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Critical thresholds in Euler-Poisson equations. (English) Zbl 0989.35110
This interesting paper contains a preliminary study of new phenomena associated with the Euler-Poisson equation called critical threshold phenomena. In these phenomena the answer to questions concerning the global smoothness versus finite time breakdown depend on whether the initial configuration crosses an intrinsic \(O(1)\) critical threshold. A typical case is the simple one-dimensional problem where the unforced inviscid Burgers’ equation has a solution that always forms a shock discontinuity except in the non-generic case of an increasing initial profile. The paper, containing five sections and an appendix emphasizes once again the way in which problems lying on the linear/nonlinear borderline lead to the Riccati equation.

MSC:
35Q35 PDEs in connection with fluid mechanics
35Q60 PDEs in connection with optics and electromagnetic theory
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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