We investigate the zero dissipation limit problem of the one-dimensional compressible isentropic Navier-Stokes equations with Riemann initial data in the case of the composite wave of two shock waves. It is shown that the unique solution to the Navier-Stokes equations exists for all time, and converges to the Riemann solution to the corresponding Euler equations with the same Riemann initial data uniformly on the set away from the shocks, as the viscosity vanishes. In contrast to previous related works, where either the composite wave is absent or the effects of initial layers are ignored, this gives the first mathematical justification of this limit for the compressible isentropic Navier-Stokes equations in the presence of both composite wave and initial layers. Our method of proof consists of a scaling argument, the construction of the approximate solution and delicate energy estimates.
We consider the partial regularity for weak solutions to superquadratic elliptic systems with controllable growth condition, under the assumption of Dini continuous coefficients. The proof relies upon an iteration scheme of a decay estimate for a new type of excess functional. To establish the decay estimate, we use the technique of A-harmonic approximation and obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particular, the proof yields directly the optimal H¨older exponent for the derivative of the weak solutions on the regular set.
In this article, we are concerned with the strong solutions of the coupled Navier-Stokes-Poisson equations for isentropic compressible fluids in a domain Ω R^3. We prove the local existence of unique strong solutions provided that the initial data u0 and u0 satisfy a nature compatibility condition. The important point in this article is that we allow the initial vacuum： the initial density may vanish in an open subset of Ω. This is achieved by getting some uniform estimates and using a Schauder fixed point theorem.
In this paper, we are concerned with the existence and non-existence of global solutions of a semi-linear heat equation with fractional Laplacian. We obtain some extem sion of results of Weissler who considered the case α = 1, and h ≡ 1.
In this paper, firstly, we study the local existence and uniqueness of mild solutions for fractional evolution systems with nonlocal in time nonlinearity. Then, we claim that such a mild solution is weak solution of this system. Finally, we prove a blow-up result under some conditions.
In this paper we derive LPS＇s criterion for the breakdown of classical solutions to the incompressible nematic liquid crystal flow, a simplified version of Ericksen-Leslie system modeling the hydrodynamic evolution of nematic liquid crystals in R^3. We show that if 0 〈 T 〈 ＋∞ is the maximal time interval for the unique smooth solution u ∈ C^∞（[0, T）,R^3）,then ｜u｜＋｜△d｜∈L^q（[0,T],L^p（R^3））,where p and q satisfy the Ladyzhenskaya-Prodi-Serrin＇s condition：3/p＋2/q=1 and p∈（3,＋∞].
In this paper, we are concerned with the asymptotic behaviour of a weak solution to the Navier-Stokes equations for compressible barotropic flow in two space dimensions with the pressure function satisfying p（g） = aglog^d（g） for large g. Here d 〉 2, a 〉 0. We introduce useful tools from the theory of Orlicz spaces and construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is also briefly discussed.
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.
