Gradient profile for the reconnection of vortex lines with the boundary in type-II superconductors

Abstract

In a recent work, Duong, Ghoul and Zaag determined the gradient profile for blowup solutions of standard semilinear heat equation with power nonlinearities in the (supposed to be) generic case. Their method refines the constructive techniques introduced by Bricmont and Kupiainen and further developed by Merle and Zaag. In this paper, we extend their refinement to the problem about the reconnection of vortex lines with the boundary in a type-II superconductor under planar approximation, a physical model derived by Chapman, Hunton and Ockendon featuring the finite time quenching for the nonlinear heat equation

$$\begin{aligned} \frac{\partial h}{\partial t}=\frac{\partial ^2 h}{\partial x^2}+e^{-h}-\frac{1}{h^\beta },\quad \beta >0 \end{aligned}$$

subject to initial boundary value conditions

$$\begin{aligned} h(\cdot ,0)=h_0>0,\quad h(\pm 1,t)=1. \end{aligned}$$

We derive the intermediate extinction profile with refined asymptotics, and with extinction time T and extinction point 0, the gradient profile behaves as $x\rightarrow 0$ like

$$\begin{aligned} \lim _{t\rightarrow T}\,(\nabla h)(x,t)\quad \sim \quad \frac{1}{\sqrt{2\beta }}\frac{x}{|x|}\frac{1}{\sqrt{|\log |x||}} \left[ \frac{(\beta +1)^2}{8\beta }\frac{|x|^2}{|\log |x||}\right] ^{\frac{1}{\beta +1}-\frac{1}{2}}, \end{aligned}$$

agreeing with the gradient of the extinction profile previously derived by Merle and Zaag. Our result holds with general boundary conditions and in higher dimensions.

Appendices

Appendix A: Direct regularity estimates via the shrinking set

We collect in this appendix some elementary estimates on $(q,\nabla q)$, B(q) and T(q) that are direct (not trivial however) consequences of the shrinking set conditions.

Lemma A.1

(Elementary estimates on $(q,\nabla q)$) For all $K_0\ge 1$ and $\varepsilon _0>0$, there exists $t_6=t_6(K_0,\varepsilon _0)<T$ such that for all $t_0\in [t_6,T)$, for all $A\ge 1$, $\alpha _0>0$, $C_0>0$, $\delta _0\le \frac{\widehat{k}(1)}{2}$ and $\eta _0\le \eta _6(\varepsilon _0)$ for some $\eta _6(\varepsilon _0)>0$, we have the following property:

If h is the solution to (1.1), generated by the initial data $h(t_0\text {;}d_0,d_1)$ defined in (5.1), that further satisfies

$$\begin{aligned} \forall \,\, t\in [t_0,T],\quad h(t)\in S^*(t_0, K_0, \varepsilon _0, A,\alpha _0,\delta _0, C_0,\eta _0,t), \end{aligned}$$

then for some $C=C(K_0,C_0)>1$

$$\begin{aligned}{} & {} \Vert q(\cdot ,s)\Vert _{L^\infty }\le C \frac{A^{{\overline{\alpha }}}\log s}{s}, \end{aligned}$$

(A.1)

$$\begin{aligned}{} & {} \left\| \frac{ q(\cdot ,s)}{1+|\cdot |^3}\right\| _{L^\infty }\le C\frac{A^{2}\log s}{ s^2}, \end{aligned}$$

(A.2)

$$\begin{aligned}{} & {} \Vert {(\nabla q)_e(\cdot ,s)}\Vert _{L^\infty }\le \frac{C}{\sqrt{s}}, \end{aligned}$$

(A.3)

where $s=-\log (T-t)$ and q is defined in (3.8).

Proof

By the definition of $h(t)\in S^*(t_0,t)$, then $q(s)\in V_{K_0,A}(s)$, and we have

$$\begin{aligned} |q_b(y,s)|{} & {} \le \chi (y,s)\left( \sum _{m=0}^2|q_m(s)||h_m(y)|+|q_-(y,s)|\right) \nonumber \\{} & {} \le C\chi (y,s)\left( \frac{A}{s^2}(1+|y|)+\frac{A^2\log s}{s^2}(1+|y|^2)+\frac{A^{{\underline{\alpha }}} \log s}{s^{\frac{5}{2}}}(1+|y|^3)\right) \nonumber \\{} & {} \le C\frac{A^2\log s}{s^2}\left( 1+(2K_0\sqrt{s})^2\right) +C\frac{A^{{\underline{\alpha }}}\log s}{s^{\frac{5}{2}}} \left( 1+(2K_0\sqrt{s})^3\right) \nonumber \\{} & {} \le C K_0^3\frac{A^{{\underline{\alpha }}}\log s}{s}, \end{aligned}$$

(A.4)

which, combined with (4.1) on $q_e$, proves (A.1). We used $K_0\ge 1$, $A\ge 1$ and ${\underline{\alpha }}\ge 2$.

Meanwhile, we infer from the derivation in (A.4) that for s large enough,

$$\begin{aligned} |q_b(y,s)|\le C\chi (y,s)\frac{A^2\log s}{s^2}(1+|y|^3), \end{aligned}$$

(A.5)

whereas by (4.1) on $q_e$

$$\begin{aligned} |q_e(y,s)|\le (1-\chi (y,s)) \frac{A^{{\overline{\alpha }}}\log s}{s}\le C(1-\chi (y,s))\frac{A^{{\overline{\alpha }}}\log s}{s^{\frac{5}{2}}}(1+|y|^3).\nonumber \\ \end{aligned}$$

(A.6)

From above two estimates, (A.2) follows immediately.

For the remaining estimates, first we observe that

$$\begin{aligned} |(\nabla (\varphi +q))_e(y,s)|\le \frac{C(K_0,C_0)}{\sqrt{s}}. \end{aligned}$$

(A.7)

This estimate is taken from the proof of [47, Lemma B.1], which uses only the Parts (ii)–(iii) information of the shrinking set (hence remains valid in our setting), together with the exterior part information of $h(\cdot ,t_0\text {;}d_0,d_1)$. Note that the modification in Part (iii) of the shrinking set is harmless. Moreover, the constraints

$$\begin{aligned} \delta _0\le \frac{\widehat{k}(1)}{2}\quad \text {and}\quad \eta _0\le \eta _6(\varepsilon _0) \end{aligned}$$

are required in this step. By the following crucial estimate

$$\begin{aligned} |(\nabla \varphi )_e(y,s)|\le \frac{C}{\sqrt{s}} \end{aligned}$$

(which holds in fact in ${\mathbb {R}}^N$ but this will not be used), (A.3) follows immediately. $\square$

Lemma A.2

(Elementary estimates on B(q)) For all $K_0\ge 1$, all $A\ge 1$, there exists $s_7=s_7(K_0,A)$ such that for all $s\ge s_7$, the condition $q(s)\in V_{K_0,A}(s)$ implies

$$\begin{aligned} |\chi (y,s)B(q)(y,s)|\le C(K_0)|q|^2 \end{aligned}$$

and

$$\begin{aligned} |B(q)|\le C|q|^{{{\bar{p}}}}, \end{aligned}$$

where ${{\bar{p}}}=\min \{p,2\}$. In particular, for some $C=C(K_0)>1$,

$$\begin{aligned} |B(q)(y,s)|\le \frac{C(1+|y|^3)}{s^{\frac{5}{2}}}\quad \text {and}\quad |B(q)(y,s)|\le \frac{C}{s}. \end{aligned}$$

(A.8)

Proof

The first part is [48, Lemma 3.15]. The second part can be found in [13, (5.18)], where $s\ge s_7(K_0,A)$ is used to have A-independent C in (A.8). $\square$

Lemma A.3

(Elementary estimates on T(q)) For all $K_0\ge 1$, $A\ge 1$ and $\varepsilon _0>0$, there exists $t_8=t_8(K_0,A,\varepsilon _0)<T$ and $\eta _8=\eta _8(\varepsilon _0)>0$ such that for all $t_0\in [t_8,T)$, $\alpha _0>0$, $C_0>0$, $\delta _0\le \frac{\widehat{k}(1)}{2}$ and $\eta _0\le \eta _8$, we have the following property:

If h is the solution to (1.1), generated by the initial data $h(t_0\text {;}d_0,d_1)$ defined in (5.1), that further satisfies

$$\begin{aligned} \forall \,\, t\in [t_0,T],\quad h(t)\in S^*(t_0, K_0, \varepsilon _0, A,\alpha _0,\delta _0, C_0,\eta _0,t), \end{aligned}$$

then for some $C=C(K_0,C_0)>1$

$$\begin{aligned}{} & {} |\chi (y,s)T(q)(s)|\le C\chi (y,s)\left( \frac{|q|}{s}+\frac{|\nabla q|}{\sqrt{s}}+|\nabla q|^2\right) , \end{aligned}$$

(A.9)

$$\begin{aligned}{} & {} |(1-\chi (y,s))T(q)(s)|\le \frac{C}{s}, \end{aligned}$$

(A.10)

where $s=-\log (T-t)$ and q is defined in (3.8).

In using (A.10), we note the following simple estimate: for $|y|\ge K_0\sqrt{s}$,

$$\begin{aligned} \frac{1}{s}\le C(K_0)\frac{|y|^3}{s^{\frac{5}{2}}}\le C(K_0)\frac{1+|y|^3}{s^{\frac{5}{2}}}. \end{aligned}$$

(A.11)

Proof

We follow the proof of Lemma B.4 in [47]. Consider

$$\begin{aligned} F(\theta )=-\frac{|\nabla (\varphi + \theta q)|^2}{\varphi + \theta q}+\frac{|\nabla \varphi |^2}{\varphi },\quad \theta \in [0,1]. \end{aligned}$$

Let us compute

$$\begin{aligned} F'(\theta )=q\frac{|\nabla (\varphi + \theta q)|^2}{(\varphi + \theta q)^2}-2\frac{(\nabla q)\cdot \nabla (\varphi + \theta q)}{\varphi + \theta q}. \end{aligned}$$

The smallness of q (which is guaranteed by Lemma A.1) and positivity of $\varphi$ imply the boundedness of $(\varphi + \theta q)^{-1}$. From $F(0)=0$ and

$$\begin{aligned} F(1)=\int _0^1F'(\theta )d\theta , \end{aligned}$$

we have the following rough estimate (where we also use $|q|\le 1$)

$$\begin{aligned} \begin{aligned} |\chi (y,s)T(q)|&\le C\sup _{\theta \in [0,1]}|\chi (y,s)F'(\theta )|\\&\le C\chi (y,s)\left( |q||\nabla \varphi |^2+|\nabla \varphi ||\nabla q|+|\nabla q|^2\right) \\&\le C\chi (y,s)\left( \frac{|q|}{s}+\frac{|\nabla q|}{\sqrt{s}}+|\nabla q|^2\right) . \end{aligned} \end{aligned}$$

Moreover, C in (A.9) depends only on $K_0$.

For (A.10), one uses the crucial estimate

$$\begin{aligned} \left| (1-\chi (y,s))\frac{|\nabla \varphi |^2}{\varphi }(y,s)\right| \le \frac{C}{s}, \end{aligned}$$

and the shrinking set conditions in ${{\mathcal {R}}}_2$ and ${{\mathcal {R}}}_3$. This is analogous to the proof of (A.3), and the reader is referred to the proof of [47, Lemma B.1] for details. Again, the constraints $\delta _0\le \frac{\widehat{k}(1)}{2}$ and $\eta _0\le \eta _8(\varepsilon _0)$ are required in this step. $\square$

Appendix B: Estimates which are independent of the shrinking set

We need estimates on V, R and L which are independent of the shrinking set.

Lemma B.1

For $s\ge s_{9}$, the potential V defined via (3.10) satisfies

$$\begin{aligned}{} & {} -p\kappa ^{p-1}\le V(y,s)\le \frac{C}{s}, \end{aligned}$$

(B.1)

$$\begin{aligned}{} & {} |V(y,s)|\le \frac{C(1+|y|^2)}{s}, \end{aligned}$$

(B.2)

$$\begin{aligned}{} & {} \left| V(y,s)+\frac{1}{4s}(|y|^2-2N)\right| \le \frac{C(1+|y|^4)}{s^2}, \end{aligned}$$

(B.3)

$$\begin{aligned}{} & {} |\nabla ^i V(y,s)|\le \frac{C}{s^\frac{i}{2}},\quad i=1,2. \end{aligned}$$

(B.4)

Proof

See [52, Lemma B.1]. $\square$

Lemma B.2

For any $s\ge 1$, the remainder R defined via (3.10) satisfies

$$\begin{aligned}{} & {} \Vert R(\cdot ,s)\Vert _{L^\infty }\le \frac{C}{s}, \end{aligned}$$

(B.5)

$$\begin{aligned}{} & {} \left| R(y,s)-\frac{\alpha _0}{s^2}\right| \le \frac{C(1+|y|^4)}{s^3}, \end{aligned}$$

(B.6)

for some $\alpha _0=\alpha _0(a,p)\in {{\mathbb {R}}}$. In particular,

$$\begin{aligned}{} & {} |R_0(s)|\le \frac{C}{s^2},\quad |R_1(s)|\le \frac{C}{s^{3}},\quad |R_2(s)|\le \frac{C}{s^{3}}, \\{} & {} \left| R_b(y,s)-\frac{\alpha _0}{s^2}\right| \le \frac{C(K_0)(1+|y|^3)}{s^{\frac{5}{2}}}\quad \left( {\text {{\,hence,\,\,}}} \left\| \frac{R_-(\cdot , s)}{1+|\cdot |^3}\right\| _{L^\infty }\le \frac{C(K_0)}{s^{\frac{5}{2}}}\right) , \\{} & {} \Vert R_e(\cdot ,s)\Vert _{L^\infty }\le \frac{C}{s} \quad \left( {\text {{ hence, }}} |R_e(y,s)|\le \frac{C(1+|y|^3)}{s^{\frac{5}{2}}}\right) . \end{aligned}$$

Proof

The first part can be found in Lemma B.5 of [47] (naturally extended to $N\ge 2$). The second part follows from (B.5) and (B.6), the decomposition (3.23) and the definitions involved in this decomposition (using in particular orthogonality). $\square$

Lemma B.3

For $\widetilde{f}$ given in (1.6), we have

$$\begin{aligned} \sup _{0\le u<\infty }|\widetilde{f}(u)|+|\widetilde{f}'(u)|\le C. \end{aligned}$$

In particular,

$$\begin{aligned} |L(q)(s)|\le \frac{C}{e^{\frac{ps}{p-1}}}. \end{aligned}$$

(B.7)

Proof

Note that as $u\rightarrow \infty$, by (1.2) we have

$$\begin{aligned} \widetilde{f}(u)=\alpha ^{\frac{\beta }{\beta +1}}u^{1+\frac{1}{\alpha }}F(\alpha ^{\frac{1}{\beta +1}}u^{-\frac{1}{\alpha }})-u^p\rightarrow 0. \end{aligned}$$

The remaining arguments (using (1.3)) can be found in Lemma B.6 of [47]. $\square$

Appendix C: Local well-posedness in ${{\textbf{H}}}$ of the quenching problem

We show in this Appendix the local well-posedness in ${{\textbf{H}}}$ of (1.1).

• (1) Suppose that $\Omega$ is a bounded domain in ${\mathbb {R}}^N$ and $e^{t\Delta }$ is the heat semigroup associated to the Dirichlet Laplacian $\Delta =\Delta _\Omega$ with boundary data $\equiv 1$. Given $0<h_0\in {{\textbf{H}}}$ with $h_0\equiv 1$ on $\partial \Omega$, we set

  $$\begin{aligned} \varepsilon _0=\inf h_0\quad \text {and}\quad M_0=\Vert h_0\Vert _{W^{1,\infty }(\Omega )}. \end{aligned}$$

  Consider

  $$\begin{aligned} K_0=K_0(\Omega ):=\left\{ {{\textbf{h}}}\in L^\infty (\Omega ): \inf {{\textbf{h}}}\ge \frac{\varepsilon _0}{2}, \Vert {{\textbf{h}}}\Vert _{W^{1,\infty }(\Omega )}\le 2CM_0\right\} , \end{aligned}$$

  where $C\ge 1$ is the operator norm of the (Dirichlet) heat semigroup $e^{t\Delta }$ on $W^{1,\infty }(\Omega )$.

First, we claim that there exists $T_0=T_0(\varepsilon _0,M_0)>0$ such that

$$\begin{aligned} \phi : h\mapsto e^{t\Delta }h_0-\int _0^t e^{(t-s)\Delta }F(h(s))ds \end{aligned}$$

is bounded and Lipschitz on $L^\infty (0,T_0\text {;}K_0)$, with a Lipschitz constant strictly less than 1. To see this, take an arbitrary $T>0$ and an arbitrary $h\in L^\infty (0,T\text {;}K_0)$. Since

$$\begin{aligned}{} & {} \frac{\varepsilon _0}{2}\le h(s)\quad \text {and}\quad \Vert h(s)\Vert _{W^{1,\infty }(\Omega )}\le 2CM_0\\{} & {} \qquad \qquad \Longrightarrow \sup _{s\in (0,T)}\Vert F(h(s))\Vert _{W^{1,\infty }}\le M \end{aligned}$$

for some $M=M(\varepsilon _0,M_0,\beta ,C)$ where $\beta$ is given in (1.2), we get

$$\begin{aligned} \left\| \int _0^t e^{(t-s)\Delta }F(h(s))ds\right\| _{W^{1,\infty }}\le C TM. \end{aligned}$$

Meanwhile, we have

$$\begin{aligned} \varepsilon _0\le e^{t\Delta }h_0\quad \text {and}\quad \Vert e^{t\Delta }h_0\Vert _{W^{1,\infty }}\le C\Vert h_0\Vert _{W^{1,\infty }}=C M_0. \end{aligned}$$

Thus

$$\begin{aligned} T\le \frac{1}{CM}\min \left\{ \frac{\varepsilon _0}{2}, M_0\right\} \Longrightarrow \phi (h)(t)\in K_0. \end{aligned}$$

Now take $h_1,h_2\in K_0$. Since

$$\begin{aligned} \Vert F(h_1(s))-F(h_2(s))\Vert _{W^{1,\infty }}\le \gamma \Vert h_1(s)-h_2(s)\Vert _{W^{1,\infty }} \end{aligned}$$

for some $\gamma =\gamma (\varepsilon _0,M_0,\beta ,C)$, we have

$$\begin{aligned} \Vert \phi (h_1)(t)-\phi (h_2)(t)\Vert _{W^{1,\infty }}\le T\gamma \sup _{s\in (0,t)}\Vert h_1(s)-h_2(s)\Vert _{W^{1,\infty }}. \end{aligned}$$

Choose $T_0$ so that $CT_0M\le \min (\frac{\varepsilon _0}{2}, M_0)$ and $T_0\gamma \le \frac{1}{2}$ proves the Lipschitz property.

By the claim just proved, for any $h_0\in {{\textbf{H}}}$ there exists a unique solution h with trajectory in $K_0$, with maximal existence time denoted by T. Moreover, either $T=\infty$ (the solution is global), or $T<\infty$, the solution escapes from $L^\infty (0,T\text {;}K_0)$ in the sense

$$\begin{aligned} \text {either}\quad \lim _{t\rightarrow T}\Vert h(t)\Vert _{W^{1,\infty }}=\infty ,\quad \text {or}\quad \lim _{t\rightarrow T}\Vert 1/h(t)\Vert _{L^\infty }=\infty . \end{aligned}$$

By the arguments above, only the second scenario is allowed, i.e., $\lim _{t\rightarrow T}\inf h(t)=0$.

• (2) Suppose that $\Omega ={\mathbb {R}}^N$ and $x_0\in \Omega$. Consider

  $$\begin{aligned} \widetilde{h}=h-\psi \quad \text {and} \quad \widetilde{h}_0=h_0-\psi , \end{aligned}$$

  where $\psi =\psi _{x_0}$ is given in Definition 1.6 for ${{\textbf{H}}}={{\textbf{H}}}_{\psi _{x_0}}$. Note that

  $$\begin{aligned}{} & {} h_0\in {{\textbf{H}}}\Longrightarrow \widetilde{h}_0\in W^{1,\infty }({\mathbb {R}}^N), \\{} & {} h(s)\in {{\textbf{H}}}\Longrightarrow \frac{1}{\widetilde{h}(s)+\psi }\in L^\infty ({\mathbb {R}}^N). \end{aligned}$$

  Moreover, (1.1) is equivalent to

  $$\begin{aligned} \frac{\partial \widetilde{h}}{\partial t}=\Delta \widetilde{h}-F(\widetilde{h}+\psi )+\Delta \psi =:\Delta \widetilde{h}-\widetilde{F}(\widetilde{h}). \end{aligned}$$

  Here $\Delta \psi \in L^\infty ({\mathbb {R}}^N)$. We then apply similar arguments as above to the new smooth nonlinear function $\widetilde{F}$ to solve $\widetilde{h}$ locally in $K_0({\mathbb {R}}^N)$. We conclude that the solution $h=\widetilde{h}+\psi$ solved this way is either global, or for some $T<\infty$, $\lim _{t\rightarrow T}\inf h(t)=0$.

Combing the two cases, we proved the local well-posedness of (1.1) in ${{\textbf{H}}}$.

Remark C.1

In the first case of bounded domains, the local well-posedness arguments extend to other types of boundary conditions.