Abstract
We analyze family of solutions to multidimensional scalar conservation law, with flux depending on the time and space explicitly, regularized with vanishing diffusion and dispersion terms. Under a condition on the balance between diffusion and dispersion parameters, we prove that the family of solutions is precompact in \({L^1_{\rm loc}}\). Our proof is based on the methodology developed in Sazhenkov (Sibirsk Math Zh 47(2):431–454, 2006), which is in turn based on Panov’s extension (Panov and Yu in Mat Sb 185(2):87–106, 1994) of Tartar’s H-measures (Tartar in Proc R Soc Edinb Sect A 115(3–4):193–230, 1990), or Gerard’s micro-local defect measures (Gerard Commun Partial Differ Equ 16(11):1761–1794, 1991). This is new approach for the diffusion–dispersion limit problems. Previous results were restricted to scalar conservation laws with flux depending only on the state variable.
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The work of J. Aleksić and S. Pilipović is supported by Ministry of Science and Technological Development, Republic of Serbia, project number 144016. The work of D. Mitrovic is supported in part by the Research Council of Norway and the local government of municipality Budva.
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Aleksić, J., Mitrovic, D. & Pilipović, S. Hyperbolic conservation laws with vanishing nonlinear diffusion and linear dispersion in heterogeneous media. J. Evol. Equ. 9, 809 (2009). https://doi.org/10.1007/s00028-009-0035-5
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DOI: https://doi.org/10.1007/s00028-009-0035-5