Weak solution of the Cauchy problem for a multi-dimensional quasi-linear equation

1967 ◽  
Vol 2 (4) ◽  
pp. 733-739 ◽  
Author(s):  
N. N. Kuznetsov
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Linear Equation ◽  
The Cauchy Problem

scholarly journals The Cauchy Problem for the Incompressible 2D-MHD with Power Law-Type Nonlinear Viscous Fluid

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Jae-Myoung Kim
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Global Existence ◽  
Viscous Fluid ◽  
Power Law ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Fluid Flows ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We investigate a motion of the incompressible 2D-MHD with power law-type nonlinear viscous fluid. In this paper, we establish the global existence and uniqueness of a weak solution u , b depending on a number q in ℝ 2 . Moreover, the energy norm of the weak solutions to the fluid flows has decay rate 1 + t − 1 / 2 .

A finite difference method for the very weak solution to a Cauchy problem for an elliptic equation

2018 ◽  
Vol 26 (6) ◽  
pp. 835-857 ◽  
Author(s):  
Dinh Nho Hào ◽  
Le Thi Thu Giang ◽  
Sergey Kabanikhin ◽  
Maxim Shishlenin
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Weak Sense ◽  
Very Weak Solution ◽  
Local Boundary ◽  
The Cauchy Problem ◽  
Non Local

Abstract We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method.

scholarly journals On the uniqueness of weak solutions to the Ericksen–Leslie liquid crystal model in ℝ2

2016 ◽  
Vol 26 (04) ◽  
pp. 803-822 ◽  
Author(s):  
Jinkai Li ◽  
Edriss S. Titi ◽  
Zhouping Xin
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Liquid Crystal ◽  
Weak Solutions ◽  
Energy Estimates ◽  
Unique Weak Solution ◽  
The Cauchy Problem ◽  
Crystal Model ◽  
Main Ideas ◽  
Leslie System

This paper concerns the uniqueness of weak solutions to the Cauchy problem to the Ericksen–Leslie system of liquid crystal models in [Formula: see text], with both general Leslie stress tensors and general Oseen–Frank density. It is shown here that such a system admits a unique weak solution provided that the Frank coefficients are close to some positive constant. One of the main ideas of our proof is to perform suitable energy estimates at the level one order lower than the natural basic energy estimates for the Ericksen–Leslie system.

scholarly journals Existence and uniqueness of weak solutions of the Cauchy problem for parabolic delay-differential equations

1980 ◽  
Vol 21 (1) ◽  
pp. 65-80 ◽  
Author(s):  
S. Nababan ◽  
K.L. Teo
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Existence And Uniqueness ◽  
A Priori ◽  
Divergence Form ◽  
Partial Delay ◽  
The Cauchy Problem ◽  
Delay Differential

In this paper, a class of systems governed by second order linear parabolic partial delay-differential equations in “divergence form” with Cauchy conditions is considered. Existence and uniqueness of a weak solution is proved and its a priori estimate is established.

scholarly journals On critical exponents for weak solutions to the Cauchy problem for one nonlinear equation with gradient non-linearity

2020 ◽  
Author(s):  
Maxim Korpusov ◽  
Alexandra Matveeva
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Weak Solution ◽  
Nonlinear Equation ◽  
Critical Exponents ◽  
Weak Sense ◽  
Order Partial Differential Equation ◽  
Third Order ◽  
The Cauchy Problem

In this paper, we consider the Cauchy problem for one nonclassical, third-order, partial differential equation with gradient non-linearity $|\nabla u(x,t)|^q$. The solution to this problem is understood in a weak sense. We show that for $1“3/2$ the existence of the only local-in-time weak solution of Cauchy’s problem.If $3/2”

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