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Inégalités de résolvante pour l’opérateur de Schrödinger avec potentiel multipolaire critique

Thomas Duyckaerts (2006)

Bulletin de la Société Mathématique de France

On étudie un opérateur de la forme - Δ + V sur ℝ d , où V est un potentiel admettant plusieurs pôles en a / r 2 . Plus précisément, on démontre l’estimation de résolvante tronquée à hautes fréquences, classique dans les cas non-captifs, et qui implique l’effet régularisant standard pour l’équation de Schrödinger correspondante. La preuve est basée sur l’introduction d’une mesure de défaut micro-locale semi-classique. On démontre également, dans le même contexte, des inégalités de Strichartz pour l’équation de Schrödinger....

Inhomogeneous parabolic Neumann problems

Robin Nittka (2014)

Czechoslovak Mathematical Journal

Second order parabolic equations on Lipschitz domains subject to inhomogeneous Neumann (or, more generally, Robin) boundary conditions are studied. Existence and uniqueness of weak solutions and their continuity up to the boundary of the parabolic cylinder are proved using methods from the theory of integrated semigroups, showing in particular the well-posedness of the abstract Cauchy problem in spaces of continuous functions. Under natural assumptions on the coefficients and the inhomogeneity the...

Initial boundary value problem for generalized Zakharov equations

Shujun You, Boling Guo, Xiaoqi Ning (2012)

Applications of Mathematics

This paper considers the existence and uniqueness of the solution to the initial boundary value problem for a class of generalized Zakharov equations in ( 2 + 1 ) dimensions, and proves the global existence of the solution to the problem by a priori integral estimates and the Galerkin method.

Initial traces of solutions to a one-phase Stefan problem in an infinite strip.

Daniele Andreucci, Marianne K. Korten (1993)

Revista Matemática Iberoamericana

The main result of this paper is an integral estimate valid for non-negative solutions (with no reference to initial data) u ∈ L1loc (Rn x (0,T)) to(0.1)   ut - Δ(u - 1)+ = 0,  in D'(Rn x (0,T)),for T > 0, n ≥ 1. Equation (0.1) is a formulation of a one-phase Stefan problem: in this connection u is the enthalpy, (u - 1)+ the temperature, and u = 1 the critical temperature of change of phase. Our estimate may be written in the form(0.2)  ∫Rn u(x,t) e-|x|2 / (2 (T - t)) dx ≤ C,   0 <...

Integral inequalities and summability of solutions of some differential problems

Lucio Boccardo (2000)

Banach Center Publications

The aim of this note is to indicate how inequalities concerning the integral of | ∇ u | 2 on the subsets where |u(x)| is greater than k ( k ∈ I R + ) can be used in order to prove summability properties of u (joint work with Daniela Giachetti). This method was introduced by Ennio De Giorgi and Guido Stampacchia for the study of the regularity of the solutions of Dirichlet problems. In some joint works with Thierry Gallouet, inequalities concerning the integral of | ∇ u | 2 on the subsets where |u(x)| is less than k ( k ∈ I R + ) or...

Interior regularity of weak solutions to the equations of a stationary motion of a non-Newtonian fluid with shear-dependent viscosity. The case q = 3 d d + 2

Jörg Wolf (2007)

Commentationes Mathematicae Universitatis Carolinae

In this paper we consider weak solutions 𝐮 : Ω → ℝ d to the equations of stationary motion of a fluid with shear dependent viscosity in a bounded domain Ω ⊂ ℝ d ( d = 2 or d = 3 ). For the critical case q = 3 d d + 2 we prove the higher integrability of ∇ 𝐮 which forms the basis for applying the method of differences in order to get fractional differentiability of ∇ 𝐮 . From this we show the existence of second order weak derivatives of u .

Interior regularity of weak solutions to the perturbed Navier-Stokes equations

Pigong Han (2012)

Applications of Mathematics

In this paper we establish interior regularity for weak solutions and partial regularity for suitable weak solutions of the perturbed Navier-Stokes system, which can be regarded as generalizations of the results in L. Caffarelli, R. Kohn, L. Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure. Appl. Math. 35 (1982), 771–831, and S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscr. Math. 69...

Invariant measures and long-time behavior for the Benjamin-Ono equation

Yu Deng, Nikolay Tzvetkov, Nicola Visciglia (2014)

Journées Équations aux dérivées partielles

We summarize the main ideas in a series of papers ([20], [21], [22], [5]) devoted to the construction of invariant measures and to the long-time behavior of solutions of the periodic Benjamin-Ono equation.

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