We give sufficient conditions on the regularity of solutions to the inhomogeneous incompressible Euler and the compressible isentropic Euler systems in order for the energy to be conserved. Our strategy relies on commutator estimates similar to those employed by P. Constantin et al. for the homogeneous incompressible Euler equations

We introduce a new concept of dissipative measure-valued solution to the compressible Navier-Stokes system satisfying, in addition, a relevant form of the total energy balance. Then we show that a dissipative measure-valued and a standard smooth classical solution originating from the same initial data coincide (weak-strong uniqueness principle) as long as the latter exists. Such a result facilitates considerably the proof of convergence of solutions to various approximations including certain numerical schemes that are known to generate a measure-valued solution. As a byproduct we show that any measure-valued solution with bounded density component that starts from smooth initial data is necessarily a classical one.

We consider unsteady flows of incompressible fluids with a general implicit constitutive equation relating the deviatoric part of the Cauchy stress S and the symmetric part of the velocity gradient D in such a way that it leads to a maximal monotone (possibly multivalued) graph and the rate of dissipation is characterized by the sum of a Young function depending on D and its conjugate being a function of S. Such a framework is very robust and includes, among others, classical power-law fluids, stress power-law fluids, fluids with activation criteria of Bingham or Herschel-Bulkley type, and shear rate-dependent fluids with discontinuous viscosities as special cases. The appearance of S and D in all the assumptions characterizing the implicit relationship G(D, S) = 0 is fully symmetric. We establish long-time and large-data existence of weak solution to such a system completed by the initial and the Navier slip boundary conditions in both the subcritical and supercritical cases. We use tools such as Orlicz functions, properties of spatially dependent maximal monotone operators, and Lipschitz approximations of Bochner functions taking values in Orlicz-Sobolev spaces.

This paper is devoted to the analysis of measure-valued solutions to a nonlinear structured population model given in the form of a nonlocal first-order hyperbolic problem on R + . We show global existence and Lipschitz continuity with respect to the model ingredients. In distinction to previous studies, where the L 1 norm was used, we apply the flat metric, similar to the Wasserstein W 1 distance. We argue that analysis using this metric, in addition to mathematical advantages, is consistent with intuitive understanding of empirical data. Lipschitz continuous dependence with respect to the model coefficients and initial data and the uniqueness of the weak solutions are shown under the assumption on the Lipschitz continuity of the kinetic functions. The proof of this result is based on the duality formula and the Gronwall-type argument.

ABSTRACT. We prove weak-strong uniqueness in the class of admissible measure-valued solutions for the isentropic Euler equations in any space dimension and for the Savage-Hutter model of granular flows in one and two space dimensions. For the latter system, we also show the complete dissipation of momentum in finite time, thus rigorously justifying an assumption that has been made in the engineering and numerical literature.