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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 99, Issue 3-4
  • January 1985, pp. 201-239

Regularity and large time behaviour of solutions of a conservation law without convexity

  • C. M. Dafermos (a1)
  • DOI: http://dx.doi.org/10.1017/S0308210500014256
  • Published online: 14 November 2011
Abstract
Synopsis

Using the method of generalized characteristics, we discuss the regularity and large time behaviour of admissible weak solutions of a single conservation law, in one space variable, with one inflection point.

We show that when the initial data are C then, generically, the solution is C except: (a) on a finite set of C arcs across which it experiences jump discontinuities (genuine shocks or left contact discontinuities); (b) on a finite set of straight line characteristic segments across which its derivatives of order m, m = 1, 2,…, experience jump discontinuities (weak waves of order m); and (c) on the finite set of points of interaction of shocks and weak waves. Weak waves of order 1 are triggered by rays grazing upon contact discontinuities. Weak waves of order m, m ≥ 2, are generated by the collision of a weak wave of order m − 1 with a left contact discontinuity.

We establish sharp decay rates for solutions with initial data of the following types: (a) with bounded primitive; (b) with primitive having sublinear growth; (c) in L1; (d) of compact support; and (e) periodic.

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20D. G. Schaeffer . A regularity theorem for conservation laws. Adv. in Math. 11 (1973), 368386.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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