Nonlinear parabolic equations with regularized derivatives
type: Item
instance of: scholarly article
title: Nonlinear parabolic equations with regularized derivatives
zbMATH Open document ID: 0926.35055
author: Ya-Guang Wang
author: Michael Oberguggenberger
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-11-16T00:00:00Z
review text: Nonlinear systems of differential equations \[ \partial_tu+\partial_xf(u)+g(u)=\Delta u\tag{1} \] are considered on \(\mathbb R_+^{n+1}\) \((t>0)\), where \(u=(u_1,\dots,u_m)^T\), \(f(u)=(f_1(u),\dots,f_n(u))^T\), \(\partial_xf(u)=\sum_{i=1}^n\partial_{x_i}f_i(u)\). This differential equation is regularized as follows: \[ \partial_tu_\varepsilon +\partial_xf(u_\varepsilon)*\varphi_\varepsilon +g(u_\varepsilon)=\Delta u_\varepsilon*\varphi_\varepsilon* \check\varphi_\varepsilon, \tag{2} \] where \((u_\varepsilon)_{\varepsilon>0}\) is a standard net of mollifiers. It should be noted that per derivative one mollifying convolution is taken. The main theorems are as follows: If \(f\) and \(g\) are smooth, globally Lipschitz, and \(f(0)=0\), \(g(0)=0\), then the Cauchy problem for (2) has a unique solution \(u_\varepsilon\) in \(C([0,\infty),L^2(\mathbb R^n))\), \((u_\varepsilon)_{\varepsilon>0}\) has a limit as \(\varepsilon\to 0\), and this limit is the weak solution of the Cauchy problem (1). For the latter convergence it is crucial to have a mollifier of the form \(\varphi_\varepsilon*\check\varphi_\varepsilon\) on the right hand side as its Fourier transform is a non-negative function. Using the results concerning (2) it is shown that the corresponding Cauchy problem in the Colombeau algebra of generalized functions has a unique solution as well. In the last section an unconditionally convergent difference scheme for numerical approximation of the solution to (1) is introduced.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 65M99
Mathematics Subject Classification ID: 65N99
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 1295270
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Colombeau algebra
zbMATH Keywords: unconditionally convergence difference scheme
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6326
OpenAlex ID: W1991104237
cites work: On well-posedness of the nonlocal boundary value problem for parabolic difference equations
cites work: A nonlinear theory of generalized functions.
cites work: Generalized solutions to Burgers' equation
cites work: Generalized Solutions to the Korteweg–de Vries and the Regularized Long-Wave Equations
cites work: On convergence to entropy solutions of a single conservation law
cites work: Q3976350
cites work: Borel's theorem for generalized functions
cites work: Multiplication of distributions
cites work: Generalized solutions to Cauchy problems
cites work: Q4289127
cites work: Q4318674
cites work: Generalized solutions to partial differential equations of evolution type
cites work: Generalized and classical solutions of nonlinear parabolic equations
cites work: Q4272018
cites work: Regularized conservation laws and nonlinear geometric optics
cites work: Q5536978
cites work: Q3960065
cites work: Q4308753
DOI: 10.1006/JMAA.1999.6326
description: scientific article; zbMATH DE number 1295270