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General Fractal Conservation Laws Arising from a Model of Detonations in Gases

We consider a model of cellular detonations in gases. They consist in conservation laws with a non-local pseudo-differential operator whose symbol is asymptotically |ξ|λ, where 0 < λ ≤ 2; it can be decomposed as the λ/2 fractional power of the Laplacian plus a convolution term. After defining the notion of entropy solution, we prove the well-posedness in the L∞ framework. In the case where 1 < λ ≤ 2 we also prove a regularizing effect. In Appendix, we show that the assumptions made to perform the mathematical study are satisfied by the considered physical model of detonations.