Uniqueness of the generalized solution of the Cauchy problem for hyperbolic systems of quasilinear equations

1964 ◽  
pp. 31-36
Author(s):  
B. L. Roždestvenskiĭ
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Systems ◽  
Generalized Solution ◽  
Quasilinear Equations ◽  
The Cauchy Problem

On the Cauchy problem associated with the Brinkman flow in

2021 ◽  
pp. 1-20
Author(s):  
Michel Molina Del Sol ◽  
Eduardo Arbieto Alarcon ◽  
Rafael José Iorio
Keyword(s):  
Cauchy Problem ◽  
Porous Media ◽  
Fluid Flow ◽  
Comparison Principle ◽  
Well Posedness ◽  
Quasilinear Equations ◽  
Brinkman Equations ◽  
The Cauchy Problem ◽  
Model Fluid ◽  
Image Position

In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space \[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \] under the assumption that the plane $z=0$ is impenetrable to the fluid. This means that we will have to introduce boundary conditions that must be attached to the Brinkman equations. We study local and global well-posedness in appropriate Sobolev spaces introduced below, using Kato's theory for quasilinear equations, parabolic regularization and a comparison principle for the solutions of the problem.

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.

scholarly journals The nonlocal solvability conditions for a system of two quasilinear equations of the first order with absolute terms

2019 ◽  
Vol 21 (3) ◽  
pp. 317-328
Author(s):  
Marina V. Dontsova
Keyword(s):  
Cauchy Problem ◽  
Finite Number ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Local Solution ◽  
Quasilinear Equations ◽  
Solvability Conditions ◽  
First Order ◽  
The Cauchy Problem ◽  
Global Estimates

The Cauchy problem for a system of two first-order quasilinear equations with absolute terms is considered. The study of this problem’s solvability in original coordinates is based on the method of an additional argument. The existence of the local solution of the problem with smoothness which is not lower than the smoothness of the initial conditions, is proved. Sufficient conditions of existence are determined for the nonlocal solution that is continued by a finite number of steps from the local solution. The proof of the nonlocal resolvability of the Cauchy problem relies on original global estimates.

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