Long time behavior of solutions of an electroconvection model in ${\mathbb {R}}^2$

Abstract

We consider a two dimensional electroconvection model which consists of a nonlinear and nonlocal system coupling the evolutions of a charge distribution and a fluid. We show that the solutions decay in time in $L^2({{\mathbb {R}}}^2)$ at the same sharp rate as the linear uncoupled system. This is achieved by proving that the difference between the nonlinear and linear evolution decays at a faster rate than the linear evolution. In order to prove the sharp $L^2$ decay we establish bounds for decay in $H^2({{\mathbb {R}}}^2)$ and a logarithmic growth in time of a quadratic moment of the charge density.

Appendix: Existence and Uniqueness of Solutions

In this appendix, we prove the existence of weak and strong solutions for the electroconvection model (1)–(5).

Definition 1

A solution (q, u) of (1)–(5) is said to be a weak solution on [0, T] if it solves (1)–(5) in the sense of distributions, u is divergence-free in the sense of distributions,

$$\begin{aligned} u \in L^{\infty }(0,T;L^2) \cap L^2(0,T; H^1) \end{aligned}$$

(179)

and

$$\begin{aligned} q \in L^{\infty }(0,T; L^2) \cap L^2(0,T; H^{1/2}). \end{aligned}$$

(180)

Theorem 5

Let $u_0\in L^2$ be divergence-free, let $q_0\in L^2$. Let $T > 0$ be arbitrary. There exists a weak solution (q, u) of the system (1)–(5) on [0, T].

Proof

We briefly sketch the main ideas of the proof. For $0 < \epsilon \le 1$, we consider a viscous approximation of (1)–(5) given by

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t q^{\epsilon } + { u^{\epsilon } \cdot \nabla q^{\epsilon } }+ \Lambda q^{\epsilon } - \epsilon \Delta q^{\epsilon } = 0 \\ \partial _t u^{\epsilon } + { u^{\epsilon } \cdot \nabla u^{\epsilon }} - \Delta u^{\epsilon } + \nabla p^{\epsilon } = -{ q^{\epsilon } Rq^{\epsilon }} \\ \nabla \cdot u^{\epsilon } = 0 \end{array}\right. } \end{aligned}$$

(181)

with smoothed out initial data $u_0^{\epsilon } = J_{\epsilon }u_0$ and $q_0^{\epsilon }= J_{\epsilon }q_0$, where $J_{\epsilon }$ is a standard mollifier operator. For each $\epsilon > 0$, we consider the map

$$\begin{aligned} (q(t), u(t)) \mapsto \Phi _{\epsilon }((q,u))(t) = (e^{\epsilon t \Delta } J_{\epsilon }q_0 - {\mathcal {A}}^{\epsilon }_t (q^{\epsilon }, u^{\epsilon }), e^{t\Delta }J_{\epsilon }u_0 - {\mathcal {B}}^{\epsilon }_t (q^{\epsilon }, u^{\epsilon })) \end{aligned}$$

(182)

where

$$\begin{aligned} {\mathcal {A}}^{\epsilon }_t (q^{\epsilon }, u^{\epsilon }) = \int _{0}^{t} e^{\epsilon (t-s)\Delta } (u^{\epsilon } \cdot \nabla q^{\epsilon })(s) ds + \int _{0}^{t} e^{\epsilon (t-s)\Delta } \Lambda q^{\epsilon }(s) ds \end{aligned}$$

(183)

and

$$\begin{aligned} {\mathcal {B}}^{\epsilon }_t (q^{\epsilon }, u^{\epsilon }) = \int _{0}^{t} e^{(t-s)\Delta } {\mathbb {P}}(u^{\epsilon } \cdot \nabla u^{\epsilon })(s) ds + \int _{0}^{t} e^{(t-s)\Delta } {\mathbb {P}}(q^{\epsilon }Rq^{\epsilon })(s) ds. \end{aligned}$$

(184)

There exists a time $T_{\epsilon } = T_{\epsilon } (\epsilon , \Vert u_0\Vert _{L^2}, \Vert q_0\Vert _{L^2}) > 0$ such that the map $\Phi _{\epsilon }$ is a contraction on the Banach space

$$\begin{aligned} X_T = L^{\infty }(0,T; \bar{B}_{L^2} (2\Vert q_0\Vert _{L^2}) \oplus L^{\infty }(0,T; \bar{B}_{L^2_{\sigma }} (2\Vert u_0\Vert _{L^2}) \end{aligned}$$

(185)

where $\bar{B}_{L^2}(r)$ is the closed ball in $L^2$, and $\bar{B}_{L^2_{\sigma }}$ is the closed ball in the space of $L^2$ divergence-free vectors. Consequently, $\Phi _{\epsilon }$ has a fixed point $(q^{\epsilon }, u^{\epsilon }) \in X_{T_{\epsilon }}$ solving (181). This solution extends to the time interval [0, T], and this can be obtained by establishing uniform-in-time bounds for $(q^{\epsilon }, u^{\epsilon })$ on [0, T]. Indeed, we have

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \left( \Vert \Lambda ^{-\frac{1}{2}}q^{\epsilon } \Vert _{L^2}^2 + \Vert u^{\epsilon } \Vert _{L^2}^2 \right) + \Vert q^{\epsilon }\Vert _{L^2}^2 + \Vert \nabla u^{\epsilon }\Vert _{L^2}^2 + \epsilon \Vert \Lambda ^{\frac{1}{2}}q^{\epsilon }\Vert _{L^2}^2 = 0 \end{aligned}$$

(186)

as shown in (11). Hence the family of mollified velocities $(u^{\epsilon })_{\epsilon }$ is uniformly bounded in $L^{\infty }(0,T; L^2) \cap L^2(0,T; H^1)$. On the other hand, the $L^2$ norm of $q^{\epsilon }$ evolves according to

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \Vert q^{\epsilon }\Vert _{L^2}^2 + \Vert \Lambda ^{\frac{1}{2}} q^{\epsilon }\Vert _{L^2}^2 + \epsilon \Vert \nabla q^{\epsilon }\Vert _{L^2}^2 = 0, \end{aligned}$$

(187)

and so the family of mollified charge densities $(q^{\epsilon })_{\epsilon }$ is uniformly bounded in $L^{\infty }(0,T; L^2) \cap L^2(0,T; H^{\frac{1}{2}})$. The $q^{\epsilon }$ and $u^{\epsilon }$ equations imply that the sequence of time derivatives $(\partial _t q^{\epsilon })_{\epsilon }$ and $(\partial _t u^{\epsilon })_{\epsilon }$ are uniformly bounded in $L^{2}(0,T; H^{-\frac{3}{2}})$ and $L^2(0,T; H^{-1})$ respectively. By the Aubin-Lions lemma, the sequence $((q^{\epsilon }, u^{\epsilon }))_{\epsilon }$ has a subsequence that converges strongly in $L^2(0,T; L^2)$ to a weak solution (q, u) of (1)–(5). We omit further details. $\square$

Definition 2

A weak solution (q, u) of (1)–(5) is said to be a strong solution on [0, T] if

$$\begin{aligned} u \in L^{\infty }(0,T;H^1) \cap L^2(0,T; H^2) \end{aligned}$$

(188)

and

$$\begin{aligned} q \in L^{\infty }(0,T; L^4) \cap L^2(0,T; H^{1/2}). \end{aligned}$$

(189)

Theorem 6

Let $u_0\in H^1$ be divergence-free and $q_0\in L^4$. Let $T>0$ be arbitrary. There exists a unique strong solution (u, q) of the system (1)–(5) on [0, T].

We take the $L^2$ inner product of the equation satisfied by $q^{\epsilon }$ in (181) with $(q^{\epsilon })^3$. In view of the divergence-free condition satisfied by $u^{\epsilon }$, the nonlinear term vanishes, that is

$$\begin{aligned} \int _{{{\mathbb {R}}}^2} u^{\epsilon } \cdot \nabla q^{\epsilon } (q^{\epsilon })^3 dx = 0. \end{aligned}$$

(190)

By the Córdoba-Córdoba inequality ( [6]), we have

$$\begin{aligned} \int _{{{\mathbb {R}}}^2} (q^{\epsilon })^3 \Lambda q^{\epsilon } dx \ge 0 \end{aligned}$$

(191)

and

$$\begin{aligned} -\int _{{{\mathbb {R}}}^2} (q^{\epsilon })^3 \Delta q^{\epsilon } dx \ge 0. \end{aligned}$$

(192)

Consequently, we obtain

$$\begin{aligned} \frac{1}{4} \frac{d}{dt} \Vert q^{\epsilon }\Vert _{L^4}^4 \le 0 \end{aligned}$$

(193)

which yields the boundedness of q in $L^{\infty } (0, T; L^4({{\mathbb {R}}}^2))$ by the Banach Alaoglu theorem and the lower semi-continuity of the norm. The $L^2$ norm of $\nabla u^{\epsilon }$ obeys the energy inequality

$$\begin{aligned} \frac{d}{dt} \Vert \nabla u^{\epsilon }\Vert _{L^2}^2 + \Vert \Delta u^{\epsilon }\Vert _{L^2}^2 \le C \Vert q^{\epsilon }\Vert _{L^4}^4 \end{aligned}$$

(194)

as shown in (81), yielding the boundedness of u in $L^{\infty }(0,T; H^1) \cap L^2(0,T; H^2)$. Now we prove the uniqueness of strong solutions. Suppose $(q_1, u_1)$ and $(q_2, u_2)$ are strong solutions of (1)–(5) with same initial data. Let $q = q_1 - q_2, u = u_1 - u_2$ and $p = p_1 - p_2$. Then q satisfies

$$\begin{aligned} \partial _t q + \Lambda q = - u_1 \cdot \nabla q - u \cdot \nabla q_2 \end{aligned}$$

(195)

and u satisfies

$$\begin{aligned} \partial _t u - \Delta u + \nabla p = - qRq_1 - q_2Rq - u_1 \cdot \nabla u - u \cdot \nabla u_2. \end{aligned}$$

(196)

We take the $L^2$ inner product of (195) with $\Lambda ^{-1}q$ and the $L^2$ inner product of (196) with u. We add the resulting energy equalities. We have a cancellation

$$\begin{aligned} - \int _{{{\mathbb {R}}}^2} (u \cdot \nabla q_2) \Lambda ^{-1} q dx - \int _{{{\mathbb {R}}}^2} (q_2Rq) \cdot u dx = 0 \end{aligned}$$

(197)

obtained from integration by parts. In view of the Ladyzhenskaya’s interpolation inequality, we estimate

$$\begin{aligned} \left| \int _{{{\mathbb {R}}}^2} (qRq_1) \cdot u dx \right|{} & {} \le C\Vert q\Vert _{L^2}\Vert q_1\Vert _{L^4}\Vert u\Vert _{L^2}^{\frac{1}{2}} \Vert \nabla u\Vert _{L^2}^{\frac{1}{2}} \le \frac{1}{4} \Vert \nabla u\Vert _{L^2}^2 + \frac{1}{4} \Vert q\Vert _{L^2}^2\nonumber \\{} & {} \quad \ + C\Vert q_1\Vert _{L^4}^4 \Vert u\Vert _{L^2}^2 \end{aligned}$$

(198)

and

$$\begin{aligned} \left| \int _{{{\mathbb {R}}}^2} (u \cdot \nabla u_2) \cdot u dx \right| \le \Vert u\Vert _{L^4}^2 \Vert \nabla u_2\Vert _{L^2} \le \frac{1}{4} \Vert \nabla u\Vert _{L^2}^2 + C\Vert \nabla u_2\Vert _{L^2}^2 \Vert u\Vert _{L^2}^2. \end{aligned}$$

(199)

Now we write

$$\begin{aligned} \int _{{{\mathbb {R}}}^2} (u_1 \cdot \nabla q) \Lambda ^{-1}q dx = \int _{{{\mathbb {R}}}^2} \left( \Lambda ^{-\frac{1}{2}}(u_1 \cdot \nabla q) - u_1 \cdot \nabla \Lambda ^{-\frac{1}{2}}q \right) \Lambda ^{-\frac{1}{2}}q dx \end{aligned}$$

(200)

via integration by parts, and we show below that

$$\begin{aligned} \left| \int _{{{\mathbb {R}}}^2} \left( \Lambda ^{-\frac{1}{2}}(u_1 \cdot \nabla q) - u_1 \cdot \nabla \Lambda ^{-\frac{1}{2}}q \right) \Lambda ^{-\frac{1}{2}}q dx \right| \le C\Vert u_1\Vert _{H^2} \Vert q\Vert _{L^2}\Vert \Lambda ^{-\frac{1}{2}}q\Vert _{L^2}. \end{aligned}$$

(201)

Putting (197)-(201) together, we obtain the energy inequality

$$\begin{aligned} \frac{d}{dt} \left[ \Vert \Lambda ^{-\frac{1}{2}}q\Vert _{L^2}^2 + \Vert u\Vert _{L^2}^2 \right] \le C\left[ \Vert u_1\Vert _{H^2}^2 + \Vert \nabla u_2\Vert _{L^2}^2 + \Vert q_1\Vert _{L^4}^4 \right] \left[ \Vert \Lambda ^{-\frac{1}{2}}q\Vert _{L^2}^2 + \Vert u\Vert _{L^2}^2 \right] \nonumber \\ \end{aligned}$$

(202)

from which we obtain uniqueness. Finally, we show that the estimate (201) holds by establishing the commutator estimate

$$\begin{aligned} \Vert \Lambda ^{-\frac{1}{2}}(u_1 \cdot \nabla q) - u_1 \cdot \nabla \Lambda ^{-\frac{1}{2}}q \Vert _{L^2} \le C\Vert u_1\Vert _{H^2} \Vert q\Vert _{L^2}. \end{aligned}$$

(203)

Indeed, let $w\in L^2({{\mathbb {R}}}^2)$. By Parseval’s identity, we have

$$\begin{aligned}{} & {} \int _{{{\mathbb {R}}}^2} (\Lambda ^{-\frac{1}{2}} (u_1 \cdot \nabla q) - u_1 \cdot \nabla \Lambda ^{-\frac{1}{2}}q)(x) w (x) dx \nonumber \\{} & {} \quad = \int _{{{\mathbb {R}}}^2} {\mathcal {F}}(\Lambda ^{-\frac{1}{2}} (u_1 \cdot \nabla q) - u_1 \cdot \nabla \Lambda ^{-\frac{1}{2}}q)(\xi ) {\mathcal {F}}w (\xi ) d\xi . \end{aligned}$$

(204)

But

$$\begin{aligned} {\mathcal {F}} (\Lambda ^{-\frac{1}{2}}(u_1 \cdot \nabla q))(\xi ) = \int _{{{\mathbb {R}}}^2} |\xi |^{-\frac{1}{2}} (\xi \cdot {\mathcal {F}}u_1 (\xi - y)) {\mathcal {F}}q (y) dy \end{aligned}$$

(205)

and

$$\begin{aligned} {\mathcal {F}} (u_1 \cdot \nabla \Lambda ^{-\frac{1}{2}}q) (\xi ) = \int _{{{\mathbb {R}}}^2} |y|^{-\frac{1}{2}} (\xi \cdot {\mathcal {F}}u_1 (\xi - y)) {\mathcal {F}}q (y) dy. \end{aligned}$$

(206)

Consequently,

$$\begin{aligned}&\left| \int _{{{\mathbb {R}}}^2} (\Lambda ^{-\frac{1}{2}} (u_1 \cdot \nabla q) - u_1 \cdot \nabla \Lambda ^{-\frac{1}{2}}q)(x) w (x) dx\right| \nonumber \\ {}&\le \int _{{{\mathbb {R}}}^2} \int _{{{\mathbb {R}}}^2} \min \left\{ |\xi |, |y| \right\} \left| |\xi |^{-\frac{1}{2}} - |y|^{-\frac{1}{2}} \right| |{\mathcal {F}} u_1 (\xi - y) | |{\mathcal {F}} q(y)| |{\mathcal {F}} w(\xi ) | dy d\xi \end{aligned}$$

(207)

where we used

$$\begin{aligned} |\xi \cdot {\mathcal {F}}u_1 (\xi - y)| \le \min \left\{ |\xi |, |y| \right\} |{\mathcal {F}} u_1 (\xi - y)| \end{aligned}$$

(208)

which holds due to the fact that the velocity is divergence-free. We note that

$$\begin{aligned} \min \left\{ |\xi |, |y| \right\} \left| |\xi |^{-\frac{1}{2}} - |y|^{-\frac{1}{2}} \right| \le \frac{\min \left\{ |\xi |, |y| \right\} }{|\xi |^{\frac{1}{2}}|y|^{\frac{1}{2}}} |\xi - y|^{\frac{1}{2}} \le |\xi - y|^{\frac{1}{2}} \end{aligned}$$

(209)

for all $\xi , y \in {{\mathbb {R}}}^2$. Therefore,

$$\begin{aligned} \left| \int _{{{\mathbb {R}}}^2} (\Lambda ^{-\frac{1}{2}} (u_1 \cdot \nabla q) - u_1 \cdot \nabla \Lambda ^{-\frac{1}{2}}q)(x) w (x) dx\right|&\le \Vert |.|^{\frac{1}{2}} {\mathcal {F}} u_1(.) \Vert _{L^1} \Vert q\Vert _{L^2}\Vert w\Vert _{L^2} \nonumber \\ {}&\le C\Vert u_1\Vert _{H^2} \Vert q\Vert _{L^2}\Vert w\Vert _{L^2} \end{aligned}$$

(210)

by Hölder’s inequality and Young’s convolution inequality. This gives (203) completing the proof of Theorem 6.