \right\Vert _{C^{0,1/2}_t L^2_x}^2 \lesssim \left\Vert u_0\right\Vert _{L^2_x}^2+\left\Vert \nabla u_0\right\Vert _{L^p_x}^p + \left\Vert b\right\Vert _{L^{2r}}^4\left\Vert L\right\Vert _{C^{0, \gamma }_tW^{1, r'}}^2. \end{aligned} \end{aligned}$$

(4.3)

Proof

Plugging the a priori bound from Lemma 4.1 back into Corollary 3.2, we obtain

$$\begin{aligned} \left\Vert u^\epsilon \right\Vert ^2_{C^{0, 1/2}_tL^2_x}&\lesssim \left\Vert u_0\right\Vert _{L^2_x}^2 + \left\Vert \nabla u_0\right\Vert _{L^p_x}^p + \int _0^T \int _\Omega \left|b_\varepsilon (u_t^{\varepsilon }(x) - w_t)\right|^2 \,\text {d}x \,\text {d}t\\&\lesssim \left\Vert u_0\right\Vert _{L^2_x}^2+\left\Vert \nabla u_0\right\Vert _{L^p_x}^p+\left\Vert b\right\Vert _{L^{2r}}^2\left\Vert L\right\Vert _{C^{0, \gamma }_tW^{1, r'}}(1+\left\Vert u^\epsilon \right\Vert _{C^{0, 1/2}_tL^2_x}). \end{aligned}$$

An application of Lemma 6.7 with $a = \left\Vert u^\epsilon \right\Vert _{C^{0, 1/2}_tL^2_x}$, $K= \left\Vert u_0\right\Vert _{L^2_x}^2+\left\Vert \nabla u_0\right\Vert _{L^p_x}^p+\left\Vert b\right\Vert _{L^{2q}_x}^2\left\Vert L\right\Vert _{C^{0, \gamma }_tW^{1, r'}_x}$ and $C =\left\Vert b\right\Vert _{L^{2q}_x}^2\left\Vert L\right\Vert _{C^{0, \gamma }_tW^{1, r'}_x}$ results in

$$\begin{aligned} \left\Vert u^\epsilon \right\Vert ^2_{C^{0, 1/2}_tL^2_x} \lesssim \left\Vert u_0\right\Vert _{L^2_x}^2+\left\Vert \nabla u_0\right\Vert _{L^p_x}^p + \left\Vert b\right\Vert _{L^{2q}}^4\left\Vert L\right\Vert _{C^{0, \gamma }_tW^{1, r'}}^2 \end{aligned}$$

(4.4)

uniformly in $\epsilon >0$. $\square$

Remark 4.4

(Refined a priori bounds) We want to stress that even stronger a priori bounds than (4.3) are available. Indeed, the a priori bounds derived in Theorem 3.1 carry over to the robustified formulation, i.e., uniformly in $\varepsilon > 0$ it holds

$$\begin{aligned} \begin{aligned}&\left\Vert u^\varepsilon \right\Vert _{L^\infty _t L^2_x}^2 + \left\Vert \nabla u^\varepsilon \right\Vert _{L^\infty _t L^p_x}^p + \left\Vert \partial _t u^\varepsilon \right\Vert _{L^2_t L^2_x}^2 + \left\Vert \textrm{div}S(\nabla u^\varepsilon )\right\Vert _{L^2_t L^2_x}^2 \\&\quad \lesssim \left\Vert u_0\right\Vert _{L^2_x}^2+\left\Vert \nabla u_0\right\Vert _{L^p_x}^p + \left\Vert b\right\Vert _{L^{2q}}^4\left\Vert L\right\Vert _{C^{0, \gamma }_tW^{1, r'}}^2. \end{aligned} \end{aligned}$$

(4.5)

By Lemma 2.4(1) we have $b_\epsilon (u^\epsilon -w)=\partial _t{\mathcal {I}}A^{u^\epsilon }$, where $A^{u^\epsilon }_{s,t}=(b*L_{s,t})(u^\epsilon _s)$ and consequently, since $u^\epsilon$ is a strong solution to (3.1) it also holds that

$$\begin{aligned} \left\Vert \partial _t {\mathcal {I}}A^{u^\varepsilon }\right\Vert _{L^2_tL^2_x}= & {} \left\Vert b_\varepsilon (u^\varepsilon - w) \right\Vert _{L^2_t L^2_x} \lesssim \left\Vert u_0\right\Vert _{L^2_x}^2+\left\Vert \nabla u_0\right\Vert _{L^p_x}^p \nonumber \\{} & {} + \left\Vert b\right\Vert _{L^{2q}}^4\left\Vert L\right\Vert _{C^{0, \gamma }_tW^{1, r'}}^2. \end{aligned}$$

(4.6)

5 Passage to the limit and proof of Theorem 1.1

In the following, we show that exploiting the a priori bounds obtained in the previous section allows a passage to the limit on the level of the robustified formulation. Toward this end, we will use the monotonicity of the p-Laplace operator as well as suitable function space embeddings under the additional assumption $p>\frac{2d}{d+2}$ given in Theorem 1.1.

So far, we constructed a family of solutions $(u^\varepsilon )_{\varepsilon }$ that satisfies, for all almost all $(t,x) \in \Omega _T$,

$$\begin{aligned} \partial _t u^\varepsilon - \textrm{div}\, S(\nabla u^\varepsilon ) = \partial _t {\mathcal {I}} A^{u^\varepsilon }. \end{aligned}$$

(5.1)

Additionally, the solutions obey the uniform bounds

$$\begin{aligned} u^\varepsilon&\in W^{1,2}_t L^2_x \cap L^\infty _t W^{1,p}_{0,x}, \end{aligned}$$

(5.2a)

$$\begin{aligned} \textrm{div}\, S(\nabla u^\varepsilon )&\in L^2_t L^2_x \cap L^\infty _t W^{-1,p'}_x, \end{aligned}$$

(5.2b)

$$\begin{aligned} {\mathcal {I}}A^{u^\epsilon }&\in W^{1,2}_t L^2_x. \end{aligned}$$

(5.2c)

Therefore, we can extract a subsequence (not relabeled) and limits

$$\begin{aligned} (u, \xi , \eta ) \in \left( W^{1,2}_t L^2_x \cap L^\infty _t W^{1,p}_{0,x} \times L^2_t L^2_x \cap L^\infty _t W^{-1,p'}_x \times W^{1,2}_t L^2_x \right) \end{aligned}$$

(5.3)

such that

$$\begin{aligned} u^\varepsilon \rightharpoonup u&\in W^{1,2}_t L^2_x, \hspace{4em} u^\varepsilon \overset{*}{\rightharpoonup }\ u \in L^\infty _t W^{1,p}_{0,x}, \end{aligned}$$

(5.4a)

$$\begin{aligned} \textrm{div}\, S(\nabla u^\varepsilon ) \rightharpoonup \xi&\in L^2_t L^2_x, \hspace{4em} \textrm{div}\, S(\nabla u^\varepsilon ) \overset{*}{\rightharpoonup }\ \xi \in L^\infty _t W^{-1,p'}_{x}, \end{aligned}$$

(5.4b)

$$\begin{aligned} {\mathcal {I}}A^{u^\epsilon } \rightharpoonup \eta&\in W^{1,2}_t L^2_x. \end{aligned}$$

(5.4c)

In order to identify the limit dynamics, we test (5.1) with $\zeta \in L^2_t L^2_x$ and integrate in space and time. Using (5.4) we pass with $\varepsilon \rightarrow 0$ and find

$$\begin{aligned} \int _0^T \int _\Omega (\partial _t u^\varepsilon - \textrm{div}\, S(\nabla u^\varepsilon ) - \partial _t {\mathcal {I}}A^{u^\varepsilon } ) \cdot \zeta \,\text {d}x \,\text {d}t \rightarrow \int _0^T \int _\Omega (\partial _t u - \xi - \partial _t \eta ) \cdot \zeta \,\text {d}x \,\text {d}t, \end{aligned}$$

i.e., the limit satisfies

$$\begin{aligned} \partial _t u - \xi - \partial _t \eta =0. \end{aligned}$$

(5.5)

It remains to identify the nonlinear terms.

First, we discuss the sewing. Since $\Omega$ is a bounded Lipschitz domain in $\mathbb {R}^d$, the assumption $p>\frac{2d}{d+2}$ allows us to use the Rellich–Kondrachov theorem and thus to conclude $W^{1,p}_{0,x}\hookrightarrow L^2_x$ compactly. Therefore, applying the Aubin–Lions Lemma 6.6 in the form of (6.14) with $X=W^{1,p}_{0,x}\cap L^2_x$ and $B = Y = L^2_x$, yields $L^\infty _t (W^{1,p}_x \cap L^2_x) \cap W^{1,2}_t L^2_x \hookrightarrow \hookrightarrow C_tL^2_x$. This allows to extract another subsequence (not relabeled) such that

$$\begin{aligned} u^\varepsilon \rightarrow u \in C_t L^2_x. \end{aligned}$$

(5.6)

Crucially, the strong convergence (5.6) is sufficient to identify the nonlinear sewing.

Lemma 5.1

Let $r\in [1,\infty )$, $q\in [r,\infty )$ and $\gamma > 1/2$. Suppose $w:[0, T]\rightarrow \mathbb {R}^N$ is continuous and admits a local time L such that $L\in C^{0, \gamma }_t W^{1, r'}_x$. Moreover, let $b, b_\varepsilon \in L^{2q}$ and $u, u^\varepsilon \in C^{0,1/2}_t L^2_x$ such that $\left\Vert b - b_\varepsilon \right\Vert _{L^{2q}} \rightarrow 0$ and $\left\Vert u-u^\varepsilon \right\Vert _{C_t L^2_x} \rightarrow 0$. Then, for

$$\begin{aligned} A^u_{s,t}=(b*L_{s,t})(u_s), \qquad A^{u^\epsilon }_{s,t}=(b_\epsilon *L_{s,t})(u^\epsilon _s) \end{aligned}$$

we have

$$\begin{aligned} \left\Vert {\mathcal {I}}A^u-{\mathcal {I}}A^{u^\epsilon }\right\Vert _{C^{0, \gamma }_t L^2_x}\rightarrow 0. \end{aligned}$$

(5.7)

Proof

Note that

$$\begin{aligned} \begin{aligned}&\left\Vert A^u_{s,t}-A^{u^\epsilon }_{s,t}\right\Vert _{L^2_x}\\&\quad \;\leqslant \;\left\Vert (b-b_\epsilon )*L_{s,t}(u^\epsilon _s)\right\Vert _{L^2_x}+\left\Vert (b*L_{s,t})(u^\epsilon _s)-(b*L_{s,t})(u_s)\right\Vert _{L^2_x}\\&\quad \lesssim \left\Vert b-b_\epsilon \right\Vert _{L^{2r}}\left\Vert L_{s,t}\right\Vert _{L^{(2r)'}}+\left\Vert b\right\Vert _{L^{2r}}\left\Vert L_{s,t}\right\Vert _{W^{1, (2r)'}}\left\Vert u^\epsilon -u\right\Vert _{C_tL^2_x}\\&\quad \lesssim \left\Vert b-b_\epsilon \right\Vert _{L^{2q}}\left\Vert L_{s,t}\right\Vert _{L^{r'}}+\left\Vert b\right\Vert _{L^{2q}}\left\Vert L_{s,t}\right\Vert _{W^{1, r'}}\left\Vert u^\epsilon -u\right\Vert _{C_tL^2_x}\\&\quad \lesssim |t-s|^{\gamma }\left( \left\Vert b-b_\epsilon \right\Vert _{L^{2q}}\left\Vert L\right\Vert _{C^{0,\gamma }_t L^{r'}}+\left\Vert b\right\Vert _{L^{2q}} \left\Vert L\right\Vert _{C^{0, \gamma }_tW^{1, r'}}\left\Vert u^\epsilon -u\right\Vert _{C_t L^2_x}\right) \end{aligned} \end{aligned}$$

(5.8)

where we exploited again that due to the compact support of L, we have $\left\Vert L\right\Vert _{L^{(2r)'}}\lesssim \left\Vert L\right\Vert _{L^{r'}}$ and similarly for the corresponding Sobolev scales. Moreover, we have accordingly

$$\begin{aligned} \begin{aligned} \left\Vert (\delta A^{u^\epsilon })_{s,r,t} \right\Vert _{L^2_x}&=\left\Vert (b_\epsilon *L_{r,t})(u^\epsilon _r)-(b_\epsilon *L_{r,t})(u^\epsilon _s)\right\Vert _{L^2_x}\\&\lesssim \left\Vert b_\epsilon \right\Vert _{L^{2q}}\left\Vert L_{r,t}\right\Vert _{W^{1, r'}}\left\Vert u^\epsilon _s-u^\epsilon _r\right\Vert _{L^2_x}\\&\lesssim \left\Vert b\right\Vert _{L^{2q}}\left\Vert L\right\Vert _{C^{0,\gamma }_tW^{1, r'}}\left\Vert u^\epsilon \right\Vert _{C^{0, 1/2}_t L^2_x}|t-s|^{\gamma +1/2}. \end{aligned}\nonumber \\ \end{aligned}$$

(5.9)

The claim now follows from Lemma 6.4. $\square$

Lemma 5.1 allows to identify $\eta = {\mathcal {I}} A^u$. Indeed, using the weak convergence (5.4c) and the strong convergence (5.7), it holds for $\zeta \in L^2_t L^2_x$

$$\begin{aligned} \int _0^T \int _\Omega \eta \cdot \zeta \,\text {d}x \,\text {d}t&= \lim _{\varepsilon \rightarrow 0} \int _0^T \int _\Omega {\mathcal {I}} A^{u^\varepsilon } \cdot \zeta \,\text {d}x \,\text {d}t = \int _0^T \int _\Omega {\mathcal {I}} A^{u} \cdot \zeta \,\text {d}x \,\text {d}t. \end{aligned}$$

Next, we identify the limit of the monotone operator by means of Minty’s Lemma 6.5. Define $\Delta _p: X \rightarrow X^*$ by

$$\begin{aligned} \langle \Delta _p v, \zeta \rangle _{X^* \times X} := - \int _0^T \int _\Omega S(\nabla v) : \nabla \zeta \,\text {d}x \,\text {d}t, \end{aligned}$$

(5.10)

where $X = L^2_t L^2_x \cap L^p_t W^{1,p}_{0,x}$ and $X^* = L^2_t L^2_x + L^{p'}_t W^{-1,p'}_x$ with norms

$$\begin{aligned} \left\Vert v\right\Vert _X&:= \left\Vert v\right\Vert _{L^2_t L^2_x} + \left\Vert \nabla v\right\Vert _{L^p_t L^p_x}, \\ \left\Vert v\right\Vert _{X^*}&:= \inf \{ \left\Vert v_1\right\Vert _{L^2_t L^2_x} + \left\Vert v_2\right\Vert _{L^{p'}_t W^{-1,p'}_x} : v = v_1 + v_2 \}. \end{aligned}$$

Notice that $L^p_t W^{1,p}_{0,x}$ is sufficient to define (5.10). However, if $\textrm{div}S(\nabla v) \in L^2_t L^2_x$ we can alternatively represent (5.10), due to integration by parts, by

$$\begin{aligned} \langle \Delta _p v, \zeta \rangle _{X^* \times X} = \int _0^T \int _\Omega \textrm{div}S(\nabla v) \cdot \zeta \,\text {d}x \,\text {d}t. \end{aligned}$$

(5.11)

Hölder’s inequality shows

$$\begin{aligned} \left\Vert \Delta _p v\right\Vert _{X^*}&\;\leqslant \;\min \{ \left\Vert \textrm{div}S(\nabla v)\right\Vert _{L^2_t L^2_x}, \left\Vert S(\nabla v)\right\Vert _{L^{p'}_t L^{p'}_x} \} \;\leqslant \;\left\Vert v\right\Vert _{L^p_t W^{1,p}_{0,x}}^{p-1} \;\leqslant \;\left\Vert v\right\Vert _X^{p-1}. \end{aligned}$$

This establishes $\Delta _p: X \rightarrow X^*$. Notice that $\Delta _p$ is monotone and hemicontinuous.

Due to (5.4a) and (5.4b), we, in particular, have

$$\begin{aligned} u^\varepsilon \rightharpoonup u \in X,\quad \Delta _p u^\varepsilon \rightharpoonup \iota _\xi \in X^*, \end{aligned}$$

(5.12)

where $\langle \iota _\xi , \zeta \rangle _{X^* \times X}:= \int _0^T \int _\Omega \xi \cdot \zeta \,\text {d}x \,\text {d}t$.

The strong convergence (5.6) of $u^\varepsilon$ and the weak convergence (5.4b) of $\textrm{div}S(\nabla u^\varepsilon )$ enable

$$\begin{aligned} \langle \Delta _p u^\varepsilon , u^\varepsilon \rangle _{X^* \times X}&= \int _0^T \int _\Omega \textrm{div}S(\nabla u^\varepsilon ) \cdot u^\varepsilon \,\text {d}x \,\text {d}t \\ {}&\quad \rightarrow \int _0^T \int _\Omega \xi \cdot u \,\text {d}x \,\text {d}t = \langle \iota _\xi , u \rangle _{X^* \times X}. \end{aligned}$$

Now, Minty’s Lemma 6.5 implies $\iota _\xi = \Delta _p u$ and Riesz representation theorem verifies $\xi = \textrm{div}S(\nabla u)$.

The a priori bounds (4.3), (4.5) and (4.6) carry over to the limit by using weak lower semi continuity of the norms.

Overall, we have thus established the existence part of Theorem 1.1.

5.1 Uniqueness

In this subsection we verify uniqueness as claimed in Theorem 1.1. In fact, we will prove a stronger result.

Lemma 5.2

(Continuous dependence on initial data) Let $r\in [1,\infty )$ and $q\in [r, \infty )$. Moreover, let $b \in L^{2q}$ satisfy (1.6) and $w:[0, T]\rightarrow \mathbb {R}^N$ be continuous and admit a local time L which satisfies $L\in C^{0,\gamma }_t W^{1, r'}$ for some $\gamma \in (1/2, 1)$. Let u, v be two robustified solutions to (1.2) started in $u_0, v_0 \in L^2_x$, respectively.

Then for all $t \in [0,T]$ it holds

$$\begin{aligned} \left\Vert u_t - v_t\right\Vert _{L^2_x} \;\leqslant \;\left\Vert u_0 - v_0\right\Vert _{L^2_x}. \end{aligned}$$

(5.13)

Proof

Let u, v be two robustified solutions to (1.2) in the sense of Definition 2.2 starting in $u_0, v_0\in L^2_x$, respectively. In particular, they belong to the regularity class

$$\begin{aligned} u, v\in \left\{ z \in C^{0, 1/2}_tL^2_x \cap L^\infty _tW^{1,p}_{0,x} \big | \, \partial _t z, \textrm{div}S(\nabla z) \in L^2_t L^2_x \right\} \end{aligned}$$

and satisfy the system of equations, for all $t\in I$ and almost all $x \in \Omega$,

$$\begin{aligned} u_t-u_0-\int _0^t\text{ div } S(\nabla u_s) \,\text {d}s=({\mathcal {I}}A^u)_{0,t}, \end{aligned}$$

(5.14a)

$$\begin{aligned} v_t-v_0-\int _0^t\text{ div } S(\nabla v_s) \,\text {d}s=({\mathcal {I}}A^v)_{0,t}, \end{aligned}$$

(5.14b)

where $A^u_{s,t}=(b*L_{s,t})(u_s)$ and $A^v_{s,t}=(b*L_{s,t})(v_s)$, respectively.

Subtract (5.14b) from  (5.14a), differentiate in time, multiply with $u_t - v_t$ and integrate in space to find

$$\begin{aligned} \begin{aligned}&\left( \partial _t u_t - \partial _t v_t, u_t -v_t \right) - \left( \textrm{div}S(\nabla u_t) - \textrm{div}S(\nabla v_t), u_t -v_t \right) \\&\quad = \left( \big (\partial _t {\mathcal {I}}A^u\big )_t -\big (\partial _t {\mathcal {I}}A^v\big )_t , u_t - v_t \right) . \end{aligned} \end{aligned}$$

(5.15)

Notice that

$$\begin{aligned} 2\left( \partial _t u_t - \partial _t v_t, u_t -v_t \right) = \partial _t \left\Vert u_t - v_t\right\Vert _{L^2}^2. \end{aligned}$$

(5.16)

Moreover, due to the monotonicity of the p-Laplace operator,

$$\begin{aligned} - \left( \textrm{div}S(\nabla u_t) - \textrm{div}S(\nabla v_t), u_t -v_t \right) \;\geqslant \;0. \end{aligned}$$

(5.17)

Next, we integrate (5.15) in time and use (5.16) and (5.17)

$$\begin{aligned} \left\Vert u_t - v_t\right\Vert _{L^2}^2 \;\leqslant \;\left\Vert u_0 - v_0\right\Vert _{L^2}^2 + 2\int _0^t \left( \big (\partial _t {\mathcal {I}}A^u\big )_s -\big (\partial _t {\mathcal {I}}A^v\big )_s , u_s - v_s \right) \,\text {d}s. \end{aligned}$$

(5.18)

The estimate (5.13) immediately follows provided we can verify

$$\begin{aligned} \int _0^t \left( \big (\partial _t {\mathcal {I}}A^u\big )_s -\big (\partial _t {\mathcal {I}}A^v\big )_s , u_s - v_s \right) \,\text {d}s \;\leqslant \;0. \end{aligned}$$

(5.19)

Since we cannot identify the time derivative of the sewing for non-smooth potentials, we will approximate the potential b by a sequence of smooth potentials $(b^n)$ that preserve the monotonicity assumption (1.6). We will use Lemma 6.4 to justify the convergence of the approximations and Lemma 2.4(1) to identify the time derivative of the sewings for smooth potentials.

Let $b^n=(\rho ^n*b)$, where $(\rho ^n)_n$ is a sequence of nonnegative mollifiers. Notice that the monotonicity assumption on b, cf. (1.6), is preserved for $b^n$. Indeed,

$$\begin{aligned}&(b^n(u)-b^n(v)) \cdot (u-v)\\&\quad =\int _{\mathbb {R}^N}\rho ^n(z) (b(u-z)-b(v-z))\cdot ((u-z)-(v-z)) \,\text {d}z \;\leqslant \;0. \end{aligned}$$

Due to integration by parts

$$\begin{aligned}&\int _0^t \left( \partial _t {\mathcal {I}}A^u_s - \partial _t {\mathcal {I}}A^v_s, u_s- v_s \right) \,\text {d}s \\&\quad = \left( {\mathcal {I}}A^u_t - {\mathcal {I}}A^v_t, u_t- v_t \right) - \int _0^t \left( {\mathcal {I}}A^u_s - {\mathcal {I}}A^v_s, \partial _t u_s- \partial _t v_s \right) \,\text {d}s \\&\quad = \left( {\mathcal {I}}A^u_t - {\mathcal {I}}A^{n,u}_t, u_t- v_t \right) - \int _0^t \left( {\mathcal {I}}A^u_s - {\mathcal {I}}A^{n,u}_s, \partial _t u_s- \partial _t v_s \right) \,\text {d}s \\&\qquad - \left( {\mathcal {I}}A^v_t - {\mathcal {I}}A^{n,v}_t, u_t- v_t \right) - \int _0^t \left( {\mathcal {I}}A^v_s - {\mathcal {I}}A^{n,v}_s, \partial _t u_s- \partial _t v_s \right) \,\text {d}s \\&\qquad + \left( {\mathcal {I}}A^{n,u}_t - {\mathcal {I}}A^{n,v}_t, u_t- v_t \right) - \int _0^t \left( {\mathcal {I}}A^{n,u}_s - {\mathcal {I}}A^{n,v}_s, \partial _t u_s- \partial _t v_s \right) \,\text {d}s \\&\quad =: \textrm{R}_u^n + \textrm{R}_v^n +\textrm{R}_\textrm{smooth}^n. \end{aligned}$$

Here $A^{n,z}_{s,t} = (b^n * L_{s,t})(z_s)$, $z \in \{u,v\}$.

By Hölder’s inequality

$$\begin{aligned} \sup _{t\in I} \left|\textrm{R}_u^n + \textrm{R}_v^n\right|&\;\leqslant \;\left( \left\Vert {\mathcal {I}}A^u - {\mathcal {I}}A^{n,u}\right\Vert _{L^\infty _t L^2_x} + \left\Vert {\mathcal {I}}A^v - {\mathcal {I}}A^{n,v}\right\Vert _{L^\infty _t L^2_x} \right) \\&\quad \left( \left\Vert u - v\right\Vert _{L^\infty _t L^2_x} + \left\Vert \partial _t u - \partial _t v\right\Vert _{L^1_t L^2_x} \right) . \end{aligned}$$

Let $z \in \{u,v\}$. Similarly, to (5.8) and (5.9) it holds

$$\begin{aligned} \left\Vert A^{n,z}_{s,t} - A^{z}_{s,t}\right\Vert _{L^2_x}&\lesssim |t-s|^{\gamma } \left\Vert b-b^n\right\Vert _{L^{2q}}\left\Vert L\right\Vert _{C^{0,\gamma }_tL^{r'}}, \\ \left\Vert (\delta A^{n,z})_{srt}\right\Vert _{L^2_x}&\lesssim \left\Vert b\right\Vert _{L^{2q}}\left\Vert L\right\Vert _{C^{0,\gamma }_tW^{1, r'}}\left\Vert z\right\Vert _{C^{0, 1/2}_tL^2_x}|t-s|^{\gamma +1/2}. \end{aligned}$$

Therefore, we can apply Lemma 6.4 and obtain

$$\begin{aligned} \left\Vert {\mathcal {I}}A^z - {\mathcal {I}}A^{n,z}\right\Vert _{C_t^0 L^2_x} \rightarrow 0 \quad \text { as } n \rightarrow \infty . \end{aligned}$$

Thus,

$$\begin{aligned} \lim _{n\rightarrow \infty } \sup _{t\in I} \left|\textrm{R}_u^n + \textrm{R}_v^n\right| = 0. \end{aligned}$$

(5.20)

Reverting the partial integration, using Lemma 2.4(1) and the monotonicity of $b^n$

$$\begin{aligned} \begin{aligned} \textrm{R}_\textrm{smooth}^n&= \int _0^t \left( \partial _t {\mathcal {I}}A^{n,u}_s - \partial _t {\mathcal {I}}A^{n,v}_s, u_s- v_s \right) \,\text {d}s \\&= \int _0^t \left( b^n(u_s - w_s) - b^n(v_s - w_s), u_s- v_s \right) \,\text {d}s \;\leqslant \;0. \end{aligned} \end{aligned}$$

(5.21)

Finally, (5.19) follows from (5.20) and (5.21) and the assertion is verified. $\square$

Appendix

1.1 Verification of Example 1.2

For convenience we recall (1.7)

$$\begin{aligned} b(u)=-|u|^{\eta -1}u {{\textbf {1}}}_{\{|u|\;\leqslant \;K\}}, \end{aligned}$$

where $\eta \in (-N/2,0)$ and $K > 0$.

1.1.1 Non-existence of weak solution

First, we verify that (1.1) does not have a weak solution started in 0 if $\eta \;\leqslant \;-1$. We proceed by contraposition. Assume that (1.1) does have a weak solution started in 0. Our goal is to use the energy equality (3.4) to derive a contradiction.

Due to the density of smooth functions in $W^{-1,p'}$, we immediately find by (2.2)

$$\begin{aligned} \left\Vert \partial _t u\right\Vert _{L^{p'}_t W^{-1,p'}_x}&\;\leqslant \;\left\Vert S(\nabla u)\right\Vert _{L^{p'}_t L^{p'}_x} + \left\Vert b(u)\right\Vert _{L^{p'}_t W^{-1,p'}_x} \\&= \left\Vert \nabla u\right\Vert _{L^{p}_t L^{p}_x}^{p-1} + \left\Vert b(u)\right\Vert _{L^{p'}_t W^{-1,p'}_x}. \end{aligned}$$

Therefore,  (2.2) implies for almost all $s \in [0,T]$

$$\begin{aligned} \partial _t u_s - \textrm{div}\,S(\nabla u_s) = b(u_s) \quad \text {in}\quad W^{-1,p'}. \end{aligned}$$

(6.1)

Moreover, for all $u \in L^{p}_t W^{1,p}_{0,x} \cap W^{-1,p'}_t L^{p'}_x$ it holds

$$\begin{aligned} 2\int _0^T \langle \partial _t u_s, u_s \rangle _{W^{-1,p'} \times W^{1,p}_0} \,\text {d}s = \left\Vert u_T\right\Vert _{L^2}^2 - \left\Vert u_0\right\Vert _{L^2}^2. \end{aligned}$$

(6.2)

Since $u\in L^p W^{1,p}_{0,x}$ we can test (6.1) with $u_s$. Using (6.2) and $u_0 = 0$, we arrive at

$$\begin{aligned} \left\Vert u_T\right\Vert _{L^2}^2 + 2\int _0^T \left\Vert \nabla u_r\right\Vert _{L^p}^p \,\text {d}r + 2\int _0^T \int _{\Omega } \left|u_r\right|^{\eta + 1} {{\textbf {1}}}_{\{|u_r|\;\leqslant \;K\}} \,\text {d}x \,\text {d}r = 0. \end{aligned}$$

(6.3)

Notice that the left-hand side of (6.3) contains only nonnegative terms. Therefore, each individual term needs to vanish. However,

$$\begin{aligned} \int _0^T \left\Vert \nabla u_r\right\Vert _{L^p}^p \,\text {d}r = 0 \Rightarrow u \equiv 0, \end{aligned}$$

(6.4)

which leads to the contradiction, since for $\eta + 1 \;\leqslant \;0$,

$$\begin{aligned} 0 = \int _0^T \int _{\Omega } \left|u_r\right|^{\eta + 1} {{\textbf {1}}}_{\{|u_r|\;\leqslant \;K\}} \,\text {d}x \,\text {d}r \;\geqslant \;T \left| \Omega \right|. \end{aligned}$$

(6.5)

1.1.2 Existence of robustified solution

Let H satisfy (1.8) and $w^H$ be a fractional Brownian motion with Hurst parameter H. We will check that the local time L associated to the fraction Brownian motion $w^H$ is sufficiently regular for us to can apply Theorem 1.1.

We recall that a fractional Brownian motion with Hurst parameter H is a continuous centered Gaussian process whose covariance is given by

$$\begin{aligned} {\mathbb {E}}\left[ w^H_s \otimes w^H_t\right] =\frac{1}{2}\left( t^{2H}+s^{2H}-|t-s|^{2H}\right) \text{ Id }. \end{aligned}$$

where $\text{ Id }$ denotes the identity in $\mathbb {R}^N$. The space–time regularity of local times associated to locally non-deterministic Gaussian processes has been addressed by Harang and Perkowksi in [27] (see also [25] for regularity results for local times associated with Volterra–Lévy processes). In particular, [27, Theorem 3.4] shows that if

$$\begin{aligned} H \in (0,1/N) \quad \text {and} \quad \lambda <\frac{1}{2H}-\frac{N}{2}, \end{aligned}$$

(6.6)

then almost any realization of the fractional Brownian motion $w^H$ admits a local time L that satisfies $L\in C^{0,\gamma }\big ([0,T];H^{\lambda }({\mathbb {R}}^N)\big )$ for all $\gamma \in [0, 1 -(\lambda + N/2)H)$.

Notice that due to (6.6)

$$\begin{aligned} 1 - (\lambda + N/2)H \;\geqslant \;1/2. \end{aligned}$$

Therefore, if (6.6) is satisfied we can always choose $\gamma > 1/2$ for the temporal regularity. However, whether the spatial regularity is sufficient is to be determined.

If $\eta \in (-N/4,0)$ it holds $b \in L^4({\mathbb {R}}^N)$. Therefore, we can choose $r = 2$ implying $r'=2$ and we have as a condition

$$\begin{aligned} 1 \;\leqslant \;\lambda< \frac{1}{2H} - \frac{N}{2} \Rightarrow H < \frac{1}{N+2}. \end{aligned}$$

(6.7)

If $\eta \in (-N/2,-N/4]$ we need to leave the Hilbert scale and work with Sobolev spaces. Let $\varepsilon \ll 1$ and define $r_\varepsilon := - \tfrac{N-\varepsilon }{2\eta }$. We observe that $b\in L^{2r_\varepsilon }({\mathbb {R}}^N)$ for all $\varepsilon >0$. Moreover, $r_\varepsilon ' = \tfrac{N - \varepsilon }{N + 2\eta - \varepsilon } \rightarrow \tfrac{N}{N + 2\eta }$ as $\varepsilon \rightarrow 0$. Next, we transfer the Hilbert scale $H^\lambda$ to the Sobolev scale $W^{1,q}$ by an application of Sobolev embeddings. Let $1 \;\leqslant \;\lambda < 1+N/2$ and

$$\begin{aligned} 1 - \frac{N}{q}\;\leqslant \;\lambda - \frac{N}{2} \Leftrightarrow q \;\leqslant \;\frac{2N}{2 + N - 2 \lambda }. \end{aligned}$$

Then $H^\lambda _{\textrm{loc}}({\mathbb {R}}^N)\hookrightarrow W^{1, q }_{\textrm{loc}}({\mathbb {R}}^N)$. In particular, if

$$\begin{aligned} \frac{2N}{2 + N - 2 \lambda } > \frac{N}{N + 2\eta }, \end{aligned}$$

then there exists $\varepsilon >0$ such that $H^\lambda _{\textrm{loc}}({\mathbb {R}}^N)\hookrightarrow W^{1, r_\varepsilon ' }_{\textrm{loc}}({\mathbb {R}}^N)$. Therefore, we find as conditions

$$\begin{aligned} 1&\;\leqslant \;\lambda < 1 + \frac{N}{2}, \end{aligned}$$

(6.8a)

$$\begin{aligned} 1&\;\leqslant \;\lambda < \frac{1}{2H} - \frac{N}{2}, \end{aligned}$$

(6.8b)

$$\begin{aligned} 1 - \frac{N + 4 \eta }{2}&< \lambda . \end{aligned}$$

(6.8c)

Since $1 - \frac{N + 4 \eta }{2} \rightarrow 1 + N/2$ as $\eta \rightarrow -N/2$ monotonically, system (6.8) has a solution $\lambda$ if

$$\begin{aligned} 1 - \frac{N + 4\eta }{2}<\frac{1}{2H} - \frac{N}{2} \Rightarrow H <\frac{1}{2- 4\eta }. \end{aligned}$$

(6.9)

In summary, we have verified that fractional Brownian motions whose Hurst parameter satisfies (1.8) are sufficiently regularizing. The existence of a robustified solution follows by Theorem 1.1.

1.2 Local time and occupation times formula

We recall for the reader the basic concepts of occupation measures, local times and the occupation times formula. A comprehensive review paper on these topics is [19].

Definition 6.1

Let $w:[0,T]\rightarrow \mathbb {R}^N$ be a measurable path. Then the occupation measure at time $t\in [0,T]$ written $\mu ^w_t$ is the Borel measure on $\mathbb {R}^d$ defined by

$$\begin{aligned} \mu ^w_t(A):=\lambda (\{ s\in [0,t]:\ w_s\in A\}), \quad A\in {\mathcal {B}}(\mathbb {R}^N), \end{aligned}$$

where $\lambda$ denotes the standard Lebesgue measure.

The occupation measure thus measures how much time the process w spends in certain Borel sets. Provided for any $t\in [0,T]$, the measure is absolutely continuous with respect to the Lebesgue measure on $\mathbb {R}^N$, we call the corresponding Radon–Nikodym derivative local time of the process w:

Definition 6.2

Let $w:[0,T]\rightarrow \mathbb {R}^N$ be a measurable path. Assume that there exists a measurable function $L^w:[0,T]\times \mathbb {R}^N\rightarrow \mathbb {R}_+$ such that

$$\begin{aligned} \mu ^w_t(A)=\int _A L^w_t(z)dz, \end{aligned}$$

for any $A\in {\mathcal {B}}(\mathbb {R}^N)$ and $t\in [0,T]$. Then we call $L^w$ local time of w.

Note that by the definition of the occupation measure, we have for any bounded measurable function $f:\mathbb {R}^N\rightarrow \mathbb {R}$ that

$$\begin{aligned} \int _0^tf(w_s)ds=\int _{\mathbb {R}^N} f(z)\mu ^w_t(dz). \end{aligned}$$

(6.10)

Equation (6.10) is called occupation times formula. Remark that, in particular, provided w admits a local time, we also have for any $u\in \mathbb {R}^N$

$$\begin{aligned} \int _0^tf(u-w_s)ds= & {} \int _{\mathbb {R}^N} f(u-z)\mu ^w_t(dz)=\int _{\mathbb {R}^N}f(u-z)L^w_t(z)dz\nonumber \\= & {} (f*L^w_t)(u). \end{aligned}$$

(6.11)

1.3 Sewing Lemma

Let us recall the Sewing Lemma due to [24] (see also [11, Lemma 4.2]). Let E be a Banach space, [0, T] a given interval. Let $\Delta _n$ denote the n-th simplex of [0, T], i.e., $\Delta _n:\{(t_1, \ldots , t_n)| 0\;\leqslant \;t_1<t_2\cdots < t_n\;\leqslant \;T \}$. For a function $A:\Delta _2\rightarrow E$ define the mapping $\delta A: \Delta _3\rightarrow E$ via

$$\begin{aligned} (\delta A)_{s,u,t}:=A_{s,t}-A_{s,u}-A_{u,t} \end{aligned}$$

Provided $A_{t,t}=0$ we say that for $\alpha , \beta >0$ we have $A\in C^{\alpha , \beta }_2(E)$ if $\left\Vert A\right\Vert _{\alpha , \beta }<\infty$ where

$$\begin{aligned} \left\Vert A\right\Vert _\alpha&:=\sup _{(s,t)\in \Delta _2}\frac{\left\Vert A_{s,t}\right\Vert _E}{|t-s|^\alpha },\quad \left\Vert \delta A\right\Vert _{\beta }:=\sup _{(s,u,t)\in \Delta _3}\frac{\left\Vert (\delta A)_{s,u,t}\right\Vert _E}{|t-s|^\beta }, \\ \left\Vert A\right\Vert _{\alpha , \beta }&:=\left\Vert A\right\Vert _\alpha +\left\Vert \delta A\right\Vert _\beta . \end{aligned}$$

In this case, we call A a germ. For a function $f:[0,T]\rightarrow E$, we note $f_{s,t}:=f_t-f_s$. Moreover, if for any sequence $({\mathcal {P}}^n([s,t]))_n$ of partitions of [s, t] whose mesh size goes to zero, the quantity

$$\begin{aligned} (I^nA)_{s,t}=\sum _{[u,v]\in {\mathcal {P}}^n([s,t])}A_{u,v} \end{aligned}$$

(6.12)

converges to the same limit, we note

$$\begin{aligned} ({\mathcal {I}} A)_{s,t}:=\lim _{n\rightarrow \infty }\sum _{[u,v]\in {\mathcal {P}}^n([s,t])}A_{u,v}. \end{aligned}$$

Lemma 6.3

(Sewing) Let $0<\alpha \;\leqslant \;1<\beta$. Then for any $A\in C^{\alpha , \beta }_2(E)$, $({\mathcal {I}} A)$ is well defined and we say that the germ A admits a sewing. Moreover, denoting $({\mathcal {I}} A)_t:=({\mathcal {I}} A)_{0,t}$, we have $({\mathcal {I}} A)\in C^\alpha ([0,T], E)$ and $({\mathcal {I}} A)_0=0$ and for some constant $c>0$ depending only on $\beta$ we have

$$\begin{aligned} \left\Vert ({\mathcal {I}} A)_{t}-({\mathcal {I}} A)_{s}-A_{s,t}\right\Vert _{E}\;\leqslant \;c\left\Vert \delta A\right\Vert _\beta |t-s|^\beta . \end{aligned}$$

Proof

For readers unfamiliar with the Sewing Lemma, let us sketch the main argument why the Riemann-type sums (6.12) converge in the case of dyadic partitions. Let us thus consider ${\mathcal {P}}^n([0,t])=\{t_k=\frac{k}{2^n}t, k=1, \ldots 2^n-1\}$ and show that the sequence $(I^nA)_t$ is Cauchy in E. Indeed, denoting by $u_k$ the midpoint of $t_k$ and $t_{k+1}$, remark that

$$\begin{aligned} (I^nA)_t-(I^{n+1}A)_t=\sum _k^{2^n} A_{t_k, t_{k+1}}-\sum _k^{2^n}A_{t_k, u_k}-\sum _k^{2^n}A_{u_k, t_{k+1}}=\sum _k^{2^n} (\delta A)_{t_k, u_k, t_{k+1}} \end{aligned}$$

Using the triangle inequality and the assumption $A\in C_2^{\alpha , \beta }$, we have

$$\begin{aligned} \left\Vert (I^nA)_t-(I^{n+1}A)_t\right\Vert _E\;\leqslant \;\left\Vert \delta A\right\Vert _\beta 2^{n(1-\beta )} \end{aligned}$$

meaning that for $\beta >1$ we can obtain convergence of $(I^nA)_t$ in E. $\square$

Let us finally cite a result allowing to commute limits and sewings.

Lemma 6.4

(Lemma A.2 [13]) For $0<\alpha \;\leqslant \;1<\beta$ and E a Banach space, let $A\in C^{\alpha , \beta }_2(E)$ and $(A^n)_n\subset C^{\alpha , \beta }_2(E)$ such that for some $R>0$ $\sup _{n\in {\mathbb {N}}}\left\Vert \delta A^n\right\Vert _\beta \;\leqslant \;R$ and such that $\left\Vert A^n-A\right\Vert _\alpha \rightarrow 0$. Then

$$\begin{aligned} \left\Vert {\mathcal {I}}(A-A^n)\right\Vert _\alpha \rightarrow 0. \end{aligned}$$

1.4 Some classical Lemmata from monotone operator theory

Suppose $u^\epsilon$ is some approximation to (1.1) (the solution to the Galerkin projected problem or the solution to the mollified problem (3.1), for example) for which uniform bounds in $L^p(0, T; W^{1, p}_0(\Omega ))$ hold, meaning we find u such that $u^\epsilon \rightharpoonup u$ along a subsequence in $L^p(0, T; W^{1, p}_0(\Omega ))$. Moreover, as $\text{ div } S(\nabla u^\epsilon )$ will be bounded in $L^{p'}(0, T; W^{-1, p'}(\Omega ))$, we find a $\xi$ such that $\text{ div } S(\nabla u^\epsilon )\rightharpoonup \xi$ along a further subsequence in $L^{p'}(0, T; W^{-1, p'}(\Omega ))$. For the identification step to work, we need an argument ensuring that $\xi =\text{ div } S(\nabla u)$. This is precisely the content of Minty’s Lemma.

Lemma 6.5

(Minty’s Lemma/Monotonicity trick [39, p. 474]) Let X be a real reflexive Banach space and $A:X\rightarrow X^*$ a monotone and hemicontinuous operator. Then, provided we have

$$\begin{aligned} u^n&\rightharpoonup u \quad in \ X\\ Au^n&\rightharpoonup \xi \quad in \ X^*\\ \langle Au^n, u^n\rangle _{X^*\times X}&\rightarrow \langle \xi , u\rangle _{X^*\times X} \end{aligned}$$

we may conclude $Au=\xi$.

Concerning the identification of the other nonlinear term, one proceeds differently. Indeed, for a generic nonlinearity $b:\mathbb {R}^N\rightarrow \mathbb {R}^N$, we can only hope for $b(u^\epsilon )\rightharpoonup b(u)$ provided that the convergence $u^\epsilon \rightarrow u$ holds in the strong topology of some function space. Toward this end, we require some refined a priori bounds and an associated compact embedding, given by the following classical result.

Lemma 6.6

(An Aubin–Lions Lemma [37, Corollary 5]) Given Banach spaces X, B and Y and assume $X \hookrightarrow \hookrightarrow B \hookrightarrow Y$. Additionally, let $q,r \in [1,\infty ]$. Then

• (1) if $q < \infty$ and $s > (1/r - 1/q) \vee 0$

  $$\begin{aligned} L^q(0,T;X) \cap W^{s,r}(0,T;Y) \hookrightarrow \hookrightarrow L^q(0,T;B), \end{aligned}$$

  (6.13)

• (2) if $q = \infty$ and $s > 1/r$

  $$\begin{aligned} L^\infty (0,T;X) \cap W^{s,r}(0,T;Y) \hookrightarrow \hookrightarrow C(0,T;B). \end{aligned}$$

  (6.14)

1.5 An algebraic inequality

Lemma 6.7

Let $a, C, K \in {\mathbb {R}}$ such that $K \;\geqslant \;-C^2/4$. Additionally, assume that

$$\begin{aligned} a^2 \;\leqslant \;K + C a. \end{aligned}$$

(6.15)

Then

$$\begin{aligned} \frac{C}{2} - \sqrt{K + \frac{C^2}{4}} \;\leqslant \;a \;\leqslant \;\frac{C}{2} + \sqrt{K + \frac{C^2}{4}}. \end{aligned}$$

(6.16)

Proof

Reordering (6.15) results in

$$\begin{aligned} a(a-C) \;\leqslant \;K. \end{aligned}$$

Completing the square ensures

$$\begin{aligned} \left( a-\frac{C}{2} \right) ^2 \;\leqslant \;K + \frac{C^2}{4}. \end{aligned}$$

This implies

$$\begin{aligned} \left|a - \frac{C}{2}\right| \;\leqslant \;\sqrt{K + \frac{C^2}{4}}. \end{aligned}$$

The assertion is verified. $\square$

If $-K\;\leqslant \;C^2/4$, then the parabola $a \mapsto a(a-C)$ and the constant function K do not intersect. Thus,  (6.15) won’t be satisfied.