Ill-posedness of the hydrostatic Euler and singular Vlasov equations

Daniel Han-Kwan and I have just submitted a paper (also, available on arxiv): “Ill-posedness of the hydrostatic Euler and singular Vlasov equations”, dedicated to Claude Bardos on the occasion of his 75th birthday, as a token of friendship and admiration.

In this paper, we develop an abstract framework to establish ill-posedness in the sense of Hadamard for some nonlocal PDEs exhibiting “ellipticity”. Examples include, but not limited to, the hydrostatic Euler equations as well as the kinetic incompressible Euler equations and the Vlasov-Dirac-Benney system. The ill-posedness is reminiscent of Lax-Mizohata ill-posedness theorem for first-order PDEs violating the hyperbolicity condition (that is, when the spectrum of the operator’s principal symbol is not included in the real line; see, for instance, discussions from the previous blog post).

1. Hydrostatic Euler equations

The Hydrostatic Euler equations arise in the context of two-dimensional incompressible ideal flows in a narrow channel. More precisely, consider the 2D incompressible Euler equations, writing in term of stream-vorticity formulation:

$\displaystyle \partial_t \omega + u \partial_x \omega + v \partial_y \omega = 0,\qquad \Delta\varphi = \omega , \qquad \varphi_{\vert{y=\pm \epsilon}} = 0, \ \ \ \ \ (1)$

in which ${ u := \partial_y \varphi}$ and ${ v := -\partial_x \varphi}$ , posed on a thin domain ${(x,y)\in \mathbb{T}\times [-\varepsilon, \varepsilon]}$ . Formally, zooming in the domain with the change of variables ${(x,y) \mapsto (x,z = \frac{y}{\varepsilon})}$ , one gets, to leading order, the Hydrostatic Euler equations:

$\displaystyle \partial_t \omega + u \partial_x \omega + v \partial_z \omega = 0,\qquad \partial_z^2 \varphi = \omega , \qquad \varphi_{\vert_{z=\pm 1}} = 0, \ \ \ \ \ (2)$

in which ${ u := \partial_z \varphi}$ and ${ v := -\partial_x \varphi}$ . Such a formal derivation was rigorously verified by Grenier, On the derivation of homogeneous hydrostatic equations. M2AN Math. Model. Numer. Anal. 33 (1999), and then Brenier, Remarks on the derivation of the hydrostatic Euler equations. Bull. Sci. Math. 127 (2003), for data that are convex with respect to the vertical variable.

Concerning the Hydrostatic Euler equations, one observes a loss of one ${x}$ -derivative in (2) through ${ v =- \partial_x \varphi}$ , as compared to ${ \omega}$ . This indicates that a standard Cauchy theory cannot be expected for this equation. Brenier was the first to develop a Cauchy theory in Sobolev spaces for data with convex profiles; this was revisited and extended recently in Masmoudi-Wong, On the Hs theory of hydrostatic Euler equations. Arch. Ration. Mech. Anal. 204 (2012), and then Kukavica-Masmoudi-Vicol-Wong, On the local well-posedness of the Prandtl and hydrostatic Euler equations with multiple monotonicity regions, SIAM J. Math. Anal. 46 (2014). A well-posedness result for data with analytic regularity was also obtained by Kukavica-Temam-Vicol-Ziane, Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain. J. Diff. Equ. (2011).

Renardy, Ill-posedness of the hydrostatic Euler and Navier-Stokes equations, Arch. Ration. Mech. Anal. 194 (2009), showed that for arbitrary odd shear flows ${U(z)}$ so that ${\frac{1}{U(z)^2}}$ is integrable, the linearized hydrostatic Euler equations (2) around ${U'}$ have unbounded unstable spectrum. Such profiles do not satisfy the convexity condition. Following an argument of Guo-Tice, Compressible, inviscid Rayleigh-Taylor instability, Indiana Univ. Math. J. 60 (2011), this property for the spectrum can be used to straightforwardly prove some ill-posedness for the nonlinear equations (as I’ve done for Prandtl equations with Gerard-Varet, here this paper, or with Guo in this paper; see also, Friedlander-Vicol for SQG equations in Nonlinearity 24 (2011)); loosely speaking, it asserts that the flow of solutions, if it exists, cannot be ${C^1(H^s, H^1)}$ , for all ${s \geq 0}$ , within a fixed positive time. In our paper, we construct a family of solutions to show that the solution map from ${H^s}$ to ${L^2}$ has unbounded Hölder norm, within arbitrarily short time. The difficulty was to establish a wellposendess theory in a more regular class of solutions. The precise result is as follows:

Theorem 1 (Ill-posedness for the hydrostatic Euler equations) There exists a stationary shear flow ${U(z)}$ such that the following holds. For all ${s \in \mathbb{N}}$ , ${\alpha \in (0,1]}$ , and ${k \in \mathbb{N}}$ , there are families of solutions ${(\omega_\varepsilon)_{\varepsilon>0}}$ of (2), times ${ t_\varepsilon = \mathcal{O}(\varepsilon |\log \varepsilon|)}$ , and ${(x_0, z_0) \in \mathbb{T} \times (-1,1)}$ such that

$\displaystyle \lim_{\varepsilon \rightarrow 0} \frac{ \| \omega_\varepsilon - U' \|_{L^2([0,t_\varepsilon] \times B(x_0, \varepsilon^k) \times B(z_0, \varepsilon^k) )}}{\| \omega_\varepsilon|_{t=0} -U' \|^\alpha_{H^s(\mathbb{T} \times (-1,1))}}=+\infty . \ \ \ \ \ (3)$

The instability is strong enough so that it occurs within a vanishing spatial domain ${\Omega_\varepsilon}$ and a vanishing time ${t_\varepsilon}$ , as ${\varepsilon \rightarrow0}$ .

2. Singular Vlasov equations

The so-called kinetic incompressible Euler equations and the Vlasov-Dirac-Benney systems are kinetic models from plasma physics, arising in the context of small Debye lengths regimes. More precisely, standard kinetic models in plasma physics for ion and electron density distributions ${f^\pm(t,x,v)}$ , respectively, read

$\displaystyle \partial_t f^\pm + v \cdot \nabla_x f^\pm \mp \nabla_x \varphi \cdot \nabla_v f^\pm = 0,$

in which collisions are ignored and the potential function ${\varphi}$ solves the Poisson equation:

$\displaystyle - \varepsilon \Delta \varphi = \rho^+ - \rho^-.$

Here, ${\rho^\pm}$ denotes the charge density of ${f^\pm}$ , respectively, and ${\varepsilon}$ is the Debye length. We look at two scenarios:

Electron dynamics: assume that the ion distributions are homogenous, with ${\rho^+ = 1}$ . The dynamics for electrons (with ${\varphi: = -\varphi}$ and ${f: = f^-}$ ) reads
$\displaystyle \partial_t f + v \cdot \nabla_x f - \nabla_x \varphi \cdot \nabla_v f = 0,\qquad - \varepsilon \Delta \varphi = \rho - 1.$

Formally, in the limit of ${\varepsilon \rightarrow 0}$ , one gets the kinetic incompressible Euler equations:

$\displaystyle \partial_t f + v \cdot \nabla_x f - \nabla_x \varphi \cdot \nabla_v f = 0,\qquad \rho = 1.\ \ \ \ \ (4)$
Ion dynamics: assume that electrons have reach their thermodynamic equilibrium, with the density distribution being a Maxwellian. Its charge density satisfies the Maxwell-Boltzmann relation: ${\rho^- = e^\varphi \approx 1 + \varphi}$ . The dynamics for ions (with ${f: = f^+}$ ) then read
$\displaystyle \partial_t f + v \cdot \nabla_x f - \nabla_x \varphi \cdot \nabla_v f = 0,\qquad - \varepsilon \Delta \varphi = \rho -\varphi - 1.$

Formally, in the limit of ${\varepsilon \rightarrow 0}$ , one gets the Vlasov-Dirac-Benney system:

$\displaystyle \partial_t f + v \cdot \nabla_x f - \nabla_x \varphi \cdot \nabla_v f = 0,\qquad \varphi = \rho - 1.\ \ \ \ \ (5)$

Looking for solutions to (4) (respectively, (5)) of the form ${f(t,x,v) = \rho(t,x) \delta_{v=u(t,x)}}$ turns out to be equivalent to finding solution ${(\rho,u)}$ of the classical incompressible Euler equations (respectively, the compressible isentropic Euler equations with ${\gamma}$ -pressure law for ${\gamma=2}$ ). This justifies the name used for (4), as also suggested by Brenier. The name Vlasov-Dirac-Benney for (5) was coined by Bardos, in connections with the Benney model for Water Waves; see, for instance, Bardos–Nouri, A Vlasov equation with Dirac potential used in fusion plasmas, J. Math. Phys. 53 (2012).

In (4), the potential ${\varphi}$ stands for a Lagrange multiplier (or, from the physical point of view, a {pressure}) related to the constraint ${\rho=1}$ . It is possible to obtain an explicit formula for the potential ${\varphi}$ , arguing as follows. Introduce the current density ${j (t,x) := \int_{\mathbb{R}^3} f(t,x,v) v \, dv}$ . We start by writing the local conservation of charge and current from the Vlasov equation:

$\displaystyle \begin{aligned}\partial_t \rho + \nabla \cdot j&= 0, \\ \partial_t j + \nabla \cdot \int f v \otimes v \, dv &= -\nabla \varphi. \end{aligned}$

By using the constraint ${\rho =1}$ , it follows that ${\nabla \cdot j=0}$ . Plugging this into the conservation of current, one gets the law

$\displaystyle -\Delta \varphi = \nabla \cdot \left(\nabla \cdot \int f v \otimes v \, dv \right). \ \ \ \ \ (6)$

Directly from the laws (6) and (5) for the potential ${\varphi}$ , one sees that there is a loss of one ${x}$ -derivative through the force ${-\nabla_x \varphi}$ , as compared to the distribution function ${f}$ . This explains why a standard Cauchy theory cannot be expected for these equations. What is known though is the existence of analytic solutions (see Han-Kwan, Jabin-Nouri, Bossy-Fontbona-Jabin-Jabir), as well as an ${H^s}$ theory for stable data (see Bardos-Besse and Han-Kwan & Rousset).

Bardos-Nouri, A Vlasov equation with Dirac potential used in fusion plasmas, J. Math. Phys. 53 (2012), show that around unstable homogeneous equilibria (for instance, double-bump equilibria), the linearized equations of (5) have unbounded unstable spectrum. This property was used to prove some ill-posedness, as in the hydrostatic Euler case; loosely speaking they show that the flow of solutions, if it exists, cannot be ${C^1(H^s, H^1)}$ , for all ${s \geq 0}$ . What we prove in this paper is that the flow cannot be ${C^\alpha({H}^s_{\mathrm{weight}}, L^2)}$ , for all ${s \geq 0}$ , ${\alpha \in (0,1]}$ , and any polynomial weight in ${v}$ . Precisely, we have:

Theorem 2 (Ill-posedness for the kinetic incompressible Euler and Vlasov-Dirac-Benney equations) There exists a stationary solution ${\mu(v)}$ such that the following holds. For all ${m,s \in \mathbb{N}}$ , ${\alpha \in (0,1]}$ , and ${k \in \mathbb{N}}$ , there are families of solutions ${(f_\varepsilon)_{\varepsilon>0}}$ of (4) (respectively, (5)), times ${ t_\varepsilon = \mathcal{O}(\varepsilon |\log \varepsilon|)}$ , and ${(x_0, v_0) \in \mathbb{T}^3 \times \mathbb{R}^3}$ , such that

$\displaystyle \lim_{\varepsilon \rightarrow 0} \frac{ \| f_\varepsilon -\mu \|_{L^2([0,t_\varepsilon] \times B(x_0, \varepsilon^k) \times B(v_0, \varepsilon^k) )}}{\| \langle v\rangle^m (f_\varepsilon|_{t=0} - \mu) \|^\alpha_{H^s(\mathbb{T}^3 \times \mathbb{R}^3)}}=+\infty . \ \ \ \ \ (7)$

The instability is strong enough so that it occurs within a vanishing spatial domain ${\Omega_\varepsilon}$ and a vanishing time ${t_\varepsilon}$ , as ${\varepsilon \rightarrow0}$ .

For more discussions and details of the proof, see our paper!

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Ill-posedness of the hydrostatic Euler and singular Vlasov equations

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