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The 2D compressible Euler equations in bounded impermeable domains with corners
About this Title
Paul Godin
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 269, Number 1313
ISBNs: 978-1-4704-4421-1 (print); 978-1-4704-6464-6 (online)
DOI: https://doi.org/10.1090/memo/1313
Published electronically: March 8, 2021
Keywords: Compressible Euler equations,
2D domains with corners
Table of Contents
Chapters
- 1. Introduction
- 2. Statement of the results
- 3. The associated linear Euler equations ($C^\infty$ coefficients)
- 4. Proof of Proposition 3.3 and of Proposition 3.4, and more estimates
- 5. The associated linear Euler equations (non-$C^\infty$ coefficients)
- 6. Proof of Theorem 2.1, Theorem 2.2, Remark 2.1, Remark 2.2
- A. Appendix A: Properties of div, curl in 2D
- B. Appendix B: Some properties of functional spaces
- C. Appendix C: Some estimates for functions with values in Sobolev spaces
Abstract
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.- Daniel Azagra and Juan Ferrera, Every closed convex set is the set of minimizers of some $C^\infty$-smooth convex function, Proc. Amer. Math. Soc. 130 (2002), no. 12, 3687–3692. MR 1920049, DOI 10.1090/S0002-9939-02-06695-9
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