Fractional BV spaces and first applications to scalar conservation laws

Christian Bourdarias (1) , Marguerite Gisclon (1, 2) , Stéphane Junca (3, 4)

1 LAMA - Laboratoire de Mathématiques
2 EDP - Equations aux Dérivées Partielles
3 COFFEE - COmplex Flows For Energy and Environment
4 EDP

The aim of this paper is to obtain new fine properties of entropy solutions of nonlinear scalar conservation laws. For this purpose, we study some ''fractional $BV$ spaces'' denoted $BV^s$, for $0 < s \leq 1$, introduced by Love and Young in 1937. The $BV^s(\mathbb{R})$ spaces are very closed to the critical Sobolev space $W^{s,1/s}(\mathbb{R})$. We investigate these spaces in relation with one-dimensional scalar conservation laws. $BV^s$ spaces allow to work with less regular functions than BV functions and appear to be more natural in this context. We obtain a stability result for entropy solutions with $BV^s$ initial data. Furthermore, for the first time we get the maximal $W^{s,p}$ smoothing effect conjectured by P.-L. Lions, B. Perthame and E. Tadmor for all nonlinear degenerate convex fluxes.

Mots clés: conservation laws; Riemann problem; entropy solution; fractional BV spaces; nonlinear convex flux; hyperbolicity

Domaines: Equations aux dérivées partielles [math.AP]

Christian Bourdarias, Marguerite Gisclon, Stéphane Junca. Fractional BV spaces and first applications to scalar conservation laws. Journal of Hyperbolic Differential Equations, 2014, 11 (4), pp.655-677. ⟨10.1142/S0219891614500209⟩. ⟨hal-00785747⟩