Solutions of Ginzburg-Landau equations with weight and minimizers of the renormalized energy

In this paper, it is proved that for any given d non-degenerate local minimum points of the renormalized energy of weighted Ginzburg-Landau eqautions, one can find solutions to the Ginzburg-Landau equations whose vortices tend to these d points. This provides the connections between solutions of a class of Ginzburg-Landau equations with weight and minimizers of the renormalized energy.     

Entropy and renormalized solutions to the general nonlinear elliptic equations in Musielak–Orlicz spaces

In this paper we mainly prove the existence and uniqueness of entropy solutions and the uniqueness of renormalized solutions to the general nonlinear elliptic equations in Musielak-Orlicz spaces. Moreover, we also obtain the equivalence of entropy solutions and renormalized solutions in the present conditions.     

一类强退化拟线性抛物方程的弱解

In this paper, we show the existence of the renormalized solutions and the entropy solutions of a class of strongly degenerate quasilinear parabolic equations.     

Equivalence of Entropy and Renormalized Solutions of Anisotropic Elliptic Problem in Unbounded Domains with Measure Data

We consider a class of anisotropic elliptic equations of second order with variable exponents of non-linearity where a special Radon measure is used as the right-hand side. We establish uniqueness of entropy and renormalized solutions of the Dirichlet problem in anisotropic Sobolev spaces with variable exponents of non-linearity for arbitrary domains and certain other their properties. In addition, we prove the equivalence of entropy and renormalized solutions of the problem under consideration.     

Existence of global weak solutions to 2D reduced gravity two-and-a-half layer model

We study the global existence of weak solutions to a reduced gravity two-and-a-half layer model appearing in oceanic fluid dynamics in two-dimensional torus. Based on Faedo–Galerkin method and weak convergence method, we construct the global weak solutions which are renormalized in velocity variable, where the technique of renormalized solutions was introduced by Lacroix-Violet and Vasseur (2018). Besides, we prove that the renormalized solutions are weak solutions, which satisfy the basic energy inequality and Bresch–Desjardins entropy inequality, but not the Mellet–Vasseur type inequality. In the proof, we use the reduced gravity two-and-a-half layer model with drag forces and capillary term as approximate system. It should be pointed out that only when the capillary term vanishes, we prove the existence of renormalized solution to the approximation system, which is different from Lacroix-Violet and Vasseur (2018) with the quantum potential.     

A new consistent discrete‐velocity model for the Boltzmann equation

This paper discusses the convergence of a new discrete‐velocity model to the Boltzmann equation. First the consistency of the collision integral approximation is proved. Based on this we prove the convergence of solutions for a modified model to renormalized solutions of the Boltzmann equation. In a numerical example, the solutions to the discrete problems are compared with the exact solution of the Boltzmann equation in the space‐homogeneous case. Copyright © 2002 John Wiley & Sons, Ltd.     

Existence of renormalized solutions to a nonlinear parabolic equation in L1 setting with nonstandard growth condition and gradient term

This paper is devoted to the study of a nonlinear parabolic p(x)‐Laplace equation with gradient term and L¹ data. The authors obtain the existence of renormalized solutions via strong convergence of truncation. Copyright © 2014 John Wiley & Sons, Ltd.     

Une définition des solutions renormalisées pour l'équation de Boltzmann sans troncature angulaire

We propose a notion of renormalized solutions for 3D Boltzmann equation, and without assuming Grad's angular cutoff. Actually, we show that P.-L. Lions's recent hypothesis about velocity averages compacity of solutions is satisfied in this framework.     

Solutions of the Vlasov–Maxwell–Boltzmann system with long-range interactions

We establish the existence of renormalized solutions of the Vlasov–Maxwell–Boltzmann system with a defect measure in the presence of long-range interactions. We also present a control of the defect measure by the entropy dissipation only, which turns out to be crucial in the study of hydrodynamic limits.     

Trend to equilibrium of renormalized solutions to reaction–cross-diffusion systems

The convergence to equilibrium of renormalized solutions to reaction–cross-diffusion systems in a bounded domain under no-flux boundary conditions is studied. The reactions model complex balanced chemical reaction networks coming from mass-action kinetics and thus do not obey any growth condition, while the diffusion matrix is of cross-diffusion type and hence nondiagonal and neither symmetric nor positive semi-definite, but the system admits a formal gradient-flow or entropy structure. The diffusion term generalizes the population model of Shigesada, Kawasaki and Teramoto to an arbitrary number of species. By showing that any renormalized solution satisfies the conservation of masses and a weak entropy–entropyproduction inequality, it can be proved under the assumption of no boundary equilibria that all renormalized solutions converge exponentially to the complex balanced equilibrium with a rate which is explicit up to a finite dimensional inequality.     

The incompressible Navier–Stokes limit of the Boltzmann equation for hard cutoff potentials

The present paper proves that all limit points of sequences of renormalized solutions of the Boltzmann equation in the limit of small, asymptotically equivalent Mach and Knudsen numbers are governed by Leray solutions of the Navier–Stokes equations. This convergence result holds for hard cutoff potentials in the sense of H. Grad, and therefore completes earlier results by the same authors [Invent. Math. 155 (2004) 81–161] for Maxwell molecules.     

Existence of a Renormalised Solutions for a Class of Nonlinear Degenerated Parabolic Problems with L1 Data

We study the existence of renormalized solutions for a class of nonlinear degenerated parabolic problem. The Carathéodory function satisfying the coercivity condition, the growth condition and only the large monotonicity. The data belongs to L^1(Ω).     

Renormalized solutions of some transport equations with partially <Emphasis Type="Italic">W</Emphasis><Superscript>1,1</Superscript> velocities and applications

We prove existence and uniqueness of renormalized solutions of some transport equations with a vector field that is not W^1,1 with respect to all variables but is of a particular form. Two specific applications of this new result are then treated, based upon the equivalence between transport equations and ordinary differential equations. The first one consists of a result about the dependance upon initial conditions for solutions of ODEs. The second one is related to some stochastic differential equations arising in the modelling of polymeric fluid flows.     

Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data

We consider the nonlinear heat equation (with Leray-Lions operators) on an open bounded subset of R^N with Dirichlet homogeneous boundary conditions. The initial condition is in L¹ and the right hand side is a smooth measure. We extend a previous notion of entropy solutions and prove that they coincide with the renormalized solutions.     

Large Time Behavior of Solutions to Parabolic Equations with Dirichlet Operators and Nonlinear Dependence on Measure Data

Potential Analysis - We study large time behavior of renormalized solutions of the Cauchy problem for equations of the form ?tu ? Lu + λu = f(x, u) + g(x, u) ? μ, where...     

Renormalized Energy with Vortices Pinning Effect

This paper is a continuation of the previous paper in the Journal of Partial Differential Equations [1]. We derive in this paper the renormalized energy to further determine the locations of vortices in some case for the variational problem related to the superconducting thin films having variable thickness.     

The Renormalized Two-Scale Method

We show that, by uniting the two-scale method with the counterterm method of renormalized perturbation, one obtains approximate solutions of nonlinear motions which hold for all times.     

$H<1/6$时次分数布朗运动赋权立变差的一个注记

本文中我们利用Malliavin计算的技巧研究了$H<1/6$时次分数布朗运动赋权立变差的$L^2$收敛性.     

Renormalized solutions in thermo-visco-plasticity for a Norton–Hoff type model. Part I: The truncated case

We prove existence of global in time strong solutions to the truncated thermo-visco-plasticity with an inelastic constitutive function of Norton–Hoff type. This result is a starting point to obtain renormalized solutions for the considered model without truncations. The method of our proof is based on Yosida approximation of the maximal monotone term and a passage to the limit.     

On Liouville Transport Equation with Force Field in BV loc

Abstract

We prove the existence and uniqueness of renormalized solutions of the Liouville equation for n particles with an interaction potential in BV _loc except at the origin. This implies the existence and uniqueness of a a.e. flow solution of the associated ODE.