Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum

We study the Cauchy problem for multi-dimensional compressible radiation hydrodynamics equations with vacuum. First, we present some sufficient conditions on the blow-up of smooth solutions in multi-dimensional space. Then, we obtain the invariance of the support of density for the smooth solutions with compactly supported initial mass density by the property of the system under the vacuum state. Based on the above-mentioned results, we prove that we cannot get a global classical solution, no matter how small the initial data are, as long as the initial mass density is of compact support. Finally, we will see that some of the results that we obtained are still valid for the isentropic flows with degenerate viscosity coefficients as well as for one-dimensional case.     

Riemann problem and elementary wave interactions in isentropic magnetogasdynamics

In this paper, we consider the Riemann problem for a quasilinear hyperbolic system of equations governing the one dimensional unsteady simple wave flow of an isentropic, inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field. This class of equations includes, as a special case, the equations of isentropic gasdynamics. We study the shock and rarefaction waves and their properties, and discuss the geometry of shock curves using the Riemann invariant coordinates. Under certain conditions, we show the existence and uniqueness of the solution to the Riemann problem for arbitrary initial data, and then discuss the vacuum state in isentropic magnetogasdynamics. Finally, we discuss numerical results for different initial data, and discuss all possible interactions of elementary waves. It is noticed that although the magnetogasdynamic system is more complex than the corresponding gasdynamic system, all the parallel results remain identical. However, unlike the ordinary gasdynamic case, the solution inside rarefaction waves in magnetogasdynamics cannot be obtained directly and explicitly; indeed, it requires an extra iteration procedure. It is also observed that the presence of a magnetic field makes both the shock and rarefaction stronger compared to what they would have been in the absence of a magnetic field.     

Stability of viscous shock wave under periodic perturbation for compressible Navier–Stokes–Korteweg system

In this paper, we study the stability of a viscous shock wave for the isentropic Navier–Stokes–Korteweg (N-S-K) equations under space-periodic perturbation. It is shown that if the initial perturbation around the shock and the amplitude of the shock are small, then the solution of the N-S-K equations tends to the viscous shock.     

Stability of Rarefaction Waves to the 1D Compressible Navier–Stokes Equations with Density-Dependent Viscosity

In this paper, we study the asymptotic stability of rarefaction waves for the compressible isentropic Navier–Stokes equations with density-dependent viscosity. First, a weak solution around a rarefaction wave to the Cauchy problem is constructed by approximating the system and regularizing the initial values which may contain vacuum states. Then some global in time estimates on the weak solution are obtained. Based on these uniform estimates, the vacuum states are shown to vanish in finite time and the weak solution we constructed becomes a unique strong one. Consequently, the stability of the rarefaction wave is proved in a weak sense. The theory holds for large-amplitudes rarefaction waves and arbitrary initial perturbations.     

Zero dissipation limit to a Riemann solution consisting of two shock waves for the 1D compressible isentropic Navier-Stokes equations

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 efects of initial layers are ignored,this gives the frst mathematical justifcation 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.     

Shock Formation and Vorticity Creation for 3d Euler

We analyze the shock formation process for the 3D nonisentropic Euler equations with the ideal gas law, in which sound waves interact with entropy waves to produce vorticity. Building on our theory for isentropic flows in [3, 4], we give a constructive proof of shock formation from smooth initial data. Specifically, we prove that there exist smooth solutions to the nonisentropic Euler equations which form a generic stable shock with explicitly computable blowup time, location, and direction. This is achieved by establishing the asymptotic stability of a generic shock profile in modulated self-similar variables, controlling the interaction of wave families via: (i) pointwise bounds along Lagrangian trajectories, (ii) geometric vorticity structure, and (iii) high-order energy estimates in Sobolev spaces. © 2022 Wiley Periodicals LLC.     

Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation

In this paper,firstly,by solving the Riemann problem of the zero-pressure flow in gas dynamics with a flux approximation,we construct parameterized delta-shock and constant density solutions,then we show that,as the flux perturbation vanishes,they converge to the delta-shock and vacuum state solutions of the zero-pressure flow,respectively.Secondly,we solve the Riemann problem of the Euler equations of isentropic gas dynamics with a double parameter flux approximation including pressure.Furthermore,we rigorously prove that,as the two-parameter flux perturbation vanishes,any Riemann solution containing two shock waves tends to a delta-shock solution to the zero-pressure flow;any Riemann solution containing two rarefaction waves tends to a two-contact-discontinuity solution to the zero-pressure flow and the nonvacuum intermediate state in between tends to a vacuum state.Finally,numerical results are given to present the formation processes of delta shock waves and vacuum states.     

On Blow-up of Regular Solutions to the Isentropic Euler and Euler-Boltzmann Equations with Vacuum*

In this paper, the authors study the Cauchy problem of n-dimensional isentropic Euler equations and Euler-Boltzmann equations with vacuum in the whole space. They show that if the initial velocity satisfies some condition on the integral J in the “isolated mass group” (see (1.13)), then there will be finite time blow-up of regular solutions to the Euler system with J ≤ 0 (n ≥ 1) and to the Euler-Boltzmann system with J < 0 (n ≥ 1) and J = 0 (n ≥ 2), no matter how small and smooth the initial data are. It is worth mentioning that these blow-up results imply the following: The radiation is not strong enough to prevent the formation of singularities caused by the appearance of vacuum,with the only possible exception in the case J = 0 and n = 1 since the radiation behaves differently on this occasion.     

Cauchy problem with general discontinuous initial data along a smooth curve for 2-d Euler system

We study the Cauchy problems for the isentropic 2-d Euler system with discontinuous initial data along a smooth curve. All three singularities are present in the solution: shock wave, rarefaction wave and contact discontinuity. We show that the usual restrictive high order compatibility conditions for the initial data are automatically satisfied. The local existence of piecewise smooth solution containing all three waves is established.     

On the Riemann problem for 2D compressible Euler equations in three pieces

The Riemann problem for two-dimensional isentropic Euler equations is considered. The initial data are three constants in three fan domains forming different angles. Under the assumption that only a rarefaction wave, shock wave or contact discontinuity connects two neighboring constant initial states, it is proved that the cases involving three shock or rarefaction waves are impossible. For the cases involving one rarefaction (shock) wave and two shock (rarefaction) waves, only the combinations when the three elementary waves have the same sign are possible (impossible).     

Uniqueness of weak solutions of the Cauchy problem for general 2 × 2 conservation laws

In this paper we prove uniqueness theorems of the Cauchy problem for general 2 × 2 genuinely nonlinear conservation laws and of isentropic gas dynamics equations, not necessarily convex. We consider solutions which are piecewise continuous and have a finite number of centered rarefaction waves in each compact set. We require the solutions to satisfy an extended entropy condition (E) which reduces to Lax's shock inequalities (L) when the system is genuinely nonlinear.     

欧拉 - 玻尔兹曼方程整体光滑解

本文证明了R^d 中具有某一类小初值的等熵欧拉 - 玻尔兹曼方程整体光滑解的存在性.本文首先构造了等熵欧拉 - 玻尔兹曼方程的局部解, 并证明了局部解的适定性. 此外,文中还构造了关于原方程的随时间 t 增加、具有良好的衰减性质的整体光滑背景解. 同时, 当方程的辐射项系数满足一定条件时, 本文建立了关于源项的估计.通过将背景解的衰减与源项的估计结合起来, 文中证明了存在整数 s>d/2 + 1 ,使得背景解与原方程解的 H^s(R^d)x L²(R₊ x S^d-1;H^s(R^d))范数之差始终是有界的, 从而保证了原方程整体光滑解的存在性.     

Global solutions to the shallow water system with a method of an additional argument

The classical system of shallow water (Saint–Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasi-linear hyperbolic system for a wide class of initial data, we find a sufficient condition on the initial data that guarantee existence of a global classical solution continued from a local solution. The sufficient conditions can be easily satisfied for the fluid flow propagating in one direction with two characteristic velocities of the same sign and two monotonically increasing Riemann invariants. We prove that these properties persist in the time evolution of the classical solutions to the shallow water equations and provide no shock wave singularities formed in a finite time over a half-line or an infinite line. On a technical side, we develop a novel method of an additional argument, which allows to obtain local and global solutions to the quasi-linear hyperbolic systems in physical rather than characteristic variables.     

The Blowup Mechanism of Small Data Solutions for the Quasilinear Wave Equations in Three Space Dimensions

For a class of three-dimensional quasilinear wave equations with small initial data, we give a complete asymptotic expansion of the lifespan of classical solutions, that is, we solve a conjecture posed by John and Hörmander. As an application of our result, we show that the solution of three-dimensional isentropic compressible Euler equations with irrotational initial data which are a small perturbation from a constant state will develop singularity in the first-order derivatives in finite time while the solution itself is continuous. Furthermore, for this special case, we also solve a conjecture of Alinhac.     

Riemann problem with delta initial data for the isentropic relativistic Chaplygin Euler equations

In this paper, we study the Riemann problem with the initial data containing the Dirac delta function for the isentropic relativistic Chaplygin Euler equations. Under suitably generalized Rankine–Hugoniot relation and entropy condition, we constructively obtain the global existence of generalized solutions including delta shock waves that explicitly exhibit four kinds of different structures. Moreover, it can be found that the solutions constructed here are stable for the perturbation of the initial data.     

Analysis on linear stability of oblique shock waves in steady supersonic flow

An attached oblique shock wave is generated when a sharp solid projectile flies supersonically in the air. We study the linear stability of oblique shock waves in steady supersonic flow under three dimensional perturbation in the incoming flow. Euler system of equations for isentropic gas model is used. The linear stability is established for shock front with supersonic downstream flow, in addition to the usual entropy condition.     

The limits of Riemann solutions to the isentropic magnetogasdynamics

The objective of this note is to prove that the Riemann solutions of the isentropic magnetogasdynamics equations converge to the corresponding Riemann solutions of the transport equations by letting both the pressure and the magnetic field vanish. The delta shock wave can be obtained as the limit of two shock waves and the vacuum state can be obtained as the limit of two rarefaction waves. Moreover the relation between the speed of formation of singular density and those of the vanishing pressure and the vanishing magnetic field is discussed in detail.     

Asymptotic behavior of solutions for the 1-D isentropic Navier–Stokes–Korteweg equations with free boundary

In this paper, we consider the problem with a gas–gas free boundary for the one dimensional isentropic compressible Navier–Stokes–Korteweg system. For shock wave, asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary, and prove that if the initial data around the shifted viscous shock profile and its strength are sufficiently small, then the problem has a unique global strong solution, which tends to the shifted viscous shock profile as time goes to infinity. Also, we show the asymptotic stability toward rarefaction wave without the smallness on the strength if the initial data around the rarefaction wave are sufficiently small.     

一维Chaplygin气体欧拉方程组中波的相互作用

研究一维Chaplygin气体欧拉方程组中波的相互作用.方程组的波包含接触间断和在密度变量以及内能变量上同时具有狄拉克函数的狄拉克激波.根据这些波的不同组合,问题被分成了7种情形.通过详细地构造每种情形的整体解,获得了各种波相互作用的完整结果.特别地,对于一类初值,两个接触间断相互作用后,产生了一个狄拉克激波;然而,对于另外一类初值,一个狄拉克激波与一个接触间断相互作用后,狄拉克激波消失.这些都是波相互作用中非常特别的现象.     

Nonlinear stability of viscous shock waves for one-dimensional nonisentropic compressible Navier–Stokes equations with a class of large initial perturbation

We study the nonlinear stability of viscous shock waves for the Cauchy problem of one-dimensional nonisentropic compressible Navier–Stokes equations for a viscous and heat conducting ideal polytropic gas. The viscous shock waves are shown to be time asymptotically stable under large initial perturbation with no restriction on the range of the adiabatic exponent provided that the strengths of the viscous shock waves are assumed to be sufficiently small. The proofs are based on the nonlinear energy estimates and the crucial step is to obtain the positive lower and upper bounds of the density and the temperature which are uniformly in time and space.