Existence of solutions for quasilinear problem with Neumann boundary conditions

My abstract is:This paper is devoted to the study of the existence of weak solutionsfor quasilinear systems of a partial dierential equations which are the combinationof the Perona-Malik equation and the Heat equation. The proof of the main resultsare based on the compactness method and the motonocity arguments.

New Results About the Lambda Constant and Ground States of the 𝑊-Functional

In this paper, we study properties of the lambda constants and the existence of ground states of Perelman’s famous W-functional from a variational formulation. We have two kinds of results. One is about the estimation of the lambda constant of G. Perelman, and the other is about the existence of ground states of his W-functional, both on a complete non-compact Riemannian manifold {(M,g)}. One consequence of our estimation is that, on an ALE (or asymptotic flat) manifold {(M,g)}, if the scalar curvature s of {(M,g)} is non-negative and has quadratical decay at infinity, then M is scalar flat, i.e., {s=0} in M. We also introduce a new constant {d(M,g)}. For the existence of the ground states, we use Lions’ concentration-compactness method.

Splitting-up scheme for the stochastic Cahn–Hilliard Navier–Stokes model

In this paper, we consider a stochastic Cahn–Hilliard Navier–Stokes system in a bounded domain of [Formula: see text] [Formula: see text]. The system models the evolution of an incompressible isothermal mixture of binary fluids under the influence of stochastic external forces. We prove the existence of a global weak martingale solution. The proof is based on the splitting-up method as well as some compactness method.

Normalized multi-bump solutions for saturable Schrödinger equations

In this paper, we are concerned with the existence of multi-bump solutions for a class of semiclassical saturable Schrödinger equations with an density function: $$\begin{array}{} \displaystyle -{\it\Delta} v +{\it\Gamma} \frac{I(\varepsilon x) + v^2}{1+I(\varepsilon x) +v^2} v =\lambda v,\, x\in{{\mathbb{R}}^{2}}. \end{array}$$ We prove that, with the density function being radially symmetric, for given integer k ≥ 2 there exist a family of non-radial, k-bump type normalized solutions (i.e., with the L2 constraint) which concentrate at the global maximum points of density functions when ε → 0+. The proof is based on a variational method in particular on a convexity technique and the concentration-compactness method.

Globalwell-posedness for a viscosity problem of the compressible Heisenberg chain equations

In this paper, we are concerned with the existence and uniqueness of global smooth solutions to a viscosity problem for the compressible Heisenberg chain equations in one dimension. Furthermore, we prove the global existence of weak solutions when the parameter A tends to zero by compactness method.

EXISTENCE OF STEADY STATES FOR THE MAXWELL–SCHRÖDINGER–POISSON SYSTEM: EXPLORING THE APPLICABILITY OF THE CONCENTRATION–COMPACTNESS PRINCIPLE

This paper reviews recent results and open problems concerning the existence of steady states to the Maxwell–Schrödinger system. A combination of tools, proofs and results are presented in the framework of the concentration–compactness method.

On application of kinetic formulation of the Le Roux system

We use the compensated compactness method coupled with some basic ideas of kinetic formulation developed by Lions, Perthame, Souganidis and Tadmor to give a refined proof for the existence of global bounded entropy solutions to the Le Roux system. This new method of the reduction of Young measures can be applied to solve other problems.