The Magneto-Hydrodynamic (MHD) system of equations governs viscous fluids subject to a magnetic field and is derived via a coupling of the Navier-Stokes equations and Maxwell's equations. Recently it has become common to study generalizations of fluids-based differential equations. Here we consider the generalized Magneto-Hydrodynamic alpha (gMHD-$α$) system, which differs from the original MHD system by including an additional non-linear terms (indexed by $α$), and replacing the Laplace operators by more general Fourier multipliers with symbols of the form $-|ξ|^γ/ g(|ξ|)$. In a paper by Pennington, the problem was considered with initial data in the Sobolev space $H^{s,2}(\mathbb{R}^n)$ with $n \geq 3$. Here we consider the problem with initial data in $H^{s,p}(\mathbb{R}^n)$ with $n \geq 3$ and $p > 2$. Our goal is to minimize the regularity required for obtaining uniqueness of a solution.
In this article we study the global regularity of 2D generalized magnetohydrodynamic equations (2D GMHD), in which the dissipation terms are $- ν(- \triangle)^α u$ and $- κ(-\triangle)^β b$. We show that smooth solutions are global in the following three cases: $α\geqslant 1 / 2, β\geqslant 1$; $0 \leqslant α< 1 / 2, 2 α+ β> 2$; $α\geqslant 2, β= 0$. We also show that in the inviscid case $ν= 0$, if $β> 1$, then smooth solutions are global as long as the direction of the magnetic field remains smooth enough.
We present an alpha-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-alpha equations, for the vortex sheet dynamics. We show that initially smooth self-avoiding vortex sheet remains smooth for all times under the alpha-regularized dynamics, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure.
We present an alpha-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-alpha equations, for the vortex sheet dynamics. We show the convergence of the solutions of Euler-alpha equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vortex sheets case. We also show that, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure, (i) an initially Lipschitz chord arc vortex sheet (curve), evolving under the BR-alpha equatio…
We introduce n/p-harmonic maps as critical points of E(v) the Lp-Norm of the alpha-laplacian of v, where pointwise v maps Rn into a sphere, and alpha = n/p. This energy combines the non-local behaviour of the fractional harmonic maps introduced by Riviere and the first author with the degenerate arguments of the n-laplacian. In this setting, we will prove Holder continuity.
It has recently become common to study many different approximating equations of the Navier-Stokes equation. One of these is the Leray-$α$ equation, which regularizes the Navier-Stokes equation by replacing (in most locations) the solution $u$ in the equation with $(1-α^2\triangle)u$ the operator $(1-α^2\triangle)$. Another is the generalized Navier-Stokes equation, which replaces the Laplacian with a Fourier multiplier with symbol of the form $|ξ|^γ$ ($γ=2$ is the standard Navier-Stokes equation), and recently in [14] Tao also considered multipliers of the form $|ξ|^γ/g(|ξ|)$, where $g$ is (e…
We study the two-dimensional generalized magnetohydrodynamics-$α$ system with fractional Laplacians in the dissipative and diffusive terms. We show that the solution pair of velocity and magnetic fields preserves their initial regularity in all cases when the powers add up to one. This settles the global regularity issue in the general case which was remarked by the authors in [33] to be a problem.
We consider nonlinear fourth order elliptic equations of double divergence type. We show that for a certain class of equations where the nonlinearity is in the Hessian, solutions that are C^{2,alpha} enjoy interior estimates on all derivatives.
C^(sigma+alpha) regularity for fractional Hamilton-Jacobi equations, and rates of convergence in periodic homogenization
Université Paris Cité, CNRS, Sorbonne Université, Laboratoire Jacques-Louis Lions (LJLL)（巴黎西岱大学，法国国家科学研究中心，索邦大学，雅克-路易·利翁斯实验室）；O. Mayer Mathematics Institute, Romanian Academy - Iaşi Branch（罗马尼亚科学院雅西分院奥·迈尔数学研究所）；Departamento de Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción（康塞普西翁大学物理与数学院数学系）；Instituto de Matemática, Universidade Federal do Rio de Janeiro（里约热内卢联邦大学数学研究所）
We study the regularity and behavior at the origin of solutions to the two-dimensional degenerate Monge-Ampere equation with homogeneous right hand side of degree alpha, alpha>-2. We show that when alpha > 0, solutions admit only two possible behaviors near the origin, radial and non-radial. We also show that the radial behavior is unstable. For alpha<0 we prove that solutions admit only the radial behavior near the origin.
In this paper we present an analytical study of a subgrid scale turbulence model of the three-dimensional magnetohydrodynamic (MHD) equations, inspired by the Navier-Stokes-alpha (also known as the viscous Camassa-Holm equations or the Lagrangian-averaged Navier-Stokes-alpha model). Specifically, we show the global well-posedness and regularity of solutions of a certain MHD-alpha model (which is a particular case of the Lagrangian averaged magnetohydrodynamic-alpha model without enhancing the dissipation for the magnetic field). We also introduce other subgrid scale turbulence models, inspired…
In this paper we consider the nonlinear beam equations accounting for rotational inertial forces. Under suitable hypotheses we prove the existence, regularity and finite dimensionality of a compact global attractor and an exponential attractor. The main purpose is to trace the behavior of solutions of the nonlinear beam equations when the effect of the rotational inertia fades away gradually. A natural question is whether there are qualitative differences would appear or not. To answer the question, we deal with the rotational inertia with a parameter alpha and consider the difference of behav…
The Neumann problem for the fractional Laplacian: optimal regularity via the Mellin transform
University of Western Australia（西澳大利亚大学）；ICREA（加泰罗尼亚高级研究院）；Universitat de Barcelona（巴塞罗那大学）；Centre de Recerca Matemàtica（数学研究中心）；University of Bonn（波恩大学）
Let L be a second order, uniformly elliptic operator, and consider the equation L u=f under the homogeneous boundary condition u=0. It is well known that f in C(Om) (Om= Omega) does not guarantee second order derivatives D^2 u in C(Om). This gap led to look for functional spaces C_*(Om), contained in C(Om), as large as possible, for which f in C_*(Om) merely guarantees the continuity of D^2 u (but nothing more, say). Hölder continuity is too restrictive to fulfill this minimal requirement since in this case D^2 u inherits the whole regularity enjoyed by f (we say that "full regularity" occurs)…
We consider a general family of regularized Navier-Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n greater than or equal to 2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier-Stokes equations, the Navier-Stokes-alpha model, the Leray-alpha model, the Modified Leray-alpha model, the Simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-alpha-like models, and certain MHD models, in addition to representi…
We investigate regularity properties of molecular one-electron densities rho near the nuclei. In particular we derive a representation rho(x)=mu(x)*(e^F(x)) with an explicit function F, only depending on the nuclear charges and the positions of the nuclei, such that mu belongs to C^{1,1}(R^3), i.e., mu has locally essentially bounded second derivatives. An example constructed using Hydrogenic eigenfunctions shows that this regularity result is sharp. For atomic eigenfunctions which are either even or odd with respect to inversion in the origin, we prove that mu is even C^{2,α}(R^3) for all alp…
We study Neumann type boundary value problems for nonlocal equations related to Lévy processes. Since these equations are nonlocal, Neumann type problems can be obtained in many ways, depending on the kind of reflection we impose on the outside jumps. To focus on the new phenomenas and ideas, we consider different models of reflection and rather general non-symmetric Lévy measures, but only simple linear equations in half-space domains. We derive the Neumann/reflection problems through a truncation procedure on the Lévy measure, and then we develop a viscosity solution theory which includes co…
We consider a stochastic perturbation of the phase field alpha-Navier-Stokes model with vesicle-fluid interaction. It consists in a system of nonlinear evolution partial differential equations modeling the fluid-structure interaction associated to the dynamics of an elastic vesicle immersed in a moving incompressible viscous fluid. This system of equations couples a phase-field equation -- for the interface between the fluid and the vesicle -- to the alpha-Navier-Stokes equation -- for the viscous fluid -- with an extra nonlinear interaction term, namely the bending energy. The stochastic pert…
