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The 2D compressible Euler equations in bounded impermeable domains with corners. (English) Zbl 1462.35189

Memoirs of the American Mathematical Society 1313. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-4421-1/pbk; 978-1-4704-6464-6/ebook). v, 72 p. (2021).
Summary: We study 2D compressible Euler flows in bounded impermeable domains whose boundary is smooth except for corners. We assume that the angles of the corners are small enough. Then we obtain local (in time) existence of solutions which keep the \(L^2\) Sobolev regularity of their Cauchy data, provided the external forces are sufficiently regular and suitable compatibility conditions are satisfied. Such a result is well known when there is no corner. Our proof relies on the study of associated linear problems. We also show that our results are rather sharp: we construct counterexamples in which the smallness condition on the angles is not fulfilled and which display a loss of \(L^2\) Sobolev regularity with respect to the Cauchy data and the external forces.

MSC:

35L50 Initial-boundary value problems for first-order hyperbolic systems
35Q31 Euler equations
35B65 Smoothness and regularity of solutions to PDEs
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35L60 First-order nonlinear hyperbolic equations
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