Strong Ill-Posedness of the 3D Incompressible Euler Equation in Borderline Spaces

For the $d$-dimensional incompressible Euler equation, the usual energy method gives local well-posedness for initial velocity in Sobolev space $H^s(\mathbb{R}^d)$, $s>s_c:=d/2+1$. The borderline case $s=s_c$ was a folklore conjecture. In the previous paper [2], we introduced a new strategy (large lagrangian deformation and high frequency perturbation) and proved strong ill-posedness in the critical space $H^1(\mathbb{R}^2)$. The main issues in 3D are vorticity stretching, lack of $L^p$ conservation, and control of lifespan. Nevertheless in this work we overcome these difficulties and show strong ill-posedness in 3D. Our results include general borderline Sobolev and Besov spaces.

Initial layer and incompressible limit for Euler–Poisson equation in nonthermal plasma

The singular limit from compressible Euler–Poisson equation in nonthermal plasma to incompressible Euler equation with an ill-prepared initial data is investigated in this paper by constructing approximate solutions of the appropriate order via an asymptotic expansion. Nonlinear asymptotic stability of initial layer approximation is established with the convergence rate.

Steady point vortex pair in a field of Stuart-type vorticity

A new family of exact solutions to the two-dimensional steady incompressible Euler equation is presented. The solutions provide a class of hybrid equilibria comprising two point vortices of unit circulation – a point vortex pair – embedded in a smooth sea of non-zero vorticity of ‘Stuart-type’ so that the vorticity $\unicode[STIX]{x1D714}$ and the stream function $\unicode[STIX]{x1D713}$ are related by $\unicode[STIX]{x1D714}=a\text{e}^{b\unicode[STIX]{x1D713}}-\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}_{0})-\unicode[STIX]{x1D6FF}(\boldsymbol{x}+\boldsymbol{x}_{0})$, where $a$ and $b$ are constants. We also examine limits of these new Stuart-embedded point vortex equilibria where the Stuart-type vorticity becomes localized into additional point vortices. One such limit results in a two-real-parameter family of smoothly deformable point vortex equilibria in an otherwise irrotational flow. The new class of hybrid equilibria can be viewed as continuously interpolating between the limiting pure point vortex equilibria. At the same time the new solutions continuously extrapolate a similar class of hybrid equilibria identified by Crowdy (Phys. Fluids, vol. 15, 2003, pp. 3710–3717).

Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus

A steady solution of the incompressible Euler equation on a toroidal surface T R , r of major radius R and minor radius r is provided. Its streamfunction is represented by an exact solution to the modified Liouville equation, ∇ T R , r 2 ψ = c e d ψ + ( 8 / d ) κ , where ∇ T R , r 2 and κ denote the Laplace–Beltrami operator and the Gauss curvature of the toroidal surface respectively, and c , d are real parameters with cd < 0. This is a generalization of the flows with smooth vorticity distributions owing to Stuart (Stuart 1967 J. Fluid Mech. 29 , 417–440. ( doi:10.1017/S0022112067000941 )) in the plane and Crowdy (Crowdy 2004 J. Fluid Mech. 498 , 381–402. ( doi:10.1017/S0022112003007043 )) on the spherical surface. The flow consists of two point vortices at the innermost and the outermost points of the toroidal surface on the same line of a longitude, and a smooth vorticity distribution centred at their antipodal position. Since the surface of a torus has non-constant curvature and a handle structure that are different geometric features from the plane and the spherical surface, we focus on how these geometric properties of the torus affect the topological flow structures along with the change of the aspect ratio α = R / r . A comparison with the Stuart vortex on the flat torus is also made.

Quasineutral limit for a model of three-dimensional Euler–Poisson system with boundary

Quasineutral limit for a model of three-dimensional Euler–Poisson system in half space with a boundary layer is studied. Based on the matched asymptotic expansion method of singular perturbation problem and the elaborate energy method, we prove that the quasineutral regime is the incompressible Euler equation.

A survey on well-balanced and asymptotic preserving schemes for hyperbolic models for chemotaxis

Chemotaxis is a biological phenomenon widely studied these last years, at the biological level but also at the mathematical level. Various models have been proposed, from microscopic to macroscopic scales. In this article, we consider in particular two hyperbolic models for the density of organisms, a semi-linear system based on the hyperbolic heat equation (or dissipative waves equation) and a quasi-linear system based on incompressible Euler equation. These models possess relatively stiff solutions and well-balanced and asymptotic-preserving schemes are necessary to approximate them accurately. The aim of this article is to present various techniques of well-balanced and asymptotic-preserving schemes for the two hyperbolic models for chemotaxis.