Well‐posedness in smooth function spaces for moving‐boundary 1‐D compressible euler equations in physical vacuum

By Daniel Coutand, Steve Shkoller. 2009

139 citations recorded by the source corpus.

The free‐boundary compressible one‐dimensional Euler equations with moving physical vacuum boundary are a system of hyperbolic conservation laws that are both characteristic and degenerate. The physical vacuum singularity (or rate of degeneracy) requires the sound speed $c^2= \gamma \rho^{ \gamma -1}$ to scale as the square root of the distance to the vacuum boundary and has attracted a great deal of attention in recent years. We establish the existence of unique solutions to this system on a short time interval, which are smooth (in Sobolev spaces) all the way to the moving boundary. The proof is founded on a new higher‐order, Hardy‐type inequality in conjunction with an approximation of the Euler equations consisting of a particular degenerate parabolic regularization. Our regular solutions can be viewed as degenerate viscosity solutions. © 2010 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/021b46a476783f3f9513ca9d435db8cf7242b0c4

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Citations in the corpus

Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum (2012, methodology): “Our solution strategy in [10] was based on a one-dimensional version of Hardy’s inequality (in higher-order form).”
Global Existence of Smooth Solutions and Convergence to Barenblatt Solutions for the Physical Vacuum Free Boundary Problem of Compressible Euler Equations with Damping (2014, methodology): “In [6], the local-in-time well-posedness of the physical vacuum free boundary problem was investigated for the one-dimensional EulerPoisson equations, adopting the methods motivated by those in [4] for the onedimensional Euler equations.”