Global classical solutions of the Cauchy problems for nonlinear vorticity equations and its applications
type: Item
instance of: scholarly article
title: Global classical solutions of the Cauchy problems for nonlinear vorticity equations and its applications
zbMATH Open document ID: 0656.76023
author: Mu Mu
published in: Chinese Annals of Mathematics. Series B
publication date: 1987-01-01T00:00:00Z
review text: The author studies a class of nonlinear vorticity equations on a compact Riemannian manifold M without boundary and proves the global existence and uniqueness of classical solutions for the associated Cauchy problems. The abstract results are applied to a simple barotropic nondivergent model of atmospheric dynamics. The Cauchy problems under consideration can be written as \[ (\partial_ t+L_{(W(u),b)})Au+L_{(R_ 0,d)}u=f\quad in\quad M\times {\mathbb{R}}_+,\quad Au|_{t=0}=Au_ 0 \] \[ (u=u(x,t)\in {\mathbb{R}},\quad x\in M,\quad t\in {\mathbb{R}}_+). \] Here, A denotes a linear selfadjoint second-order elliptic operator; \(L_{(W(u),b)}\) and \(L_{(R_ 0,d)}\) are linear first-order operators defined in terms of time-dependent vector fields W(u), \(R_ 0\) and functions b, d on M; W(u) depends on u and grad u. The data \(u_ 0\) and f are functions on M and \(M\times {\mathbb{R}}_+\), respectively. In the important case that 0 is an eigenvalue of A, compatibility conditions are required. The method of proof is to solve a family of linearized problems, establish certain a-priori estimates, and apply Schauder's fixed point theorem to find a solution of the nonlinear problem. Most of the proofs are merely outlined and rely heavily on earlier work of the same author.
Mathematics Subject Classification ID: 76B47
Mathematics Subject Classification ID: 35G25
Mathematics Subject Classification ID: 86A10
zbMATH DE Number: 4072569
zbMATH Keywords: nonlinear vorticity equations
zbMATH Keywords: compact Riemannian manifold
zbMATH Keywords: global existence
zbMATH Keywords: uniqueness
zbMATH Keywords: classical solutions
zbMATH Keywords: Cauchy problems
zbMATH Keywords: barotropic nondivergent model of atmospheric dynamics
zbMATH Keywords: selfadjoint second-order elliptic operator
zbMATH Keywords: a-priori estimates
zbMATH Keywords: Schauder's fixed point theorem
reviewed by: P. G. Schmidt
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4072569