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A theory of \(L ^{1}\)-dissipative solvers for scalar conservation laws with discontinuous flux. (English) Zbl 1261.35088
The authors study \(L^1\) contractive semigroups of solutions to conservation laws \(u_t+f(x,u)_x=0\) with discontinuous flux \(f(x,u)=f^l(u)\) for \(x<0\), \(f(x,u)=f^r(u)\) for \(x>0\). It is known that such conservation laws usually admit many different \(L^1\) contractive semigroups, and additional admissibility (entropy) criteria are necessary to extract unique solutions. The authors propose a general framework to these criteria, based on selection of a special family of piecewise constant weak solutions, which is called germ. For any given germ the authors formulate the corresponding “germ-based” admissibility criteria in the form of interface conditions on the discontinuity line \(x=0\). They characterize the germs that lead to \(L^1\) contractive semigroups. The suggested unified approach allows to provide new uniqueness results for conservation laws with discontinuous flux as well as to recover the known results under the weaker assumptions. The existence of admissible solutions is also discussed and is established for fluxes satisfying some additional conditions.

MSC:
35L65 Hyperbolic conservation laws
47H20 Semigroups of nonlinear operators
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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