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Global Existence of Near-Affine Solutions to the Compressible Euler Equations

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Abstract

We establish the global existence of solutions to the compressible Euler equations, in the case that a finite volume of ideal gas expands into a vacuum. Vacuum states can occur with either smooth or singular sound speed, the latter corresponding to the so-called physical vacuum singularity when the enthalpy vanishes on the vacuum wave front like the distance function. In this instance, the Euler equations lose hyperbolicity and form a degenerate system of conservation laws, for which a local existence theory has only recently been developed. Sideris (Arch Ration Mech Anal 225(1):141–176, 2017) found a class of expanding finite degree-of-freedom global-in-time affine solutions, obtained by solving nonlinear ODEs. In three space dimensions, the stability of these affine solutions, and hence the global existence of solutions, was established by Hadžić and Jang (Expanding large global solutions of the equations of compressible fluid mechanics, 2016) with the pressure-density relation \(p = \rho ^\gamma \) with the constraint that \(1< \gamma \leqslant {\frac{5}{3}} \); they asked if a different approach could go beyond the \(\gamma > {\frac{5}{3}} \) threshold. We provide an affirmative answer to their question, and prove the stability of affine flows and global existence for all \(\gamma >1\), thus also establishing global existence for the shallow water equations when \(\gamma =2\).

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Notes

  1. By (63), which we write as with \( {\mathcal {L}} \) and \( {\mathcal {Q}} \) respectively denoting linear and quadratic functions of their arguments. The identity (65) then shows that , with \( {\mathcal {C}} \) denote a cubic function of its argument. Hence, .

  2. We note that in the term \({\text {div}} _\eta \nabla ^2 \eta \), the operator \( {\text {div}} _\eta \) acts on each component of the Hessian \( \nabla ^2 \eta \). In particular \( |{\text {div}} _\eta \ \nabla ^2 \eta |^2 = {\text {div}} _\eta \eta ^i,_{r_1 r_2} \, {\text {div}} _\eta \eta ^i,_{r_1 r_2}\).

  3. Note well that the weighted embedding theorems are codimension-1 results; only the behavior in the direction normal to the boundary of the domain determines the inequality.

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Acknowledgements

SS was supported by the National Science Foundation Grant DMS-1301380, and by the Department of Energy Advanced Simulation and Computing (ASC) Program. We thank the anonymous referee for many useful suggestions which have significantly improved the presentation. Funding sources have been acknowledged.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Davis, Davis, CA, 95616, USA

    Steve Shkoller

  2. Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA, 93106, USA

    Thomas C. Sideris

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  1. Steve Shkoller
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  2. Thomas C. Sideris
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Correspondence to Steve Shkoller.

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Shkoller, S., Sideris, T.C. Global Existence of Near-Affine Solutions to the Compressible Euler Equations. Arch Rational Mech Anal 234, 115–180 (2019). https://doi.org/10.1007/s00205-019-01387-4

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