Nonlinear stability of relativistic vortex sheets in three-dimensional Minkowski spacetime (Q1732568)

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Nonlinear stability of relativistic vortex sheets in three-dimensional Minkowski spacetime
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    Nonlinear stability of relativistic vortex sheets in three-dimensional Minkowski spacetime (English)
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    25 March 2019
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    It is a very interesting paper, concerning the relativistic Euler equations. The spacetime coordinates $(t, x_1, x_2)$ are used and a smooth hypersurface $ \Sigma(t)$ is considered, where a jump of the tangential velocity is considered. The fluid is barotropic, with some restrictions on the pressure derivative. The weak solution for density and velocity verify the Rankine-Hugoniot conditions at every point of $\Sigma(t)$, whose simpler form is given by the equations (2.8). A piecewise smooth weak solution with such discontinuities is called a relativistic vortex-sheet. A trivial vortex-sheet solutions exists, given by two constant states separated by a rectilinear front. The aim of the paper is to study the local-in-time existence and nonlinear stability of relativistic vortex-sheets with initial data close to the above trivial solutions. New appropriate functions are introduced as primary unknowns, depending on the light speed $\epsilon$ and on the particle number density $N$. The equivalent form (2.18) of the initial problem is obtained, which is appropriate for computing the roots of the Lopatinskii determinant and allows to the authors' ``to obtain the linearized vorticity equation through which the loss of derivatives can be compensated in the higher-order energy estimates''. \par It is proved that a relativistic vortex sheet is a characteristic discontinuity and the relativistic vortex-sheet problem is a free boundary problem. New unknowns are introduced and the problem is reformulated in the domain $x_2>0$. The existence result is obtained assuming the condition (2.25), which for the non-relativistic limit $\epsilon \rightarrow 0$ reduces to the classical condition $\text{Mach number }> \sqrt 2$ for the compressible case. Certain weighted Sobolev spaces are used in order to prove the existence result. \par The constant and variable coefficient linearized problems are studied in sections 3-4. Some results of \textit{G. Métivier} [Prog. Nonlinear Differ. Equ. Appl. 47, 25--103 (2001; Zbl 1017.35075)] are used. The linearized problem with variable coefficient is transformed into a problem with a constant and diagonal boundary matrix. Energy estimates for the linearized problem are given in section 5. For this, some results on paradifferential calculus are used. In Section 7, a smooth ``approximate solution'' is obtained, by imposing necessary compatible conditions on the initial data. Then the original problem is reduced to a nonlinear problem with zero initial data. In the last part, the construction of the solution of the reduced problem is given, by using the Nash-Moser iteration scheme.
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    Euler relativistic equations
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    vortex sheets
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    Lopatinskii determinant
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    Nash-Moser iteration scheme
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    Kreiss symmetrizers
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