GLOBAL WEAK SOLUTIONS FOR AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM WITH p-LAPLACIAN DIFFUSION AND LOGISTIC SOURCE

This paper is concerned with the following attraction-repulsion chemotaxis system with p-Laplacian diffusion and logistic source:■The system here is under a homogenous Neumann boundary condition in a bounded domainΩ ■ R^(n)(n≥2),with χ,ξ,α,β,γ,δ,k_(1),k_(2)> 0,p> 2.In addition,the function f is smooth and satisfies that f(s)≤κ-μs~l for all s≥0,with κ ∈ R,μ> 0,l> 1.It is shown that(ⅰ)if l> max{2k_(1),(2k_(1)n)/(2+n)+1/(p-1)},then system possesses a global bounded weak solution and(ⅱ)if k_(2)> max{2k_(1)-1,(2k_(1)n)/(2+n)+(2-p)/(p-1)} with l> 2,then system possesses a global bounded weak solution.

GLOBAL WEAK SOLUTIONS TO ONE-DIMENSIONAL COMPRESSIBLE VISCOUS HYDRODYNAMIC EQUATIONS

In this article, we are concerned with the global weak solutions to the 1D com- pressible viscous hydrodynamic equations with dispersion correction δ2ρ（（φ（ρ））xxφ′（ρ））x with φ（ρ） = ρα. The model consists of viscous stabilizations because of quantum Fokker-Planck operator in the Wigner equation and is supplemented with periodic boundary and initial con- ditions. The diffusion term εuxx in the momentum equation may be interpreted as a classical conservative friction term because of particle interactions. We extend the existence result in [1] （α=1/2） to 0 〈 α ≤ 1. In addition, we perform the limit ε→0 with respect to 0 〈 α ≤1/2.

GLOBAL WEAK SOLUTIONS TO THE α-MODEL REGULARIZATION FOR 3D COMPRESSIBLE EULER-POISSON EQUATIONS

Global in time weak solutions to the α-model regularization for the three dimensional Euler-Poisson equations are considered in this paper. We prove the existence of global weak solutions to α-model regularization for the three dimension compressible EulerPoisson equations by using the Fadeo-Galerkin method and the compactness arguments on the condition that the adiabatic constant satisfies γ >4/3.

GLOBAL WEAK SOLUTIONS FOR A NONLINEAR HYPERBOLIC SYSTEM

In this paper, we study the global existence of weak solutions for the Cauchy problem of the nonlinear hyperbolic system of three equations(1.1) with bounded initial data(1.2). When we fix the third variable s, the system about the variables ρ and u is the classical isentropic gas dynamics in Eulerian coordinates with the pressure function P(ρ, s) = ese-1/ρ,which, in general, does not form a bounded invariant region. We introduce a variant of the viscosity argument, and construct the approximate solutions of(1.1) and(1.2) by adding the artificial viscosity to the Riemann invariants system(2.1). When the amplitude of the first two Riemann invariants(w1(x, 0), w2(x, 0)) of system(1.1) is small,(w1(x, 0), w2(x, 0)) are nondecreasing and the third Riemann invariant s(x, 0) is of the bounded total variation, we obtained the necessary estimates and the pointwise convergence of the viscosity solutions by the compensated compactness theory. This is an extension of the results in [1].

GLOBAL WEAK SOLUTIONS TO A GENERALIZED BENJAMIN-BONA-MAHONY-BURGERS EQUATION

The existence of global weak solutions for a generalized Benjamin-Bona-MahonyBurgers equation is established in the space C（[0, ∞） × R） ∩ L~∞（[0, ∞）; H1（R）） under the condition that its initial value belongs to the space H1（R）. A one-sided super bound estimate and a space-time higher-norm estimate on the first order derivatives of the solution with respect to the space variable are derived to prove the existence.

Global Weak Solutions of Initial Boundary Value Problem for Boltzmann-Poisson System with Absorbing Boundary

This paper deals with the initial boundary value problem for the Boltzmann-Poisson system, which arises in semiconductor physics, with absorbing boundary. The global existence of weak solutions is proved by using the stability of velocity averages and the compactness results on L1-theory under weaker conditons on initial boundary values.

Global Weak Solutions to a Fluid-particle System of an Incompressible Non-Newtonian Fluid and the Vlasov Equation

The purpose of this work is to investigate the existence and uniqueness of weak solutions to the initial-boundary value problem for a coupled system of an incompressible non-Newtonian fluid and the Vlasov equation.The coupling arises from the acceleration in the Vlasov equation and the drag force in the incompressible viscous non-Newtonian fluid with the stress tensor of a power-law structure for p≥11/5.The main idea of the existence analysis is to reformulate the coupled system by means of a so-called truncation function.The advantage of the new formulation is to control the external force term G=-∫Rd(u-v)fdc(d=2,3).The global existence of weak solutions to the reformulated system is shown by using the Faedo-Galerkin method and weak compactness techniques.We further prove the uniqueness of weak solutions to the considered system.

GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE NON-ISOTHERMAL NEMATIC LIQUID CRYSTALS IN 2D

In this article, we prove the global existence of weak solutions to the non- isothermal nematic liquid crystal system on T2, on the basis of a new approximate system which is different from the classical Ginzburg-Landau approximation. Local in space energy inequalities are employed to recover the estimates on the second order spatial derivatives of the director fields locally in time, which cannot be derived from the basic energy balance. It is shown that these weak solutions satisfy the temperature equation, and also the total energy equation but away from at most finite many ＂singular＂ times, at which the energy concentration occurs and the director field losses its second order derivatives.

GLOBAL WEAK SOLUTION TO COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY,VACUUM AND GRAVITATIONAL FORCE

Recently,the global existence of weak solutions to the compressible Navier-Stokes equations with vacuum has attracted much attention.In this paper,we study the one-dimension isentropic Navier-Stokes equations with gravitational force and fixed boundary condition when the density connects with vacuum discontinuously.We prove the global existence and the uniqueness of weak solution,requiring less regularity of the initial data.

Global Weak Solutions for the Weakly Dissipative Camassa-Holm Equation

In this paper we prove the existence and uniqueness of global weak solutions to the weakly dissipative Camassa-Holm equation.

NEW PROOFS OF THE DECAY ESTIMATE WITH SHARP RATE OF THE GLOBAL WEAK SOLUTION OF THE n-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Consider the Cauchy problem for the n-dimensional incompressible NavierStokes equations:??tu-α△u+(u·?)u+?p = f(x, t), with the initial condition u(x, 0) = u_0(x) and with the incompressible conditions ? · u = 0, ? · f = 0 and ? · u_0= 0. The spatial dimension n ≥ 2.Suppose that the initial function u_0∈ L1(Rn) ∩ L^2(Rn) and the external force f ∈ L^1(Rn× R+) ∩ L^1(R+, L^2(Rn)). It is well known that there holds the decay estimate with sharp rate:(1 + t)1+n/2∫Rn|u(x, t)|2 dx ≤ C, for all time t > 0, where the dimension n ≥ 2, C > 0 is a positive constant, independent of u and(x, t).The main purpose of this paper is to provide two independent proofs of the decay estimate with sharp rate, both are complete, systematic, simplified proofs, under a weaker condition on the external force. The ideas and methods introduced in this paper may have strong influence on the decay estimates with sharp rates of the global weak solutions or the global smooth solutions of similar equations, such as the n-dimensional magnetohydrodynamics equations, where the dimension n ≥ 2.

GLOBAL WEAK SOLUTION TO THE NONLINEAR SCHRDINGER EQUATIONS WITH DERIVATIVE

In this paper, we study the nonlinear Schr¨odinger equations with derivative. By using the Gal¨erkin method and a priori estimates, we obtain the global existence of the weak solution.

GLOBAL WEAK SOLUTIONS OF THE RELATIVISTIC VLASOV-KLEIN-GORDON SYSTEM IN TWO DIMENSIONS

In this paper we consider the system of classical particles coupled with a Klein-Gordon field in two dimensions. We establish a-priori-bounds on the solutions of this system with initial data satisfying a size restriction derived from conservation of energy. This result, together with the smoothing of ＂velocity averaging＂, yields the existence of global weak solutions to the corresponding restricted initial value problem. The size restriction is necessary since energy of the system is indefinite. Finally, we show that the weak solutions preserve the total mass.

ON THE EXISTENCE OF ALMOST GLOBAL WEAK SOLUTION TO MULTIDIMENSIONAL VLASOV-POISSON EQUATION

The authors prove the existence of almost global weak solution to multidimensional Vlasov Poisson equation with a class of Randon measure as initial data.

GLOBAL EXISTENCE OF WEAKLY DISCONTINUOUS SOLUTIONS TO A KIND OF MIXED INITIAL-BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS

In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {（t, x） ｜ t ≥ 0, x ≥0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.

EXISTENCE AND UNIQUENESS OF THE WEAK SOLUTION TO THE INCOMPRESSIBLE NAVIER-STOKES-LANDAU-LIFSHITZ MODEL IN 2-DIMENSION

In this paper, we prove the existence and uniqueness of the weak solution to the incompressible Navier-Stokes-Landau-Lifshitz equations in two-dimension with finite energy.The main techniques is the Faedo-Galerkin approximation and weak compactness theory.

Global existence of weak solutions to the drift-flux system for general pressure laws

The initial value problem of the multi-dimensional drift-flux model for two-phase flow is investigated in this paper,and the global existence of weak solutions with finite energy is established for general pressure-density functions without the monotonicity assumption.

Global Weak Entropy Solution of Nonlinear Ideal Reaction Chromatography System and Applications

The ideal reaction chromatography model can be regarded as a semi-coupled system of two hyperbolic partial differential equations, in which, one is a self-closed nonlinear equation for the reactant concentration and another is a linear equation coupling the reactant concentration for the resultant concentration. This paper is concerned with the initial-boundary value problem for the above model. By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial initial-boundary value problem for Riemann type of initial-boundary data. Moreover, as examples, we apply the obtained results to the cases of head-on and wide pulse injections and give the expression of the global weak entropy solution.

具有非线性边界条件的不可压MHD-Boussinesq方程组初边值问题的整体适定性

The global well-posedness of another class of initial-boundary value problem on two/three-dimensional incompressible MHD-Boussinesq equations in the bounded domain with the smooth boundary is studied. The existence of a class of global weak solution to the initial boundary value problem for two/three-dimensional incompressible MHD-Boussinesq equation with the given pressure-velocity’s relation boundary condition for the fluid field,one generalized perfectly conducting boundary condition for the magnetic field and one density/temperature-velocity’s relation boundary condition for the density/temapture at the boundary is obtained, and the global existence and uniqueness of the smooth solution to the corresponding problem in two-dimensional case for the smooth initial data is also proven.

GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE THREE-DIMENSIONAL FULL COMPRESSIBLE QUANTUM EQUATIONS

We consider the quantum Navier-Stokes equations for the viscous, compressible, heat conducting fluids on the three-dimensional torus T3. The model is based on a system which is derived by Jungel, Matthes and Milisic [15]. We made some adjustment about the relation of the viscosities of quantum terms. The viscosities and the heat conductivity coefficient are allowed to depend on the density, and may vanish on the vacuum. By several levels of approxima- tion we prove the global-in-time existence of weak solutions for the large initial data.