The sharp form of Oleĭnik’s entropy condition in several space variables
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- by David Hoff PDF
- Trans. Amer. Math. Soc. 276 (1983), 707-714 Request permission
Abstract:
We investigate the conditions under which the Volpert-Kruzkov solution of a single conservation law in several space variables with flux $F$ will satisfy the simplified entropy condition $\operatorname {div} F’(u) \leqslant 1/t$, and when this condition guarantees uniqueness for given ${L^\infty }$ Cauchy data. We show that, when $F$ is ${C^1}$, our condition guarantees uniqueness iff $F$ is isotropic, and that, for such $F$, the Volpert-Kruzkov solution always satisfies our condition.References
- Michael Crandall and Andrew Majda, The method of fractional steps for conservation laws, Numer. Math. 34 (1980), no. 3, 285–314. MR 571291, DOI 10.1007/BF01396704 S. N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR-Sb. 10 (1970), 217-242.
- O. A. Oleĭnik, Discontinuous solutions of non-linear differential equations, Amer. Math. Soc. Transl. (2) 26 (1963), 95–172. MR 0151737, DOI 10.1090/trans2/026/05
- A. I. Vol′pert, Spaces $\textrm {BV}$ and quasilinear equations, Mat. Sb. (N.S.) 73 (115) (1967), 255–302 (Russian). MR 0216338
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 276 (1983), 707-714
- MSC: Primary 35L65
- DOI: https://doi.org/10.1090/S0002-9947-1983-0688972-6
- MathSciNet review: 688972