# zbMATH — the first resource for mathematics

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
Full Text: