A priori estimates and analytical construction of radially symmetric solutions in the gas dynamics (Q2927871)

This paper is concerned with the construction of shock wave solutions to the compressible isentropic Euler equations in the spherically or cylindrically symmetric case with a reasonable lower bound on the time of existence. The author considers only 2-shocks, and the general framework is the one of a Bethe-Weyl gas (for example, Van der Waals gas or perfect gas) in the particular case of spherical or cylindrical symmetry. Assuming that initially one has a piecewise regular solution with a discontinuity jump at radius \(R_0\) satisfying the Lax compatibility conditions, and that, prolongating each regular piece of this initial condition into a regular global in space function, the author finds that a regular solution of the equations with this initial condition admitting a long time of existence. Moreover, under some expansivity hypotheses on the Riemann invariant, the author obtains that the time of existence of the cylindrical or spherical symmetry solution is proportional to the radius \(R_0\) of the initial discontinuity. The main ingredient in the proof is to establish \({\mathcal C}^1\)-a priori estimates on the Riemann invariants in the case of cylindrical or spherical symmetry which allow one to construct shock waves with a time of existence proportional to the distance to the origin at the initial time.