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Commutator estimates and the Euler and Navier-Stokes equations. (English) Zbl 0671.35066
The Cauchy problem for the Euler and Navier-Stokes equations is solved by an abstract method in the Lebesgue spaces $$L^ p_ s({\mathbb{R}}^ m)$$, $$1<p<\infty$$, $$s>1+m/p$$. This method gives quick proofs for unique existence of local solutions, continuous dependence of the solution on the initial data and viscosity jointly (in particular, convergence for vanishing viscosity), breakdown criteria in terms of the vorticity and displacement tensor, etc. It was made possible by deducing new estimates in $$L^p$$-norm for commutators of the form $$J^ sf-fJ^ s$$, where $$J=(1- \Delta)^{1/2}$$, $$s\in {\mathbb{R}}$$, and $$f$$ is a multiplication operator.
Reviewer: Tosio Kato

##### MSC:
 35Q30 Navier-Stokes equations 35G25 Initial value problems for nonlinear higher-order PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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