Global well‐posedness for the compressible magnetohydrodynamic system in the critical Lp framework

The present work is dedicated to the well‐posedness issue of strong solutions (away from vacuum) to the compressible viscous magnetohydrodynamic (MHD) system in (d ≥ 2). We aim at extending those results in previous studies to more general L^p critical framework. Precisely, by recasting the whole system in Lagrangian coordinates, we prove the local existence and uniqueness of solutions by means of Banach fixed‐point theorem. Furthermore, with the aid of effective velocity, we employ the energy argument to establish global a priori estimates, which lead to the unique global solution near constant equilibrium. Our results hold in case of small data but large highly oscillating initial velocity and magnetic field.     

A Lagrangian Approach for the Incompressible Navier‐Stokes Equations with Variable Density

We investigate the Cauchy problem for the inhomogeneous Navier‐Stokes equations in the whole n‐dimensional space. Under some smallness assumption on the data, we show the existence of global‐in‐time unique solutions in a critical functional framework. The initial density is required to belong to the multiplier space of \input amssym $\dot {B}^{n/p‐1}_{p,1}({\Bbb R}^n)$ . In particular, piecewise‐constant initial densities are admissible data provided the jump at the interface is small enough and generate global unique solutions with piecewise constant densities. Using Lagrangian coordinates is the key to our results, as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence. © 2012 Wiley Periodicals, Inc.     

Global weak solutions for a periodic two‐component μ‐Camassa–Holm system

In this paper, we consider the global existence of weak solutions for a two‐component μ‐Camassa–Holm system in the periodic setting. Global existence for strong solutions to the system with smooth approximate initial value is derived. Then, we show that the limit of approximate solutions is a global‐in‐time weak solution of the two‐component μ‐Camassa–Holm system. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Caginalp system with dynamic boundary conditions and singular potentials

This article is devoted to the study of the Caginalp phase field system with dynamic boundary conditions and singular potentials. We first show that, for initial data in H ², the solutions are strictly separated from the singularities of the potential. This turns out to be our main argument in the proof of the existence and uniqueness of solutions. We then prove the existence of global attractors. In the last part of the article, we adapt well-known results concerning the Lojasiewicz inequality in order to prove the convergence of solutions to steady states.      

Global axisymmetric classical solutions of full compressible magnetohydrodynamic equations with vacuum free boundary and large initial data

In this paper,the global existence of the classical solution to the vacuum free boundary problem of full compressible magnetohydrodynamic equations with large initial data and axial symmetry is studied.The solutions to the system(1.6)–(1.8) are in the class of radius-dependent solutions,i.e.,independent of the axial variable and the angular variable.In particular,the expanding rate of the moving boundary is obtained.The main difficulty of this problem lies in the strong coupling of the magnetic field,velocity,temperature and the degenerate density near the free boundary.We overcome the obstacle by establishing the lower bound of the temperature by using different Lagrangian coordinates,and deriving the uniform-in-time upper and lower bounds of the Lagrangian deformation variable r;by weighted estimates,and also the uniform-in-time weighted estimates of the higher-order derivatives of solutions by delicate analysis.     

Global classical solution to the 3D isentropic compressible Navier–Stokes equations with general initial data and a density‐dependent viscosity coefficient

In this paper, we study the global existence of classical solutions to the three‐dimensional compressible Navier–Stokes equations with a density‐dependent viscosity coefficient (λ = λ(ρ)). For the general initial data, which could be either vacuum or non‐vacuum, we prove the global existence of classical solutions, under the assumption that the viscosity coefficient μ is large enough. Copyright © 2014 John Wiley & Sons, Ltd.     

Entropy solutions for linearly degenerate hyperbolic systems of rich type

Consider a linearly degenerate hyperbolic system of rich type. Assuming that each eigenvalue of the system has a constant multiplicity, we construct a representation formula of entropy solutions in L^∞ to the Cauchy problem. This formula depends on the solution of an autonomous system of ordinary differential equations taking x as parameter. We prove that for smooth initial data, the Cauchy problem for such an autonomous system admits a unique global solution. By using this formula together with classical compactness arguments, we give a very simple proof on the global existence of entropy solutions. Moreover, in a particular case of the system, we obtain an another explicit expression and the uniqueness of the entropy solution. Applications include the one-dimensional Born–Infeld system and linear Lagrangian systems.     

Global analysis for strong solutions to the equations of a ferrofluid flow model

In this paper we are concerned with the differential system proposed by Shliomis to describe the motion of an incompressible ferrofluid submitted to an external magnetic field. The system consists of the Navier-Stokes equations, the magnetization equations and the magnetostatic equations. No regularizing term is added to the magnetization equations. We prove the local existence of unique strong solution for the Cauchy problem and establish a finite time blow-up criterion of strong solutions. Under the smallness assumption of the initial data and the external magnetic field, we prove the global existence of strong solutions and derive a decay rate of such small solutions in L²-norm.     

Global strong solutions for a class of compressible non‐Newtonian fluids with vacuum

We study a class of compressible non‐Newtonian fluids in one space dimension. We prove, by using iterative method, the global time existence and uniqueness of strong solutions provided that the initial data satisfy a compatibility condition and the initial density is small in its H¹‐norm. The main difficulty is due to the strong nonlinearity of the system and the initial vacuum. Copyright © 2010 John Wiley & Sons, Ltd.     

Existence and continuous dependence of large solutions for the magnetohydrodynamic equations

Global solutions of the nonlinear magnetohydrodynamic (MHD) equations with general large initial data are investigated. First the existence and uniqueness of global solutions are established with large initial data in H ¹. It is shown that neither shock waves nor vacuum and concentration are developed in a finite time, although there is a complex interaction between the hydrodynamic and magnetodynamic effects. Then the continuous dependence of solutions upon the initial data is proved. The equivalence between the well-posedness problems of the system in Euler and Lagrangian coordinates is also showed.     

Global Existence and Convergence Rates of Smooth Solutions for the 3-D Compressible Magnetohydrodynamic Equations Without Heat Conductivity

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations without heat conductivity, which is a hyperbolic-parabolic system. The global solutions are obtained by combining the local existence and a priori estimates if H³-norm of the initial perturbation around a constant states is small enough and its L¹-norm is bounded. A priori decay-in-time estimates on the pressure, velocity and magnetic field are used to get the uniform bound of entropy. Moreover, the optimal convergence rates are also obtained.     

On the regularization mechanism for the periodic Korteweg–de Vries equation

In this paper we develop and use successive averaging methods for explaining the regularization mechanism in the the periodic Korteweg–de Vries (KdV) equation in the homogeneous Sobolev spaces Ḣ^s for s ≥ 0. Specifically, we prove the global existence, uniqueness, and Lipschitz‐continuous dependence on the initial data of the solutions of the periodic KdV. For the case where the initial data is in L₂ we also show the Lipschitz‐continuous dependence of these solutions with respect to the initial data as maps from Ḣ^s to Ḣ^s for s ∈(−1,0]. © 2010 Wiley Periodicals, Inc.     

On a McKean‐Vlasov Stochastic Integro‐differential Evolution Equation of Sobolev‐Type

Abstract

We establish the global existence and uniqueness of mild solutions for a class of first‐order abstract stochastic Sobolev‐type integro‐differential equations in a real separable Hilbert space in which we allow the nonlinearities at a given time t to depend not only on the state of the solution at time, t, but also on the corresponding probability distribution at time t. Results concerning the continuous dependence of solutions on the initial data and almost sure exponential stability, as well as an extension of the existence result to the case in which the classical initial condition is replaced by a so‐called nonlocal initial condition, are also discussed. Finally, an example is provided to illustrate the applicability of the general theory.     

On the global weak solutions for a modified two‐component Camassa‐Holm equation

In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})$ and some a priori estimates on the first‐order derivatives of approximation solutions.     

Nonlinear Stability of Expanding Star Solutions of the Radially Symmetric Mass‐Critical Euler‐Poisson System

We prove nonlinear stability of compactly supported expanding star solutions of the mass‐critical gravitational Euler‐Poisson system. These special solutions were discovered by Goldreich and Weber in 1980. The expansion rate of such solutions can be either self‐similar or non‐self‐similar (linear), and we treat both types. An important outcome of our stability results is the existence of a new class of global‐in‐time radially symmetric solutions, which are not homologous and therefore not encompassed by the existing works. Using Lagrangian coordinates we reformulate the associated free‐boundary problem as a degenerate quasilinear wave equation on a compact spatial domain. The problem is mass‐critical with respect to an invariant rescaling and the analysis is carried out in similarity variables. © 2017 Wiley Periodicals, Inc.     

On the cauchy problem for a one‐dimensional full compressible non‐Newtonian fluids

This paper is concerned with global existence and asymptotic behavior of H¹ solutions to the Cauchy problem of one‐dimensional full non‐Newtonian fluids with the weighted small initial data. We then obtain the global existence of Hⁱ(i = 2,4) solutions and their asymptotic behavior for the system. Copyright © 2015 John Wiley & Sons, Ltd.     

Global regularity for a two‐dimensional nonlinear Boussinesq system

In this paper, we first utilize the vanishing diffusivity method to prove the existence of global quasi‐strong solutions and get some higher order estimates, and then prove the global well‐posedness of the two‐dimensional Boussinesq system with variable viscosity for H³ initial data. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and uniqueness of bounded weak solution for non‐linear parabolic boundary value problem with equivalued surface

In this paper, we discuss the existence and uniqueness of bounded weak solution for non‐linear parabolic boundary value problem with equivalued surface and correct the mistake in Zhang Xu (Math. Meth. Appl. Sci. 1999; 22 : 259). The approach is based on L^∞ estimate of solution. Copyright © 2004 John Wiley & Sons, Ltd.     

Self-Gravitating Klein–Gordon Fields in Asymptotically Anti-de-Sitter Spacetimes

We initiate the study of the spherically symmetric Einstein–Klein–Gordon system in the presence of a negative cosmological constant, a model appearing frequently in the context of high-energy physics. Due to the lack of global hyperbolicity of the solutions, the natural formulation of dynamics is that of an initial boundary value problem, with boundary conditions imposed at null infinity. We prove a local well-posedness statement for this system, with the time of existence of the solutions depending only on an invariant H ²-type norm measuring the size of the Klein–Gordon field on the initial data. The proof requires the introduction of a renormalized system of equations and relies crucially on r-weighted estimates for the wave equation on asymptotically AdS spacetimes. The results provide the basis for our companion paper establishing the global asymptotic stability of Schwarzschild-Anti-de-Sitter within this system.     

Large time behavior for the non‐isentropic Navier–Stokes–Maxwell system

In this paper, we are concerned with the system of the non‐isentropic compressible Navier–Stokes equations coupled with the Maxwell equations through the Lorentz force in three space dimensions. The global existence of solutions near constant steady states is established, and the time‐decay rates of perturbed solutions are obtained. The proof for existence is due to the classical energy method, and the investigation of large‐time behavior is based on the linearized analysis of the non‐isentropic Navier–Stokes–Poisson equations and the electromagnetic part for the linearized isentropic Navier–Stokes–Maxwell equations. In the meantime, the time‐decay rates obtained by Zhang, Li, and Zhu [J. Differential Equations, 250(2011), 866‐891] for the linearized non‐isentropic Navier–Stokes–Poisson equations are improved. Copyright © 2016 John Wiley & Sons, Ltd.