Measure-valued solution for non-Newtonian compressible isothermal monopolar fluid

The global existence of measure-valued solutions of initial boundary-value problems in bounded domains to systems of partial differential equations for viscous non-Newtonian isothermal compressible monopolar fluid and the global existence of the weak solution for multipolar fluid is proved.     

Global Solvability of One-Dimensional Axially-Symmetric Micropolar Fluid Equations

Journal of Applied and Industrial Mathematics - We prove the theorem of global existence of a weak solution to an one-dimensional initial-boundary value problem for the micropolar fluid equations...     

非凸单个守恒律初边值问题的整体弱熵解的构造

本文研究具有两段常数的初始值和常数边界值的非凸单个守恒律的初边值问题.在流函数具有一个拐点的条件下,由相应的初始值问题弱熵解的结构和Bardos-Leroux-Nedelec提出的边界熵条件,给出初边值问题整体弱熵解的一个构造方法,澄清弱熵解在边界附近的结构.与严格凸的单个守恒律初边值问题相比,非凸单个守恒律初边值问题的弱熵解中包括下列新的相互作用类型:一个接触或非接触激波碰到边界,边界弹回一个非接触激波.     

Asymptotic behaviour of the density for one-dimensional Navier-Stokes equations

The existence of a (global in time) solution to the Navier-Stokes equations for barotropic compressible fluids in a bounded interval is already known in the case of vanishing external force field. In this paper we consider these equations for time-independent forces and prove that: (i) there exists a global solution to the usual initial-boundary value problem; (ii) the density of the fluid is bounded and its infimum is greater than zero for infinite time only if the external forces and the pressure satisfy a compatibility condition (which is the same derived in [2] for the existence of a stationary solution having bounded and strictly positive density).     

Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: a local existence theorem

An initial-boundary value problem for 1-D flow of a compressible viscous heat-conducting micropolar fluid is considered; the fluid is assumed thermodynamically perfect and polytropic. The original problem is transformed into homogeneous one and studied the Faedo-Galerkin method. A local-in-time existence of generalized solution is proved.      

STABILITY OF BOUNDARY LAYER TO AN OUTFLOW PROBLEM FOR A COMPRESSIBLE NON-NEWTONIAN FLUID IN THE HALF SPACE

This paper investigates the large-time behavior of solutions to an outflow problem for a compressible non-Newtonian fluid in a half space. The main concern is to analyze the phenomena that happens when the compressible non-Newtonian fluid blows out through the boundary. Based on the existence of the stationary solution, it is proved that there exists a boundary layer(i.e., the stationary solution) to the outflow problem and the boundary layer is nonlinearly stable under small initial perturbation.     

Measure Valued Solutions for Non-Newtonian Compressible Isothermal Monopolar Fluids in a Finite Channel with Nonzero Input and Output

The global existence of measure valued solutions of initial boundary value problems in a bounded domain with nonzero input and output in a finite channel for the systems of partial differential equations for viscous non-Newtonian isothermal compressible monopolar fluids and the global existence of a weak solution for multipolar fluids is proved.     

On Solvability of Some Initial-Boundary Problems for Mathematical Models of the Motion of Nonlinearly Viscous and Viscoelastic Fluids

In the paper, the settings of initial-boundary and initial value problems arising in a number of models of movement of nonlinearly viscous or viscoelastic incompressible fluid are considered, and existence theorems for these problems are presented. In particular, the settings of initial-boundary value problems appearing in the regularized model of the movement of viscoelastic fluid with Jeffris constitutive relation are described. The theorems for the existence of weak and strong solutions for these problems in bounded domains are given. The initial value problem for a nonlinearly viscous fluid on the whole space is considered. The estimates on the right-hand side and initial conditions under which there exist local and global solutions of this problem are presented. The modification of Litvinov's model for laminar and turbulent flows with a memory is described. The existence theorem for weak solutions of initial-boundary value problem appearing in this model is given.     

TWO PHASE COMPRESSIBLE FLOW IN POROUS MEDIA

We study the mathematical model of two phase compressible flows through porous media. Under the condition that the compressibility of rock, oil, and water is small, we prove that the initial-boundary value problem of the nonlinear system of equations admits a weak solution.     

非线性势力与轴向力联合作用下梁方程的初边值问题

考虑非线性势力的作用下梁方程的初边值问题,利用Galerkin方法证明了非线性项在适当条件下,在不高于6维的空间中该初边值问题整体弱解的存在及唯一性.     

Global solutions for a one-dimensional problem in conducting fluids

A mathematical model of the motion of conducting fluids is studied in this paper. The dynamics of such fluids is described by the equations of compressible fluids coupled to the Maxwell’s equations. We prove global existence of strong solution for a one-dimensional initial-boundary value problem of this model (plane conducting flows) with general large data.     

Initial-boundary value problem of three phase flow in porous media

We study the mathematical model of three phase compressible flows through porous media. Under the condition that the rock, water and oil are incompressible, and the compressibility of gas is small, we present a finite element scheme to the initial-boundary value problem of the nonlinear system of equations, then by the convergence of the scheme we prove that the problem admits a weak solution.     

Global strong solution to the three-dimensional density-dependent incompressible magnetohydrodynamic flows

An initial-boundary value problem is considered for the density-dependent incompressible viscous magnetohydrodynamic flow in a three-dimensional bounded domain. The homogeneous Dirichlet boundary condition is prescribed on the velocity, and the perfectly conducting wall condition is prescribed on the magnetic field. For the initial density away from vacuum, the existence and uniqueness are established for the local strong solution with large initial data as well as for the global strong solution with small initial data. Furthermore, the weak-strong uniqueness of solutions is also proved, which shows that the weak solution is equal to the strong solution with certain initial data.     

Global Weak Solutions for Compressible Navier-Stokes-Vlasov-Fokker-Planck System

The one-dimensional compressible Navier-Stokes-Vlasov-Fokker-Planck system with density-dependent viscosity and drag force coefficients is investigated in the present paper. The existence, uniqueness, and regularity of global weak solution to the initial value problem for general initial data are established in spatial periodic domain. Moreover, the long time behavior of the weak solution is analyzed. It is shown that as the time grows, the distribution function of the particles converges to the global Maxwellian, and both the fluid velocity and the macroscopic velocity of the particles converge to the same speed.     

The Asymptotic Behavior and the Quasineutral Limit forthe Bipolar Euler-Poisson System withBoundary Effects and a Vacuum

In this paper, a one-dimensional bipolar Euler-Poisson system(a hydrodynamic model) from semiconductors or plasmas with boundary efects is considered. This system takes the form of Euler-Poisson with an electric field and frictional damping added to the momentum equations. The large-time behavior of uniformly bounded weak solutions to the initial-boundary value problem for the one-dimensional bipolar Euler-Poisson system is firstly presented. Next, two particle densities and the corresponding current momenta are verified to satisfy the porous medium equation and the classical Darcy’s law time asymptotically. Finally, as a by-product, the quasineutral limit of the weak solutions to the initial-boundary value problem is investigated in the sense that the bounded L∞entropy solution to the one-dimensional bipolar Euler-Poisson system converges to that of the corresponding one-dimensional compressible Euler equations with damping exponentially fast as t → +∞. As far as we know, this is the first result about the asymptotic behavior and the quasineutral limit for the one-dimensional bipolar Euler-Poisson system with boundary efects and a vacuum.     

Global regularity estimates for multidimensional equations of compressible non-Newtonian fluids

The present paper is devoted to the problem of global existence of sufficiently regular solutions to two- and three-dimensional equations of a compressible non-Newtonian fluid. In the case of the potential stress tensor, we develop a technique for deriving energy identities that do not contain derivatives of density. On the basis of these identities, in the case of sufficiently rapidly increasing potentials, we obtain an extended system ofa priori estimates for the equations mentioned above. We also study the related problem of estimating solutions to the nonlinear elliptic system generated by the stress tensor. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 360–376, September, 2000.     

On a non-isothermal,non-Newtonian lubrication problem with Tresca law: Existence and the behavior of weak solutions

We consider a problem describing the motion of an incompressible, non-isothermal, and non-Newtonian fluid in a three-dimensional thin domain. We first establish an existence result for weak solutions of this problem. Then we study the asymptotic analysis when one dimension of the fluid domain tends to zero. A specific weak Reynolds equation, the limit of Tresca fluid–solid boundary conditions, and the limit boundary conditions for the temperature are obtained. The uniqueness result for the limit problem is also proved.     

Large-time behavior of the motion of a viscous heat-conducting one-dimensional gas coupled to radiation

We study the large-time behavior of the solution of an initial-boundary value problem for the equations of 1D motions of a compressible viscous heat-conducting gas coupled to radiation through a radiative transfer equation. Assuming suitable hypotheses on the transport coefficients and adapted boundary conditions, we prove that the unique strong solution of this problem converges toward a well-determined equilibrium state at exponential rate.     

The Global Solution of the Scalar Nonconvex Conservation Law with Boundary Condition (continuation)

Using the Kruskov's method [1], we show the uniqueness for the global weak solution of the initial-boundary value problem (1.1)-(1.3) (in the class of bounded and measurable functions).