We obtain new fine properties of entropy solutions to scalar nonlinear conservation laws. For this purpose, we study the "fractional BV spaces" denoted by BV s (R) (for 0 < s ≤ 1), which were introduced by Love and Young in 1937 and closely related to the critical Sobolev space W s,1/s (R). We investigate these spaces in connection with one-dimensional scalar conservation laws. The BV s spaces allow one 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 Lions, Perthame and Tadmor for all nonlinear (possibly degenerate) convex fluxes.
In this paper, the main objective is to generalize to the Navier-Stokes-Korteweg (with density dependent viscosities satisfying the BD relation) and Euler-Korteweg systems a recent relative entropy [proposed by D. Bresch, P. Noble and J.-P. Vila, (2016)] introduced for the compressible Navier-Stokes equations with a linear density dependent shear viscosity and a zero bulk viscosity. As a concrete application, this helps to justify mathematically the convergence between global weak solutions of the quantum Navier-Stokes system [recently obtained simultaneously by I. Lacroix-Violet and A. Vasseur (2017)] and dissipative solutions of the quantum Euler system when the viscosity coefficient tends to zero: This selects a dissipative solution as the limit of a viscous system. We also get weak-strong uniqueness for the Quantum-Euler and for the Quantum-Navier-Stokes equations. Our results are based on the fact that Euler-Korteweg systems and corresponding Navier-Stokes-Korteweg systems can be reformulated through an augmented system such as the compressible Navier-Stokes system with density dependent viscosities satisfying the BD algebraic relation. This was also observed recently [by D. Bresch, F. Couderc, P. Noble and J.-P. Vila, (2016)] for the Euler-Korteweg system for numerical purposes. As a by-product of our analysis, we show that this augmented formulation helps to define relative entropy estimates for the Euler-Korteweg systems in a simplest way compared to recent works [See D. Donatelli, E. Feireisl, P. Marcati (2015) and J. Giesselmann, C. Lattanzio, A.-E. Tzavaras (2017)] with less hypothesis required on the capillary coefficient.
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