Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions
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- by Donald P. Ballou PDF
- Trans. Amer. Math. Soc. 152 (1970), 441-460 Request permission
Abstract:
This paper is concerned with the existence of weak solutions to certain nonlinear hyperbolic Cauchy problems. A condition on the curves of discontinuity is used which guarantees uniqueness in the class of piecewise smooth weak solutions. The method of proof is geometric in nature and is constructive in the manner of A. Douglis and Wu Cho-Chün; that is, for certain types of initial data the method of characteristics is employed to construct piecewise smooth weak solutions. A limiting process is then used to obtain existence for bounded, measurable initial data. The solutions in some cases exhibit interesting, new phenomena. For example, a certain class of initial data having one jump gives rise to a solution having a curving contact discontinuity which does not enter the region of intersecting characteristics.References
- Edward Conway and Joel Smoller, Clobal solutions of the Cauchy problem for quasi-linear first-order equations in several space variables, Comm. Pure Appl. Math. 19 (1966), 95–105. MR 192161, DOI 10.1002/cpa.3160190107
- Avron Douglis, An ordering principle and generalized solutions of certain quasi-linear partial differential equations, Comm. Pure Appl. Math. 12 (1959), 87–112. MR 104899, DOI 10.1002/cpa.3160120106
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
- 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
- Zhuo-qun Wu, On the existence and uniqueness of the generalized solutions of the Cauchy problem for quasilinear equations of first order without convexity conditions, Chinese Math. 4 (1964), 561–577. MR 168890
Additional Information
- © Copyright 1970 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 152 (1970), 441-460
- MSC: Primary 35L45
- DOI: https://doi.org/10.1090/S0002-9947-1970-0435615-3
- MathSciNet review: 0435615