We investigate boundary regularity of solutions of generalized Stokes equations. The problem is complemented with perfect slip boundary conditions and we assume that the nonlinear elliptic operator satisfies non-standard ϕ-growth conditions. We show the existence of second derivatives of velocity and their optimal regularity.

We consider the stabilization of Maxwell’s equations with space-time variable coefficients in a bounded region with a smooth boundary by means of linear or nonlinear Silver–Müller boundary condition. This is based on some stability estimates that are obtained using the “standard” identity with multiplier and appropriate properties of the feedback. We deduce an explicit decay rate of the energy, for instance exponential, polynomial or logarithmic decays are available for appropriate feedbacks.

We consider the stabilization of Maxwell's equations with space-time variable coefficients in a bounded region with a smooth boundary by means of linear or nonlinear Silver–Müller boundary condition. This is based on some stability estimates that are obtained using the “standard" identity with multiplier and appropriate properties of the feedback. We deduce an explicit decay rate of the energy, for instance exponential, polynomial or logarithmic decays are available for appropriate feedbacks. ...

This paper is concerned with the two-species chemotaxis-Navier–Stokes system with Lotka–Volterra competitive kinetics $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\left(1\right)}_t+u·1=𝔻1-{\chi }_1·\left(1c\right)+{\mu }_{1}1(1-1-a_{1}2)\hfill & \text{in}\times (0,\infty ),\hfill \\ {\left(2\right)}_t+u·2=𝔻2-{\chi }_2·\left(2c\right)+{\mu }_{2}2(1-a_{2}1-2)\hfill & \text{in}\times (0,\infty ),\hfill \\ c_t+u·c=𝔻c-(\alpha 1+\beta 2)c\hfill & \text{in}\times (0,\infty ),\hfill \\ u_t+(u·)u=𝔻u+\nabla P+(\gamma 1+2)\Phi ,·u=0\hfill & \text{in}\times (0,\infty )\hfill \end{array}\right.\end{array}$$ under homogeneous Neumann boundary conditions and initial conditions, where is a bounded domain in R3 with smooth boundary. Recently, in the 2-dimensional setting, global existence and stabilization of classical solutions to the above system were first established. However, the 3-dimensional case has not been studied: Because of difficulties in the Navier–Stokes system, we can...

We consider the cubic Nonlinear Schrödinger Equation (NLS) and the Korteweg-de Vries equation in one space dimension. We prove that the solutions of NLS satisfy a-priori local in time $H^s$ bounds in terms of the $H^s$ size of the initial data for $s\ge -\frac{1}{4}$ (joint work with D. Tataru, [15, 14]) , and the solutions to KdV satisfy global a priori estimate in $H^{-1}$ (joint work with T. Buckmaster [2]).

We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one-dimensional waves, and consider the case of a flat bottom. Following the method presented in [J.L. Bona, T. Colin and D. Lannes, Arch. Ration. Mech. Anal. 178 (2005) 373–410] for the one-layer case, we introduce a new family of symmetric hyperbolic models, that are equivalent to the classical Boussinesq/Boussinesq system displayed in [W. Choi...

We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one-dimensional waves, and consider the case of a flat bottom. Following the method presented in [J.L. Bona, T. Colin and D. Lannes, Arch. Ration. Mech. Anal.178 (2005) 373–410] for the one-layer case, we introduce a new family of symmetric hyperbolic models, that are equivalent to the classical Boussinesq/Boussinesq system displayed in [W. Choi...

Ω being a bounded open set in R∙, with regular boundary, we associate with Navier-Stokes equation in Ω where the velocity is null on ∂Ω, a non-linear branching process (Yt, t ≥ 0). More precisely: Eω0(〈h,Yt〉) = 〈ω,h〉, for any test function h, where ω = rot u, u denotes the velocity solution of Navier-Stokes equation. The support of the random measure Yt increases or decreases in one unit when the underlying process hits ∂Ω; this stochastic phenomenon corresponds to the creation-annihilation of vortex...

We consider superconductors of Type II near the transition from the ‘bulk superconducting’ to the ‘surface superconducting’ state. We prove a new $L^{\infty }$ estimate on the order parameter in the bulk, i.e. away from the boundary. This solves an open problem posed by Aftalion and Serfaty [AS].