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Global multidimensional shock waves of 2-dimensional and 3-dimensional unsteady potential flow equations. (English) Zbl 1395.35146

Although local existence results of multidimensional planar shock waves have been established in some fundamental well known references there are few results on the global existence of those waves except the ones for potential flow equations in much higher space dimensions or in special unbounded space-time domains with some suitable boundary conditions (of Dirichlet or Neumann type). In this paper the authors are concerned with both the local and global multidimensional conic wave shock problem of an unsteady potential flow equation when a pointed piston or an explosive wave expands fast in 2 or 3-dimensional polytropic gases. This problem is reduced to a BVP of a multidimensional unsteady potential equation in a region which is bounded by two infinite cones, and thus the domain boundary includes a twofold conic point with strong singularity and the corresponding second order hyperbolic equation has no initial data in this domain. The resulting nonlinear boundary conditions are of almost of Neumann-type. In physical experiments or in numerical computations it is shown that this multi-dimensional shock wave solution not only exits locally but also exists globally in the whole time space and tends to a self-similar solution as the time increases indefinitely.

MSC:

35L67 Shocks and singularities for hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
76N15 Gas dynamics (general theory)
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