Logarithmic Sobolev inequalities for some nonlinear PDE’s
نویسنده
چکیده
The aim of this paper is to study the behavior of solutions of some nonlinear partial differential equations of Mac Kean-Vlasov type. The main tools used are, on one hand, the logarithmic Sobolev inequality and its connections with the concentration of measure and the transportation inequality with quadratic cost; on the other hand, the propagation of chaos for particle systems in mean field interaction.
منابع مشابه
Best Constants for Gagliardo-nirenberg Inequalities and Applications to Nonlinear Diiusions ?
In this paper, we nd optimal constants of a special class of Gagliardo-Nirenberg type inequalities which turns out to interpolate between the classical Sobolev inequality and the Gross logarithmic Sobolev inequality. These inequalities provide an optimal decay rate (measured by entropy methods) of the intermediate asymptotics of solutions to nonlinear diiusion equations.
متن کاملBest constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions
In this paper, we find optimal constants of a special class of Gagliardo-Nirenberg type inequalities which turns out to interpolate between the classical Sobolev inequality and the Gross logarithmic Sobolev inequality. These inequalities provide an optimal decay rate (measured by entropy methods) of the intermediate asymptotics of solutions to nonlinear diffusion equations.
متن کاملLogarithmic Sobolev inequalities and Nash-type inequalities for sub-markovian symmetric semigroups
1 We study relationships between Logarithmic Sobolev inequalities with one parameter of Davies-Simon type, energy-entropy inequality, Nash-type inequality and Sobolev-type inequalities. The inequalities of Sobolev-type apply in the general setting of symmetric sub-Markovian semigroups (and some generalizations). We provide several examples of application of theses results for ultracontractive s...
متن کاملOn a class of stochastic semilinear PDE’s
We consider stochastic semilinear partial differential equations with Lipschitz nonlinear terms. We prove existence and uniqueness of an invariant measure and the existence of a solution for the corresponding Kolmogorov equation in the space L(H ; ν), where ν is the invariant measure. We also prove the closability of the derivative operator and an integration by parts formula. Finally, under bo...
متن کاملModi!ed logarithmic Sobolev inequalities for some models of random walk!
Logarithmic Sobolev inequalities are a well-studied technique for estimating rates of convergence of Markov chains to their stationary distributions. In contrast to continuous state spaces, discrete settings admit several distinct log Sobolev inequalities, one of which is the subject of this paper. Here we derive modi!ed log Sobolev inequalities for some models of random walk, including the ran...
متن کامل