Vortex Motion for the Lake Equations

Abstract

The lake equations

$$\begin{aligned} \left\{ \begin{aligned}&\nabla \cdot \big ( b \, {\mathbf {u}}\big ) = 0&\text {on}\ {\mathbb {R}}\times D , \\&\partial _t{\mathbf {u}} + ({\mathbf {u}}\cdot \nabla ){\mathbf {u}} = -\nabla h&\text {on}\ {\mathbb {R}}\times D , \\&{\mathbf {u}} \cdot \varvec{\nu } = 0&\text {on}\ {\mathbb {R}}\times \partial D \end{aligned} \right. \end{aligned}$$

model the vertically averaged horizontal velocity in an inviscid incompressible flow of a fluid in a basin whose variable depth $b : D \rightarrow [0, + \infty )$ is small in comparison to the size of its two-dimensional projection $D \subset {\mathbb {R}}^2$. When the depth b is positive everywhere in D and constant on the boundary, we prove that the vorticity and energy of solutions of the lake equations whose initial vorticity concentrates at an interior point behaves asympotically a multiple of a Dirac mass whose motion is governed by the depth function b.

Appendices

Appendix A: Weak Solutions of the Transport Equation

A first interesting fact is that for a transport equation with no flux through the boundary, it is equivalent to test the equation against compactly supported smooth functions or functions that are smooth up to the boundary.

Proposition A.1

Assume that ${\mathbf {u}}\in L^\infty (W^{1, 1} (D))$ and that ${\mathbf {u}}\cdot \varvec{\nu }= 0$ on $\partial D$ in the sense of traces. If $f_0 \in L^\infty (D)$ and $f \in L^\infty ([0, +\infty ) \times D)$ satisfy for every $\varphi \in C^1_c ([0, +\infty )\times D)$ the identity

$$\begin{aligned} \int _0^{+\infty }\int _{D} f\,(\partial _t \varphi + {\mathbf {u}}\cdot \nabla \varphi ) + \int _{D} f_0 \,\varphi (0, \cdot ) = 0, \end{aligned}$$

then the identity holds for every $\varphi \in C^1_c ([0, +\infty ) \times \bar{D})$.

Proof

We consider a map $\theta \in C^1 ((0, + \infty ))$ such that $\theta = 0$ on $(0,\frac{1}{2})$ and $\theta (t) = 1$ on $[1, +\infty )$ and we define for each $n \in {{\mathbb {N}}}$, the function $\chi _n : D \rightarrow {{\mathbb {R}}}$ for each $x \in D$ by $\chi _n (x) \triangleq \theta (n {{\,\mathrm{dist}\,}}(x, \partial D))$. By the smoothness assumption on $D$, $\chi _n \in C^1_c (D)$. Since ${\mathbf {u}}\cdot \varvec{\nu }= 0$ in the sense of traces, we have for every $T \in [0, +\infty )$,

$$\begin{aligned} \int _0^T \int _{D} |{\mathbf {u}}\cdot \nabla \chi _n | \le C_{1} \int _0^T \int \limits _{\begin{array}{c} x \in D\\ {{\,\mathrm{dist}\,}}(x, \partial D) \le \frac{1}{n} \end{array}} |\nabla {\mathbf {u}} |, \end{aligned}$$

and thus by Lebesgue’s dominated convergence theorem,

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _0^{T} \int _{D} |{\mathbf {u}}\cdot \nabla \chi _n | =0. \end{aligned}$$

(A.1)

For each $\varphi \in C^1_c ([0, +\infty ) \times {\bar{D}})$ and every $n \in {{\mathbb {N}}}$, we take $\chi _n \varphi \in C^1_c ([0, +\infty ) \times D)$ as test function, and we obtain by assumption

$$\begin{aligned} \int _0^{+\infty } \int _{D} \chi _n \, f \, (\partial _t \varphi + {\mathbf {u}}\cdot \nabla \varphi ) + \int _{D} \chi _n \, f_0 \, \varphi (0, \cdot ) = - \int _0^{+\infty } \int _{D} \varphi \, f\, {\mathbf {u}}\cdot \nabla \chi _n; \end{aligned}$$

the conclusion follows by letting $n \rightarrow \infty$, and using (A.1). $\quad \square$

The flow ${\mathbf {u}}$ can be integrated following DiPerna and P.-L. Lions [17] in order to provide solutions to the corresponding transport problem.

Proposition A.2

Let $b \in C^1({\bar{D}}, (0, +\infty ))$ and assume that the velocity field satisfies ${\mathbf {u}}\in L^1_{\mathrm {loc}} ([0, +\infty ), W^{1, 1} (D) \cap L^\infty (D))$. If ${\mathbf {u}}\cdot \varvec{\nu }= 0$ in the sense of traces and if $\nabla \cdot (b\, {\mathbf {u}}) = 0$ in $D$ almost everywhere, then there is a unique Borel-measurable function $X : [0, +\infty ) \times [0, +\infty ) \times D\rightarrow D$, such that

• (i) the map $(s, t) \in [0, +\infty )^2 \mapsto X (s, t, \cdot )$ is continuous for the convergence in measure,

• (ii) for every $r, s, t \in [0, + \infty )$ and almost every $x \in D$, one has $X (s, t, x)= X (s, r, X (r, t, x))$,

• (iii) for every function $f_0 \in L^\infty (D)$ and every $s, t \in [0, +\infty )$, one has

  $$\begin{aligned} \int _{D} f_0 (X (s, t, x)) \, b (x) {\,{{\,\mathrm{d}\,}}}x = \int _{D} f_0 (x) \, b (x) {\,{{\,\mathrm{d}\,}}}x , \end{aligned}$$

• (iv) for almost every $x \in D$,

  $$\begin{aligned} X (s, t, x) = x + \int _t^s {\mathbf {u}}(r, X (r, t, x)) {\,{{\,\mathrm{d}\,}}}r. \end{aligned}$$

Moreover, for every $f_0 \in L^\infty (D)$, $f_0 \circ X(0, \cdot )$ is the unique function $f \in L^\infty ([0, +\infty ) \times D)$ that satisfies for every $\varphi \in C^1_c ([0, +\infty )\times D)$,

$$\begin{aligned} \int _0^{+\infty }\int _{D} f(\partial _t \varphi + {\mathbf {u}}\cdot \nabla \varphi ) + \int _{D} f_0 \,\varphi (0, \cdot ) = 0 \end{aligned}$$

and $f \in C([0, +\infty ), L^1 (D))$.

As a corollary of the above representation formula, the potential vorticity $\omega (t)/b$ at any time $t\ge 0$ is a rearrangement of the initial potential vorticity $\omega _0/b$, in the sense of the weighted Lebesgue measure ${\,{{\,\mathrm{d}\,}}}\mu (x)=b(x){\,{{\,\mathrm{d}\,}}}x$.

Note that, a priori, the statements only make sense when the function $f_0$ is Borel measurable; the proposition implies then that $X (s, t, \cdot )$ preserves Lebesgue null sets and thus allows one to extend the statement to Lebesgue-measurable functions.

Proof of Proposition A.2

We first observe that $\nabla \cdot {\mathbf {u}}= {\mathbf {u}}\cdot \nabla (\ln b)$ almost everywhere on $D$ and thus $\nabla \cdot {\mathbf {u}}\in L^\infty (D)$. The existence and the properties (i), (ii) and (iv) of X follow from the DiPerna–Lions theory [17, Theorem III.2], as does the characterization of solutions to the transport equations and the continuity of the latter [17, Corollary II.2]. By Proposition A.1, the transport equation holds for each test function $\varphi \in C^1_c ([0, +\infty ) \times {\bar{D}})$.

Given $f_0 \in L^\infty (D)$ as an initial data, we observe that the function $f : [0, +\infty ) \times D\rightarrow {{\mathbb {R}}}$ defined for each $(t, x) \in [0, +\infty ) \times D$ by $f (t, x)\triangleq f_0 (X(t, 0, x))$ satisfies the transport equation. By taking $\varphi \in C^1_c ([0, +\infty ) \times {\bar{D}})$ defined for each $t, x \in [0, +\infty ) \times {\bar{D}})$ by $\varphi (t, x) \triangleq b (x) \theta (t)$, with $\theta \in C^1_c ((0, + \infty ))$ as test function we have

$$\begin{aligned}&\int _{0}^{+\infty } \theta ' (t) \biggl (\int _{D} f_0 (X (t, 0, \cdot )) \, b\biggr ) {\,{{\,\mathrm{d}\,}}}t + \theta (0) \int _{D} f_0 \,b\nonumber \\&\quad = -\int _{0}^{+\infty } \biggl ( \int _{D} f (X (t, 0, \cdot ))\, \nabla \cdot (b\,{\mathbf {u}}(t)) \biggr ){\,{{\,\mathrm{d}\,}}}t = 0, \end{aligned}$$

(A.2)

since $\nabla \cdot (b\, {\mathbf {u}}(t)) = 0$ almost everywhere in $D$ for almost every $t \in [0, +\infty )$ and the conclusion follows. $\quad \square$

Appendix B: Regularity of Solutions with Smooth Initial Data

We prove that when the initial vorticity is smooth enough, then weak solutions of the vorticity formulation of the lake equations have some regularity.

Proposition B.1

Assume that $k \in {{\mathbb {N}}}$ and $\alpha \in (0, 1)$, that $D$ is of class $C^{k + 1}$ and that $b \in (C^2 \cap C^{k + 1, \alpha }) (D)$. If $\omega _0 \in C^{k, \alpha } ({\bar{D}}, {{\mathbb {R}}})$ and if $(\omega , {\mathbf {u}}) \in L^\infty ([0, +\infty ) \times D, {{\mathbb {R}}}) \times L^\infty ( [0, +\infty ), L^2 (D, {{\mathbb {R}}}^2))$ is a weak solution to the vortex formulation of the lake equations, then for every $T > 0$, $\omega \in C^{k, \alpha } ([0, T] \times {\bar{D}})$ and ${\mathbf {u}}\in L^\infty ([0, T], C^{k + 1, \alpha } ({{{\bar{D}}}}, {{\mathbb {R}}}^2)) \cap C^{k, \alpha } ([0, T] \times {\bar{D}}, {{\mathbb {R}}}^2)$.

When $k = 0$, Proposition B.1 is due to Huang [24, Theorem 4.1].

Our proof follows the same strategy as proofs of the regularity of solutions of the planar Euler equations [39, §2.4] (see also [26, §3.1]).

The first tool that we use is the fact that the velocity field generated by a bounded vorticity field satisfies a bound known as quasi-Lipschitz bound [26, Lemma 1.4]; [39, Lemma 3.1] or logarithmically Lipschitz [22].

Lemma B.2

There exists a constant $C > 0$ such that for every $\omega \in L^\infty (D)$ and every $x, y \in D$, one has

$$\begin{aligned} \bigl | \nabla {\mathcal {K}}_b [\omega ] (x) - \nabla {\mathcal {K}}_b [\omega ] (y) \bigr | \le C\, |y - x | \ln \frac{2{{\,\mathrm{diam}\,}}D}{|y - x |}. \end{aligned}$$

Proof

By Proposition 3.3, we have

$$\begin{aligned} \nabla {\mathcal {K}}_b[\omega ] (x)= & {} \int _{D} \nabla G_D (x, z) \, \omega (z) \, \sqrt{b (x) \, b (z)} {\,{{\,\mathrm{d}\,}}}z\nonumber \\&+\, \frac{1}{2} \int _{D} G_D (x, z) \, \omega (z) \, \sqrt{\frac{b (z)}{b (x)}} \, \nabla b (x) {\,{{\,\mathrm{d}\,}}}z \nonumber \\&+\, \int _{D} \nabla R_b (x, z) \, \omega (z) {\,{{\,\mathrm{d}\,}}}z. \end{aligned}$$

(B.1)

We first have the estimate

$$\begin{aligned}&\biggl | \int _{D} \nabla G_D (y, z) \, \omega (z) \, \sqrt{b (y)\, b (z)} {\,{{\,\mathrm{d}\,}}}z - \int _{D} \nabla G_D (x, z) \, \omega (z) \, \sqrt{b (x)\, b (z)} {\,{{\,\mathrm{d}\,}}}z \biggr |\nonumber \\&\quad \le C_{1} \, \Vert {\omega }\Vert _{L^\infty (D)} \, |y - x | \ln \frac{2{{\,\mathrm{diam}\,}}(D)}{|y - x |} \end{aligned}$$

(B.2)

(see [39, Lemma 2.3.1 and Appendix 2.3]). Next, we have

$$\begin{aligned}&\biggl | \int _{D} G_D (y, z) \, \omega (z) \, \sqrt{\frac{b (z)}{b (x)}} \, \nabla b (y) {\,{{\,\mathrm{d}\,}}}z - \int _{D} G_D (x, z) \, \omega (z) \, \sqrt{\frac{b (z)}{b (x)}} \, \nabla b (x) {\,{{\,\mathrm{d}\,}}}z \biggr |\nonumber \\&\quad \le \frac{|\nabla b (y) - \nabla b (x) |}{\sqrt{b (x)}} \, \int _{D} G_D (x, z)\, |\omega (z) | \, \sqrt{b (z)} {\,{{\,\mathrm{d}\,}}}z\nonumber \\&\qquad +\, C_{2} \frac{|\nabla b (x) |}{\sqrt{b (x)}} \int _{D} \bigl |G_D (y, z) - G_D (x, z) \bigr | \, |\omega (z) | \, \sqrt{b (z)} {\,{{\,\mathrm{d}\,}}}z . \end{aligned}$$

(B.3)

We compute now by Proposition 3.5,

$$\begin{aligned} \begin{aligned} \int _{D} |G_D (y, z) - G_D (x, z) |{\,{{\,\mathrm{d}\,}}}z&\le \int _0^1 \int _{D} |\nabla G_D ((1 - s) x + s y, z) | \,|y - x |{\,{{\,\mathrm{d}\,}}}z {\,{{\,\mathrm{d}\,}}}t\\&\le \int _0^1 \int _{D} \frac{C_{3}\, |y - x |}{|(1 - s) x + s y - z |} {\,{{\,\mathrm{d}\,}}}z {\,{{\,\mathrm{d}\,}}}s \le C_{4} \, |y - x |. \end{aligned} \end{aligned}$$

(B.4)

By (B.3) and (B.4), we deduce, since the derivative of $\nabla b$ is bounded, that the gap

$$\begin{aligned} \biggl | \int _{D} G_D (y, z) \, \omega (z) \, \sqrt{\frac{b (z)}{b (y)}} \, \nabla b (y) {\,{{\,\mathrm{d}\,}}}z - \int _{D} G_D (x, z) \, \omega (z) \, \sqrt{\frac{b (z)}{b (x)}} \, \nabla b (x) {\,{{\,\mathrm{d}\,}}}z \biggr | \end{aligned}$$

(B.5)

is bounded by

$$\begin{aligned} C_{5} \, \Vert {\omega }\Vert _{L^\infty (D)} \, |y - x |. \end{aligned}$$

In order to control the variation of $\nabla R_b(\cdot ,z)$, we recall that by the Proof of Proposition 3.3, $R_b=S_b+Q_b$ for some function $Q_b\in C^2(D\times D)$ defined in Proposition 2.4 and for some function $S_b$ constructed in the Proof of Proposition 3.1 in such a way that for each $z\in D$, the function $S_b (\cdot , y) \in W^{1, 2}_0 (D)$ is the unique solution of the elliptic problem

Hence, in order to conclude the proof, it is sufficient to focus on the $S_b$-term. We recall that for all $z\in D$ the function $S_b$ admits the integral representation

$$\begin{aligned} S_b(x,z) \end{aligned}$$

and moreover, $S_b$ is continuous and symmetric on $D\times D$ (Proposition 3.1). Therefore, we have for all $x,z\in D$:

or equivalently, using the symmetry of the Green’s function :

In particular, a direct application of Fubini’s theorem shows that, for almost-every $x\in D$, we have

(B.6)

Since the $L^p$-norms of Green’s functions are uniformly bounded on as y varies in $D$ [54], we may apply estimates (B.2) and (B.5) to obtain

$$\begin{aligned}&\biggl | \int _{D} \nabla S_b (x, z) \, \omega (z) {\,{{\,\mathrm{d}\,}}}z - \int _{D} \nabla S_b (x, z) \, \omega (z) {\,{{\,\mathrm{d}\,}}}z \biggr | \\&\quad \le C_{6} \, \Vert {\omega }\Vert _{L^\infty (D)} \, |y - x | \ln \frac{2{{\,\mathrm{diam}\,}}(D)}{|y - x |} , \end{aligned}$$

and therefore

$$\begin{aligned}&\biggl | \int _{D} \nabla R_b (x, z) \, \omega (z) {\,{{\,\mathrm{d}\,}}}z - \int _{D} \nabla R_b (x, z) \, \omega (z) {\,{{\,\mathrm{d}\,}}}z \biggr | \nonumber \\&\quad \le C_{7} \, \Vert {\omega }\Vert _{L^\infty (D)} \, |y - x | \ln \frac{2{{\,\mathrm{diam}\,}}(D)}{|y - x |} . \end{aligned}$$

(B.7)

The conclusion follows from (B.1), (B.2), (B.5) and (B.7). $\quad \square$

The next tool is Grönwall type estimate for a logarithmic perturbation of linear growth.

Lemma B.3

Let $A, B, C \in [0, +\infty )$, $A<C$ and let f be a continuous function from $[0,+\infty )$ to (0, C), that is: $f \in C([0,+\infty ), (0, C))$. If for every $t \in [0, + \infty )$,

$$\begin{aligned} f (t) \le A + B \int _0^t f (s) \ln \frac{C}{f (s)} {\,{{\,\mathrm{d}\,}}}s, \end{aligned}$$

then for every $t \in [0, + \infty )$,

$$\begin{aligned} f (t) \le C \exp \Bigl (- \ln \tfrac{C}{A} e^{-B t}\Bigr ). \end{aligned}$$

Proof

We observe that if the function $u \in C^1 ({{\mathbb {R}}},(0, +C))$ satisfies for every $t \in I$ the equation

$$\begin{aligned} u'(t) = B \, u (t) \ln \frac{C}{u (t)}, \end{aligned}$$

then

$$\begin{aligned} u(t) = C \exp \bigl (- e^{-Bt}\ln \tfrac{C}{u (0)} \bigr ) \end{aligned}$$

and the conclusion follows then by comparison. $\quad \square$

We finally rely on the next classical regularity property of Lagrangian flows.

Lemma B.4

If ${\mathbf {u}}\in C^{k - 1} ([0, +\infty )\times {\bar{D}}, {{\mathbb {R}}}^2) \cap C ([0, +\infty ), C^{k} ({\bar{D}}, {{\mathbb {R}}}^2))$ and if the function $X \in C^1 ([0, +\infty ), C({\bar{D}}, {\bar{D}}))$ satisfies

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t X (t, x) = {\mathbf {u}}(t, X (t, x))&\text {if }t \in [0, +\infty ) \text { and }x \in D,\\&X (0, x) = x&\text {if }x \in D, \end{aligned} \right. \end{aligned}$$

then $X \in C^k ([0, +\infty ) \times D)$. If moreover ${\mathbf {u}}\in C^{k - 1, \alpha } ([0, +\infty )\times {\bar{D}}, {{\mathbb {R}}}^2) \cap L^\infty ([0, +\infty ), C^{k, \alpha } ({\bar{D}}, {{\mathbb {R}}}^2))$, then $X \in C^{k, \alpha } ([0, +\infty ) \times D)$.

Proof

The first part is classical (see for example [10, §1.7]). For the second part, we first have by the first part $X \in C^{k} ({{\mathbb {R}}}\times {\bar{D}}, {\bar{D}})$ and thus by the chain rule for Hölder continuous functions (see for example [11, Theorem 16.31]) $\partial _t X \in C^{k - 1, \alpha } ({{\mathbb {R}}}\times {\bar{D}}, {\bar{D}})$.

Next we observe that for every $T > 0$ and $t \in [0, T]$, we have

$$\begin{aligned}&|D_x^k X (t, y) - D_x^k X (t, x) |\\&\quad \le \int _0^t |D^k f (X (s, y)) [D_x^k X (s, y)] - D^k f (X (s, x)) [D_x^k X (s, x)] | {\,{{\,\mathrm{d}\,}}}s + C_{1} |y - x |\\&\quad \le C_{2} \int _0^t |D_x^k X (s, y) - D^k_x X (s, x) | {\,{{\,\mathrm{d}\,}}}s + C_{3} |y - x |^\alpha , \end{aligned}$$

and it follows then from the classical Grönwall inequality that

$$\begin{aligned} |D_x^k X (t, y) - D_x^k X (t, x) | \le C_{4} |y - x |^\alpha . \end{aligned}$$

$\square$

Proof of Proposition B.1

By Propositions 2.6, A.2 is applicable to $f_0 = \omega _0/b$ and implies that for every $t \in {{\mathbb {R}}}$,

$$\begin{aligned} \omega (t, x) = \frac{b (x)}{b (X (t, x))} \, \omega _0 (X (t, x)). \end{aligned}$$

By Proposition A.2 and Lemma B.2, we have for every $x, y \in D$,

$$\begin{aligned}&|X (t, y) - X (t, x) |\\&\quad \le |y - x | + C_{1} \, \Bigg ( \bigl (\Vert {\omega _0}\Vert _{L^\infty (D)} + \Vert {\Gamma }\Vert _{}\bigr ) \, \int _0^t |X (s, y) \\&\qquad -\, X (s, x) | \ln \frac{2 {{\,\mathrm{diam}\,}}D}{|X (s, y) - X (s, x) |}{\,{{\,\mathrm{d}\,}}}s\Bigg ). \end{aligned}$$

It follows then by Lemma B.3, that

$$\begin{aligned} |X (t, y) - X (t, x) | \le C_{2} \exp \Bigl (- \alpha e^{-C_{3} t} \ln \frac{2 {{\,\mathrm{diam}\,}}D}{|y - x |}) \Bigr ) =C_{4} \Bigl (\frac{|y - x |}{2 {{\,\mathrm{diam}\,}}D}\Bigr )^{ \alpha \exp ({- C_{3} t})}. \end{aligned}$$

This implies thus that

$$\begin{aligned} |\omega (t, y) - \omega (t, x) | \le C_{5} \Bigl (\frac{|y - x |}{2 {{\,\mathrm{diam}\,}}D}\Bigr )^{\alpha \exp ({-C_{3} t})}. \end{aligned}$$

By the representation formula for the velocity (2.7) and classical regularity estimates [21, Theorem 6.8], it follows then that we have the inclusion ${\mathbf {u}}\in C ([0, +\infty ), C^1 (D))$. By classical regularity theory of the Lagrangian flow, this implies that $X \in C^1 ([0, + \infty ) \times {\bar{D}}, {\bar{D}})$ and thus by composition $\omega \in C^{0, \alpha } ([0, T] \times D)$. By regularity estimates [21, Theorem 6.8] we have then ${\mathbf {u}}\in L^\infty ([0, T], C^{1, \alpha } ({{{\bar{D}}}}))$ and $u \in C^{0, \alpha } ([0, T] \times {\bar{D}})$.

We assume now that ${\mathbf {u}}\in L^\infty ([0, T], C^{k, \alpha } ({{{\bar{D}}}})) \cap C^{k - 1, \alpha } ([0, T] \times {\bar{D}})$ and that $\omega _0 \in C^{k, \alpha } ({\bar{D}})$. By regularity of the Lagrangian flow (Lemma B.4), we have $X \in C^{k, \alpha } (D)$ and thus $\omega \in C^{k, \alpha } ([0, T] \times D)$. By classical regularity estimates, this implies that ${\mathbf {u}}\in L^\infty ([0, T], C^{k + 1, \alpha } ({{{\bar{D}}}})) \cap C^{k, \alpha } ([0, T] \times {\bar{D}})$. $\quad \square$