On the critical behavior in nonlinear evolutionary PDEs with small viscosity. (English) Zbl 1263.35019
Summary: The problem of general dissipative regularization of the quasilinear transport equation is studied. We argue that the local behavior of solutions to the regularized equation near the point of gradient catastrophe for the transport equation is described by the logarithmic derivative of the Pearcey function; this statement generalizes a result of A. M. Il’in [Matching of asymptotic expansions of solutions of boundary value problems. Providence, RI: American Mathematical Society (AMS) (1992; Zbl 0754.34002)]. We provide some analytic arguments supporting the conjecture and test it numerically.
MSC:
| 35B25 | Singular perturbations in context of PDEs |
Keywords:
one space dimension; general dissipative regularization; quasilinear transport equation; gradient catastrophe; Pearcey functionCitations:
Zbl 0754.34002Software:
FreeFem++References:
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