Global smooth solutions and singularity formation for the relativistic Euler equations with radial symmetry

In this paper, we mainly study the initial value problem of a two-component Hunter-Saxton system. By the method of characteristics, we first show that the system has global smooth solutions and blowing up smooth solutions. By using the obtained a priori estimates on smooth solutions, we then prove the existence of global weak solutions to the system.

The partial null conditions and global smooth solutions of the nonlinear wave equations on [formula omitted] with d = 2,3

Global smooth solution of multi-dimensional non-homogeneous conservation laws *

We consider Euler equations for a perfect gas in R, where d ≥ 1. We state that global smooth solutions exist under the hypotheses (H1)-(H3) on the initial data. We choose a small smooth initial density, and a smooth enough initial velocity which forces particles to spread out. We also show a result of global in time uniqueness for these global solutions. Introduction. We consider Euler equations for a perfect gas:  ∂tρ+ div(ρu) = 0, ρ(∂tu+ (u · ∇)u) +∇p = 0, ∂tS + u · ∇S = 0, (1) where t ∈ R+, x ∈ R and u : R × R+ → R stands for the velocity, ρ : R×R+ → R+ for the density, p = (γ−1)ρe for the pressure, with e the internal energy of the gas and S : R×R+ → R+ for the entropy. The adiabatic constant of the gas is denoted by γ > 1 and d ≥ 1 is the dimension of the space. We are interested in the existence of global smooth solutions to the Cauchy problem for (1) with (ρ0, u0, S0) as initial data. There exist few results concerning this problem, especially when d is strictly larger than one. The choice of initial data is decisive for this problem and it depends on whether one wants to prove or to disprove global existence. We aim at finding conditions on (ρ0, u0, S0) as weak as possible which ensure the existence of a global smooth solution. In [4], T. Sideris has shown a result of non global existence: the initial density is close to a constant at infinity—the constant should be different from 0—and some global quantities have to be large. For d = 1, in the isentropic case, we have a 2 × 2 system. In this case, some results can be proved using P. D. Lax’s works [3]. In the same case with less restrictive conditions, J. Y. Chemin [2] has also proved a result of non global existence: the initial velocity has to be smaller than the initial sound speed in each point—this quantity depends mostly on ρ0—. In [1], 1397

This paper is concerned with the large time behavior of global smooth solutions to the Cauchy problem of the $p-$system with relaxation. Former results in this direction indicate that such a problem possesses a global smooth solution provided that the first derivative of the solutions with respect to the space variable $x$ are sufficiently small. Under the same small assumption on the global smooth solution, we show that it converges to the corresponding nonlinear rarefaction wave and in our analysis, we do not ask the rarefaction wave to be weak and the initial error can also be chosen arbitrarily large.

In this paper, we are concerned with the global smooth solution problem for 3-D compressible isentropic Euler equations with vanishing density in an infinitely expanding ball. It is well-known that the classical solution of compressible Euler equations generally forms the shock as well as blows up in finite time due to the compression of gases. However, for the rarefactive gases, it is expected that the compressible Euler equations will admit global smooth solutions. We now focus on the movement of compressible gases in an infinitely expanding ball. Because of the conservation of mass, the fluid in the expanding ball becomes rarefied meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. We will confirm this interesting phenomenon from the mathematical point of view. Through constructing some anisotropy weighted Sobolev spaces, and by carrying out the new observations and involved analysis on the radial speed and angular speeds together with the divergence and rotations of velocity, the uniform weighted estimates on sound speed and velocity are established. From this, the pointwise time-decay estimate of sound speed is obtained, and the smooth gas fluids without vacuum are shown to exist globally.

Global smooth solutions for the one-dimensional spin-polarized transport equation

In this paper, we investigate the large‐time decay and stability to any given global smooth solutions of the 3‐D incompressible inhomogeneous Navier‐Stokes equations. In particular, we prove that given any global smooth solution (a,u) of (1.2), the velocity fieldudecays to 0 with an explicit rate, which coincides with theL2norm decay for the weak solutions of the 3‐D classical Navier‐Stokes system [26,29] astgoes to ∞. Moreover, a small perturbation to the initial data of (a,u) still generates a unique global smooth solution to (1.2), and this solution keeps close to the reference solution (a,u) fort> 0. We should point out that the main results in this paper work for large solutions of (1.2). © 2010 Wiley Periodicals, Inc.

The study of spherically symmetric motion is important for the theory of explosion waves. In this paper, we consider a ‘spherical piston’ problem for the relativistic Euler equations, which describes the wave motion produced by a sphere expanding into an infinite surrounding medium. We use the reflected characteristics method to construct a global piecewise smooth solution with a single shock of this spherical piston problem, provided that the speed of the sphere is a small perturbation of a constant speed.

One considers Euler equations for an isentropic perfect gas in IR d where d ≥ 1. One shows that there exist global smooth solutions, provided that the initial data satisfy: \( \left\{ {\begin{array}{*{20}{c}} {{{\partial }_{t}}\bar{u} + (\bar{u} \cdot \nabla )\bar{u} = 0\quad {\text{on}}\quad {{\mathbb{R}}^{d}} \times {{\mathbb{R}}_{ + }},} {\bar{u}(x,0)\, = \;{{u}_{0}}(x)\quad {\text{on}}\quad {{\mathbb{R}}^{d}}.} \end{array} } \right. \). Thanks to the hypotheses, this problem has a global solution ū. Using this approximate solution and a symmetrisation of the system, one obtains a local solution, such that \(({\rho ^{\frac{{\gamma - 1}}{2}}},u - \bar u) \in \mathcal{C}{^j}([0,\infty [;{H^{m - j}}({\mathbb{R}^d}))\) for j ∈ {0,1}. With some accurate energy estimates, one shows that this solution is global in time

On the stability of global solutions to the 3D Boussinesq system

Global smooth axisymmetric solutions to 2D compressible Euler equations of Chaplygin gases with non-zero vorticity

In this paper we introduce the ( I , J ) similar method for incompressible two-dimensional Euler equations, and obtain a series of explicit ( I , J ) similar solutions to the incompressible two-dimensional Euler equations. These solutions include all of the twin wave solutions, some new singularity solutions, and some global smooth solutions with a finite energy. We also reveal that the twin wave solution and an affine solution to the two-dimensional incompressible Euler equations are, respectively, a plane wave and constant vector. We prove that the initial boundary value problem of the incompressible two-dimensional Euler equations admits a unique solution and discuss the stability of the solution. Finally, we supply some explicit piecewise smooth solutions to the incompressible three-dimensional Euler case and an example of the incompressible three-dimensional Navier-Stokes equations which indicates that the viscosity limit of a solution to the Navier-Stokes equations does not need to be a solution to the Euler equations. MSC:35Q30, 76D05, 76D10.

Global weak and smooth solutions of the equation for timelike extremal surface in Minkowski space

In order to construct global piecewise smooth solutions to two-dimensional (2D) Riemann problems for the compressible Euler equations, it is important to investigate 2D elementary wave interactions. Recently, 2D shock interactions and 2D Riemann problems for the compressible Euler equations for Chaplygin gas have been investigated. In this paper, we study several types of 2D elementary wave interactions of the Chaplygin gas Euler equations. These elementary waves include shock waves, simple waves, and delta waves. Using these elementary wave interactions, global piecewise smooth solutions to several 2D Riemann problems are constructed.