Non-existence of global smooth solutions to symmetrizable nonlinear hyperbolic systems

2003 ◽  
Vol 133 (3) ◽  
pp. 719-728 ◽  
Author(s):  
Tong Yang ◽  
Changjiang Zhu
Keyword(s):  
Blow Up ◽  
Dimensional Space ◽  
Hyperbolic Systems ◽  
Shock Formation ◽  
Sufficient Condition ◽  
Smooth Solutions ◽  
Physical Systems ◽  
Global Smooth Solutions ◽  
The Cauchy Problem ◽  
Nonlinear Hyperbolic Systems

In this paper, we consider the Cauchy problem of general symmetrizable hyperbolic systems in multi-dimensional space. When some components of the initial data have compact support, we give a sufficient condition on the non-existence of global C1 solutions. This non-existence theorem can be applied to some physical systems, such as Euler equations for compressible flow in multi-dimensional space. The blow-up phenomena here can come from the singularity developed at the interface, such as vacuum boundary, rather than the shock formation as studied in the previous works on strictly hyperbolic systems. Therefore, the systems considered here include those which are non-strictly hyperbolic.

Global regularity and formation of singularities of solutions to first order quasilinear hyperbolic systems

1981 ◽  
Vol 87 (3-4) ◽  
pp. 255-261 ◽  
Author(s):  
Lee Da-tsin(Li Ta-tsien)
Keyword(s):  
Global Regularity ◽  
Hyperbolic Systems ◽  
The Other ◽  
Sufficient Condition ◽  
Smooth Solutions ◽  
Quasilinear Hyperbolic Systems ◽  
First Order ◽  
Global Smooth Solutions ◽  
Special Cases ◽  
The Cauchy Problem

SynopsisFor the Cauchy problem for strictly hyperbolic systems with general eigenvalues, we obtain existence of global smooth solutions under certain conditions on the composition of the eigenvalues and the initial data; on the other hand, we give a sufficient condition which guarantees that singularities of the solution must occur in a finite time and describe certain applications. The present paper includes the corresponding results in earlier papers by several authors as special cases.

Singularities of solutions for first order quasilinear hyperbolic systems

1983 ◽  
Vol 94 (1-2) ◽  
pp. 137-147 ◽  
Author(s):  
Lee Da-tsin(Li Ta-tsien) ◽  
Shi Jia-hong
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Hyperbolic Systems ◽  
Smooth Solutions ◽  
Small Initial Data ◽  
Quasilinear Hyperbolic Systems ◽  
First Order ◽  
Global Smooth Solutions ◽  
The Cauchy Problem

SynopsisIn this paper, the existence of global smooth solutions and the formation of singularities of solutions for strictly hyperbolic systems with general eigenvalues are discussed for the Cauchy problem with essentially periodic small initial data or nonperiodic initial data. A result of Klainerman and Majda is thus extended to the general case.

Global smooth solutions for a class of quasilinear hyperbolic systems with dissipative terms

1997 ◽  
Vol 127 (6) ◽  
pp. 1311-1324 ◽  
Author(s):  
Tong Yang ◽  
Changjiang Zhu ◽  
Huijiang Zhao
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Existence Theorem ◽  
Hyperbolic Systems ◽  
A Priori ◽  
Smooth Solutions ◽  
Quasilinear Hyperbolic Systems ◽  
Global Smooth Solutions ◽  
The Cauchy Problem ◽  
Dissipative Terms

In this paper we prove an existence theorem of global smooth solutions for the Cauchy problem of a class of quasilinear hyperbolic systems with nonlinear dissipative terms under the assumption that only the C0-norm of the initial data is sufficiently small, while the C1-norm of the initial data can be large. The analysis is based on a priori estimates, which are obtained by a generalised Lax transformation.

Global smooth solutions to 3D irrotational Euler equations for Chaplygin gases

2020 ◽  
Vol 17 (03) ◽  
pp. 613-637
Author(s):  
Changhua Wei ◽  
Yu-Zhu Wang
Keyword(s):  
Euler Equations ◽  
Compressible Euler Equations ◽  
Wave System ◽  
Smooth Solutions ◽  
Small Perturbations ◽  
Global Smooth Solutions ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Constant State ◽  
New Formulation

We study here the Cauchy problem associated with the isentropic and compressible Euler equations for Chaplygin gases. Based on the new formulation of the compressible Euler equations in J. Luk and J. Speck [The hidden null structure of the compressible Euler equations and a prelude to applications, J. Hyperbolic Differ. Equ. 17 (2020) 1–60] we show that the wave system satisfied by the modified density and the velocity for Chaplygin gases satisfies the weak null condition. We then prove the global existence of smooth solutions to the irrotational and isentropic Chaplygin gases without introducing a potential function, when the initial data are small perturbations to a constant state.

scholarly journals The global existence issue for the compressible Euler system with Poisson or Helmholtz couplings

2021 ◽  
Vol 18 (01) ◽  
pp. 169-193
Author(s):  
Xavier Blanc ◽  
Raphaël Danchin ◽  
Bernard Ducomet ◽  
Šárka Nečasová
Keyword(s):  
Initial Density ◽  
Euler System ◽  
Decay Estimates ◽  
Smooth Solutions ◽  
Perfect Gas ◽  
Poisson Equations ◽  
Global Smooth Solutions ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Whole Space

We consider the Cauchy problem for the barotropic Euler system coupled to Helmholtz or Poisson equations, in the whole space. We assume that the initial density is small enough, and that the initial velocity is close to some reference vector field [Formula: see text] such that the spectrum of [Formula: see text] is positive and bounded away from zero. We prove the existence of a global unique solution with (fractional) Sobolev regularity, and algebraic time decay estimates. Our work extends Grassin and Serre’s papers [Existence de solutions globales et régulières aux équations d’Euler pour un gaz parfait isentropique, C. R. Acad. Sci. Paris Sér. I 325 (1997) 721–726, 1997; Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math. J. 47 (1998) 1397–1432; Solutions classiques globales des équations d’Euler pour un fluide parfait compressible, Ann. Inst. Fourier Grenoble 47 (1997) 139–159] dedicated to the compressible Euler system without coupling and with integer regularity exponents.

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