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.
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.
We consider steady flows of incompressible fluids with power-law-like rheology given by an implicit constitutive equation relating the Cauchy stress and the symmetric part of the velocity gradient in such a way that it leads to a maximal monotone (possibly multivalued) graph. Such a framework includes standard Navier–Stokes and power-law fluids, Bingham fluids, Herschel–Bulkley fluids, and shear-rate dependent fluids with discontinuous viscosities as special cases. We assume that the fluid adheres to the boundary.
Using tools such as the Young measures, properties of spatially dependent maximal monotone operators and Lipschitz approximations of Sobolev functions, we are able to extend the results concerning large data existence of weak solutions to those values of the power-law index that are of importance from the point of view of engineering and physical applications.
Abstract. Our research is directed to a quasi-static evolution of the thermo-visco-elastic model. We assume that the material is subject to two kinds of mechanical deformations: elastic and inelastic. Moreover, our analysis captures the influence of the temperature on the visco-elastic properties of the body. The novelty of the paper is the consideration of the thermodynamically complete model to describe this kind of phenomena related with a hardening rule of Norton-Hoff type. We provide the proof of existence of solutions to thermovisco-elastic model in a simplified setting, namely the thermal expansion effects are neglected. Consequently, the coupling between the temperature and the displacement occurs only in the constitutive function for the evolution of the visco-elastic strain.
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