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Kato’s perturbation theory and well-posedness for the Euler equations in bounded domains. (English) Zbl 0672.35044

This paper gives a new (and simpler than previous) proof of the well- posedness for the solution to the Euler equations on a bounded domain. A previous proof may be found in D. G. Ebin and J. E. Marsden [Ann. Math. 92, 102-163 (1970; Zbl 0211.574)].
The method of proof is to use the perturbation theory of T. Kato. However, it is not a straightforward application; it is necessary to introduce a suitable modification.
The paper establishes estimates first for the stationary problem before proceeding to the unsteady one. The main result is given in Theorem 5.3 which establishes existence on a bounded (sufficiently small) interval and then shows continuous dependence on the initial data.
Reviewer: B.Straughan

MSC:

35L60 First-order nonlinear hyperbolic equations
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
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35Q99 Partial differential equations of mathematical physics and other areas of application

Citations:

Zbl 0211.574
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References:

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[10] T. Kato, ?Quasi-linear equations of evolution, with applications to partial differential equations?, Lecture Notes in Mathematics, No. 448, pp. 25-70. Springer, New York, 1975. · Zbl 0315.35077
[11] T. Kato & C. Y. Lai, ?Nonlinear evolution equations and the Euler flow?, J. Funct. Analysis 56 (1984), 15-28. · Zbl 0545.76007 · doi:10.1016/0022-1236(84)90024-7
[12] T. Kato & G. Ponce, ?Commutator estimates and the Euler and Navier-Stokes equations?, (to appear). · Zbl 0671.35066
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