Well-posedness Of Classical Solutions Research Articles

On existence of global classical solutions to the 3D compressible MHD equations with vacuum

In this paper, the existence of global classical solutions is justified for the three-dimensional compressible magnetohydrodynamic (MHD) equations with vacuum. The main goal of this paper is to obtain a unique global classical solution on mathbb{R}^{3}times [0, T] with any Tin (0, infty ), provided that the initial magnetic field in the L^{3}-norm and the initial density are suitably small. Note that the first result is obtained under the condition of rho _{0}in L^{gamma }cap W^{2, q} with qin (3, 6) and gamma in (1, 6). It should be noted that the initial total energy can be arbitrarily large, the initial density allowed to vanish, and the system does not satisfy the conservation law of mass (i.e., rho _{0} notin L^{1}). Thus, the results obtained particularly extend the one due to Li–Xu–Zhang (Li et al. in SIAM J. Math. Anal. 45:1356–1387, 2013), where the global well-posedness of classical solutions with small energy was proved.

On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum

In this paper we present a sufficient condition for the global well-posedness of classical solutions to an initial value problem of compressible isentropic Navier-Stokes equations in the whole space $\mathbb{R}^3$. As an immediate result, the main theorem obtained implies that the Cauchy problem of compressible Navier-Stokes equations with vacuum has a global unique classical solution, provided the initial energy is sufficiently small, or the shear viscosity coefficient is sufficiently large, or the upper bound of the initial density is suitably small and the adiabatic exponent $\gamma\in (1,3/2)$. These results particularly extend the recent ones due to Huang-Li-Xin [7], where the global well-posedness of classical solutions with small initial energy was established.

The well-posedness theory for Euler–Poisson fluids with non-zero heat conduction

This paper is devoted to the Euler–Poisson equations for fluids with non-zero heat conduction, arising in semiconductor science. Due to the thermal effect of the temperature equation, the local well-posedness theory by Xu and Kawashima (2014) for generally symmetric hyperbolic systems in spatially critical Besov spaces does not directly apply. To deal with this difficulty, we develop a generalized version of the Moser-type inequality by using Bony's decomposition. With a standard iteration argument, we then establish the local well-posedness of classical solutions to the Cauchy problem for intial data in spatially Besov spaces.

Global Well-posedness of Classical Solutions with Large Initial Data to the Two-dimensional Isentropic Compressible Navier-Stokes Equations

Global Well-posedness of Classical Solutions with Large Initial Data to the Two-dimensional Isentropic Compressible Navier-Stokes Equations

Local Existence of Solutions of Self Gravitating Relativistic Perfect Fluids

This paper deals with the evolution of the Einstein gravitational fields which are coupled to a perfect fluid. We consider the Einstein–Euler system in asymptotically flat spacestimes and therefore use the condition that the energy density might vanish or tend to zero at infinity, and that the pressure is a fractional power of the energy density. In this setting we prove local in time existence, uniqueness and well-posedness of classical solutions. The zero order term of our system contains an expression which might not be a C ∞ function and therefore causes an additional technical difficulty. In order to achieve our goals we use a certain type of weighted Sobolev space of fractional order. In Brauer and Karp (J Diff Eqs 251:1428–1446, 2011) we constructed an initial data set for these of systems in the same type of weighted Sobolev spaces. We obtain the same lower bound for the regularity as Hughes et al. (Arch Ratl Mech Anal 63(3):273–294, 1977) got for the vacuum Einstein equations. However, due to the presence of an equation of state with fractional power, the regularity is bounded from above.

Global Classical Solutions for Partially Dissipative Hyperbolic System of Balance Laws

This work is concerned with ($N$-component) hyperbolic system of balance laws in arbitrary space dimensions. Under entropy dissipative assumption and the Shizuta-Kawashima algebraic condition, a general theory on the well-posedness of classical solutions in the framework of Chemin-Lerner's spaces with critical regularity is established. To do this, we first explore the functional space theory and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous Chemin-Lerner's spaces. Then this fact allows to prove the local well-posedness for general data and global well-posedness for small data by using the Fourier frequency-localization argument. Finally, we apply the new existence theory to a specific fluid model-the compressible Euler equations with damping, and obtain the corresponding results in critical spaces.