Related papers: Superlinear gradient growth for 2D Euler equation …
We consider solutions to the Navier-Stokes equations with Navier boundary conditions in a bounded domain in the plane with a C^2-boundary. Navier boundary conditions can be expressed in the form w = (2 K - A) v . T and v . n = 0 on the…
This paper studies the existence and singularity formation of supersonic expanding waves for the radially symmetric non-isentropic compressible Euler equations of polytropic gases. We introduce a suitable pair of gradient variables to…
We study a free boundary problem which is motivated by a particular case of the flow of a non-Newtonian fluid, with a pressure depending yield stress given by a Drucker-Prager plasticity criterion. We focus on the steady case and…
In this paper, we prove global regularity for all smooth, axisymmetric, swirl-free solutions of the Euler equation in four dimensions. Previous works establishing global regularity for certain axisymmetric, swirl-free solutions of the Euler…
Coherent vortices are often observed to persist for long times in turbulent 2D flows even at very high Reynolds numbers and are observed in experiments and computer simulations to potentially be asymptotically stable in a weak sense for the…
We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. We assume that $u_0 \in H^{2.5+\delta }$ is such that $\mathrm{curl}\,u_0 \in H^{2+\delta }$ in an arbitrarily small neighborhood of…
We study the long-time behavior of scale-invariant solutions of the 2d Euler equation satisfying a discrete symmetry. We show that all scale-invariant solutions with bounded variation on $\mathbb{S}^1$ relax to states that are piece-wise…
Any classical solution of the 2D incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all…
A class of singular 3D-velocity vector fields is constructed which satisfy the incompressible 3D-Euler equation. It is shown that such a solution scheme does not exist in dimension 2. The solutions constructed are bounded and smooth up to…
In this paper we construct solutions to the Euler and gSQG equations that are concentrated near unstable stationary configurations of point-vortices. Those solutions are themselves unstable, in the sense that their localization radius grows…
Analysis of non-compact manifolds almost always requires some controlled behavior at infinity. Without such, one neither can show, nor expect, strong properties. On the other hand, such assumptions restrict the possible applications and…
We study the small-scale behavior of generalized two-dimensional turbulence governed by a family of model equations, in which the active scalar $\theta=(-\Delta)^{\alpha/2}\psi$ is advected by the incompressible flow $\u=(-\psi_y,\psi_x)$.…
Chemin has shown that solutions of the Navier-Stokes equations in the plane for an incompressible fluid whose initial vorticity is bounded and lies in L^2 converge in the zero-viscosity limit in the L^2-norm to a solution of the Euler…
We construct examples and provide a classification of self-similar solutions to the two-dimensional incompressible Euler equations whose pseudo-velocity fields possess more than one stagnation point. These solutions are also homogeneous…
We consider a two-dimensional, two-layer, incompressible, steady flow, with vorticity which is constant in each layer, in an infinite channel with rigid walls. The velocity is continuous across the interface, there is no surface tension or…
We study whether the solutions of a fully nonlinear, uniformly parabolic equation with superquadratic growth in the gradient satisfy initial and homogeneous boundary conditions in the classical sense, a problem we refer to as the classical…
We consider a steady state $v_{0}$ of the Euler equation in a fixed bounded domain in $\mathbf{R}^{n}$. Suppose the linearized Euler equation has an exponential dichotomy of unstable and center-stable subspaces. By rewriting the Euler…
The second-grade fluid equations are a model for viscoelastic fluids, with two parameters: $\alpha > 0$, corresponding to the elastic response, and $\nu > 0$, corresponding to viscosity. Formally setting these parameters to $0$ reduces the…
Steady fluid flows have very special topology. In this paper we describe necessary and sufficient conditions on the vorticity function of a 2D ideal flow on a surface with or without boundary, for which there exists a steady flow among…
We study positive solutions of the pseudoparabolic equation with a sublinear source in $\mathbb{R}^n$. In this work, the source coefficient could be unbounded and time-dependent. Global existence of solutions to the Cauchy problem is…