Abstract. We design arbitrarily high-order accurate entropy stable schemes for systems of conservation laws. The schemes, termed TeCNO schemes, are based on two main ingredients: (i) high-order accurate entropy conservative fluxes and (ii) suitable numerical diffusion operators involving ENO reconstructed cell-interface values of scaled entropy variables. Numerical experiments in one and two space dimensions are presented to illustrate the robust numerical performance of the TeCNO schemes.

There is no theory for the initial value problem for compressible flows in two space dimensions once shocks show up, much less in three space dimensions. This is a scientific scandal and a challenge." P. D. Lax, 2007 Gibbs Lecture [48] Abstract Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in [17,18] have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that demonstrates that state of the art numerical schemes may not necessarily converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions.Furthermore, we propose a more general notion, that of entropy measure valued solutions, as an appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure, which constructs stable approximations to entropy measure valued solutions and provide sufficient conditions that guarantee that these approximations converge to an entropy measure valued solution as the mesh is refined, thus providing a viable numerical framework for systems of conservation laws in several space dimensions. A large number of numerical experiments that illustrate the proposed schemes are presented and are utilized to examine several interesting properties of the computed entropy measure valued solutions.

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A standard paradigm for the existence of solutions in fluid dynamics is based on the construction of sequences of approximate solutions or approximate minimizers. This approach faces serious obstacles, most notably in multidimensional problems, where the persistence of oscillations at ever finer scales prevents compactness. Indeed, these oscillations are an indication, consistent with recent theoretical results, of the possible lack of existence/uniqueness of solutions within the standard framework of integrable functions. It is in this context that Young measures -parametrized probability measures which can describe the limits of such oscillatory sequences -offer the more general paradigm of measure-valued solutions for these problems. . † Supported in part by ERC STG. N 306279, SPARCCLE. ‡ Supported in part by NSF grants DMS10-08397, RNMS11-07444 (KI-Net) and ONR grant N00014-1512094. 568 U. S. Fjordholm, S. Mishra and E. TadmorWe present viable numerical algorithms to compute approximate measurevalued solutions, based on the realization of approximate measures as laws of Monte Carlo sampled random fields. We prove convergence of these algorithms to measure-valued solutions for the equations of compressible and incompressible inviscid fluid dynamics, and present a large number of numerical experiments which provide convincing evidence for the viability of the new paradigm. We also discuss the use of these algorithms, and their extensions, in uncertainty quantification and contexts other than fluid dynamics, such as non-convex variational problems in materials science.

We seek to define statistical solutions of hyperbolic systems of conservation laws as time-parametrized probability measures on p-integrable functions. To do so, we prove the equivalence between probability measures on L p spaces and infinite families of correlation measures. Each member of this family, termed a correlation marginal, is a Young measure on a finite-dimensional tensor product domain and provides information about multi-point correlations of the underlying integrable functions. We also prove that any probability measure on a L p space is uniquely determined by certain moments (correlation functions) of the equivalent correlation measure.We utilize this equivalence to define statistical solutions of multi-dimensional conservation laws in terms of an infinite set of equations, each evolving a moment of the correlation marginal. These evolution equations can be interpreted as augmenting entropy measure-valued solutions, with additional information about the evolution of all possible multi-point correlation functions. Our concept of statistical solutions can accommodate uncertain initial data as well as possibly non-atomic solutions even for atomic initial data.For multi-dimensional scalar conservation laws we impose additional entropy conditions and prove that the resulting entropy statistical solutions exist, are unique and are stable with respect to the 1-Wasserstein metric on probability measures on L 1 .Global well-posedness (existence, uniqueness and continuous dependence on initial data) of entropy solutions of scalar conservation laws (N = 1 in (1.1)), was established in the pioneering work of Kruzkhov [38]. For one-dimensional systems (d = 1, N > 1 in (1.1)), global existence, under the assumption of small initial total variation, was shown by Glimm in [32] and by Bianchini and Bressan in [6]. Uniqueness and stability of entropy solutions for one-dimensional systems has also been shown; see [8] and references therein.Although existence results have been obtained for some very specific examples of multi-dimensional systems (see [4] and references therein), there are no global existence results for any generic class of multi-dimensional systems. In fact, De Lellis, Székelyhidi et al. have recently been able to construct infinitely many entropy solutions for prototypical multi-dimensional systems such as the Euler equations for polytropic gas dynamics (see [16,17] and references therein). Their construction involves a novel iterative procedure where oscillations at smaller and smaller scales are successively added to suitably constructed sub-solutions of (1.1).Given the lack of global existence and uniqueness results for entropy solutions of multi-dimensional systems of conservation laws, it is natural to seek alternative solution paradigms. One option, advocated for instance in [3], is to augment entropy solutions with further admissibility criteria, such as the vanishing viscosity limit, in order to rule out "unphysical" solutions. However, given the difficulties of obtaining existence results f...