Behavior of the solutions of the Cauchy problem for certain quasilinear equations for unbounded increase of the time

1964 ◽  
pp. 19-23 ◽  
Author(s):  
A. M. Il′ln ◽  
O. A. Oleĭnik
Keyword(s):  
Cauchy Problem ◽  
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.

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.

Sufficient conditions of a nonlocal solvability for a system of two quasilinear equations of the first order with constant terms

2020 ◽  
Vol 55 ◽  
pp. 60-78
Author(s):  
M.V. Dontsova
Keyword(s):  
Cauchy Problem ◽  
Finite Number ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Local Solution ◽  
Quasilinear Equations ◽  
Uniqueness Of Solutions ◽  
First Order ◽  
The Cauchy Problem ◽  
Global Estimates

We consider a Cauchy problem for a system of two quasilinear equations of the first order with constant terms. The study of the solvability of the Cauchy problem for a system of two quasilinear equations of the first order with constant terms in the original coordinates is based on the method of an additional argument. Theorems on the local and nonlocal existence and uniqueness of solutions to the Cauchy problem are formulated and proved. We prove the existence and uniqueness of the local solution of the Cauchy problem for a system of two quasilinear equations of the first order with constant terms, which has the same smoothness with respect to $x$ as the initial functions of the Cauchy problem. Sufficient conditions for the existence and uniqueness of a nonlocal solution of the Cauchy problem for a system of two quasilinear equations of the first order with constant terms are found; this solution is continued by a finite number of steps from the local solution. The proof of the nonlocal solvability of the Cauchy problem for a system of two quasilinear equations of the first order with constant terms relies on global estimates.

scholarly journals On the Cauchy problem associated to the Brinkman flow in Rn

2012 ◽  
Vol 6 (2) ◽  
pp. 214-237
Author(s):  
Del Molina ◽  
Alarcon Arbieto ◽  
Iorio José
Keyword(s):  
Cauchy Problem ◽  
Porous Media ◽  
Fluid Flow ◽  
Sobolev Spaces ◽  
Well Posedness ◽  
Quasilinear Equations ◽  
The Cauchy Problem

In this work we deal with the Cauchy problem associated to the Brinkman flow, which models fluid flow in certain types of porous media. We study local and global well-posedness in Sobolev spaces Hs(Rn), s>n/2+1, using Kato's theory for quasilinear equations and parabolic regularization.

scholarly journals Analyticity of solutions for quasilinear wave equations and other quasilinear systems

2014 ◽  
Vol 144 (6) ◽  
pp. 1155-1169 ◽  
Author(s):  
Sergei Kuksin ◽  
Nikolai Nadirashvili
Keyword(s):  
Cauchy Problem ◽  
Classical Solution ◽  
Wave Equations ◽  
Cauchy Data ◽  
Classical Solutions ◽  
Quasilinear Equations ◽  
Quasilinear Systems ◽  
The Cauchy Problem ◽  
Quasilinear Wave Equations

We prove the persistence of analyticity for classical solutions of the Cauchy problem for quasilinear wave equations with analytic data. Our results show that the analyticity of solutions, stated by the Cauchy–Kowalewski and Ovsiannikov–Nirenberg theorems, lasts until a classical solution exists. Moreover, they show that if the equation and the Cauchy data are analytic only in a part of the space variables, then a classical solution is also analytic in these variables. The approach applies to other quasilinear equations and implies the persistence of the space analyticity (and the partial space analyticity) of their classical solutions.

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