Related papers: Regularity of Stagnation Point-form Solutions of t…
We study global regularity for solutions of quasilinear elliptic equations of the form $\div \A(x,u,\nabla u) = \div \F $ in rough domains $\Omega$ in $\R^n$ with nonhomogeneous Dirichlet boundary condition. The vector field $\A$ is assumed…
In this paper, we provide a classification of steady solutions to two-dimensional incompressible Euler equations in terms of the set of flow angles. The first main result asserts that the set of flow angles of any bounded steady flow in the…
Singularity formation of the 3D incompressible Euler equations is known to be extremely challenging. In [18], Elgindi proved that the 3D axisymmetric Euler equations with no swirl and $C^{1,\alpha}$ initial velocity develops a finite time…
We consider in a smooth and bounded two dimensional domain the convergence in the $L^2$ norm, uniformly in time, of the solution of the stochastic second-grade fluid equations with transport noise and no-slip boundary conditions to the…
We introduce a new wave formulation for the relativistic Euler equations with vacuum boundary conditions that consists of a system of non-linear wave equations in divergence form with a combination of acoustic and Dirichlet boundary…
We present a preliminary study of a new phenomena associated with the Euler-Poisson equations -- the so called critical threshold phenomena, where the answer to questions of global smoothness vs. finite time breakdown depends on whether the…
In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian.
In this paper, we discuss the Cauchy Problem for the compressible isentropic Euler-Boltzmann equations with vacuum in radiation hydrodynamics. Firstly, we establish the local existence of regular solutions by the fundamental methods in the…
Established in the 30's, Schauder {\it a priori} estimates are among the most classical and powerful tools in the analysis of problems ruled by 2nd order elliptic PDEs. Since then, a central problem in regularity theory has been to…
Under the assumption that a solution to the 3D incompressible Euler equations blows up at a time $T_\ast$ and that $T_\ast $ is the first such time, we establish lower bounds on the rate of blow-up of the maximum norm of the vorticity. In…
In three previous papers by the two first authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be…
We study the regularity of solutions of elliptic second order boundary value problems on a bounded domain $\Omega$ in $\mathbb R^3$. The coefficients are not necessarily continuous and the boundary conditions may be mixed, i.e. Dirichlet on…
We consider the three-dimensional incompressible free-boundary Euler equations in a bounded domain and with surface tension. Using Lagrangian coordinates, we establish a priori estimates for solutions with minimal regularity assumptions on…
In this article, we study the break-down of smooth and continuous solutions to isentropic Euler system in multi dimension. Sideris [Comm. Math. Phys. 1985] proved the blow up of smooth solutions when initial data satisfies an `integral…
Fix a bounded, analytic, and simply connected domain $\Omega\subset\mathbb{R}^2.$ We show that all analytic steady states of the Euler equations with stream function $\psi$ are either radial or solve a semi-linear elliptic equation of the…
It is known that smooth solutions to the non-isentropic Navier-Stokes equations without heat-conductivity may lose their regularities in finite time in the presence of vacuum. However, in spite of the recent progress on such blowup…
We propose a novel framework for stabilization, with an estimate of the region of attraction, of quasilinear parabolic partial differential equations (PDEs) that exhibit finite-time blow-up phenomena when null boundary inputs are imposed.…
Inspired by the numerical evidence of a potential 3D Euler singularity \cite{luo2014potentially,luo2013potentially-2}, we prove finite time blowup of the 2D Boussinesq and 3D axisymmetric Euler equations with smooth initial data of finite…
In this paper, we study the existence and uniqueness of solutions to the Euler equations with initial conditions that exhibit analytic regularity near the boundary and Sobolev regularity away from it. A key contribution of this work is the…
In the junction $\Omega$ of several semi-infinite cylindrical waveguides we consider the Dirichlet Laplacian whose continuous spectrum is the ray $[\lambda_\dagger, +\infty)$ with a positive cut-off value $\lambda_\dagger$. We give two…