Optimal regularity for all time for entropy solutions of conservation laws in $$BV^s$$ B
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Nonlinear Differential Equations and Applications NoDEA
Optimal regularity for all time for entropy solutions of conservation laws in BV s Shyam Sundar Ghoshal , Billel Guelmame, Animesh Jana and St´ephane Junca Abstract. This paper deals with the optimal regularity for entropy solutions of conservation laws. For this purpose, we use two key ingredients: (a) fine structure of entropy solutions and (b) fractional BV spaces. We show that optimality of the regularizing effect for the initial value problem from L∞ to fractional Sobolev space and fractional BV spaces is valid for all time. Previously, such optimality was proven only for a finite time, before the nonlinear interaction of waves. Here for some well-chosen examples, the sharp regularity is obtained after the interaction of waves. Moreover, we prove sharp smoothing in BV s for a convex scalar conservation law with a linear source term. Next, we provide an upper bound of the maximal smoothing effect for nonlinear scalar multi-dimensional conservation laws and some hyperbolic systems in one or multi-dimension. Mathematics Subject Classification. 35L65, 35B45, 35L45. Keywords. Conservation laws, Entropy solutions, Shocks, Smoothing effect, Fractional BV spaces BV s .
Contents 1. Introduction 2. Fractional BV spaces, BV s , 0 < s ≤ 1 3. Sharp regularity for scalar 1D entropy solutions 4. The scalar multi-D case 5. A class of 2 × 2 triangular systems 6. The multi-D Keyfitz–Kranzer system Acknowledgements References
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NoDEA
1. Introduction For nonlinear conservation laws, it is known since Lax–Ole˘ınik [36,41] that the entropy solution can have a better regularity than the initial data for Burgers type fluxes. Such smoothing effect has been obtained in fractional Sobolev spaces [37] and recently in fractional BV space [13] for more general fluxes. The optimality of such regularization is largely open in general. For scalar 1-D conservation laws, there are some optimal results proven up to finite time [18,25,38]. The aim of this article is to obtain the same optimality for all time. We start with the one-dimensional scalar conservation laws which reads as follow: ∂u ∂f (u) + =0 forx ∈ R, t > 0, (1) ∂t ∂x u(x, 0) = u0 (x) for x ∈ R. (2) The classical well-posedness theory for the Cauchy problem (1)–(2) is available for L∞ and BV initial data [35,36,41]. BV -regularizing effect on entropy solutions has been established in [36,41] for uniformly convex fluxes. It is well know that if the flux function is not uniformly convex then in general, the entropy solution of (1) may not have a finite total variation, [4,21]. It can be shown that in one dimension if f vanishes at some point then there exists a class of initial data such that f can not regularize the corresponding entropy solution up to BV for all time [28]. Hence, to understand the optimal regularity of the entropy solution of (1), one works with more general space like fractional Sobolev space W s,p and fractional BV spaces BV s , 0 < s < 1, 1 ≤ p. The advantag
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