Related papers: A note on weak solutions of conservation laws and …
In this article we focus our attention on the principle of energy conservation within the context of systems of fluid dynamics. We give an overview of results concerning the resolution of the famous Onsager conjecture - which states…
In this work we consider companion conservation laws to general systems of conservation laws. We investigate sufficient regularity for weak solutions to satisfy companion laws, assuming the fluxes to be $C^{1,\gamma}$, $0<\gamma<1$,…
Onsager's conjecture, which relates the conservation of energy to the regularity of weak solutions of the Euler equations, was completely resolved in recent years. In this work, we pursue an analogue of Onsager's conjecture in the context…
We consider a hydrodynamic model of self-organized evolution of agents, with singular interaction kernel $\phi_\alpha(x)=1/|x|^{1+\alpha}$ ($0<\alpha<2$), in the presence of an additional external force. Well-posedness results are already…
We investigate the relation between several generalized solution concepts for nonlinear PDE systems from fluid dynamics. More precisely, we study measure-valued solutions, dissipative weak solutions, and energy-variational solutions. For…
We consider the compressible isentropic Euler equations on $\mathbb{T}^d\times [0,T]$ with a pressure law $p\in C^{1,\gamma-1}$, where $1\le \gamma <2$. This includes all physically relevant cases, e.g.\ the monoatomic gas. We investigate…
We show that weak solutions of general conservation laws in bounded domains conserve their generalized entropy, and other respective companion laws, if they possess a certain fractional differentiability of order 1/3 in the interior of the…
We prove that bounded weak solutions of the compressible Euler equations will conserve thermodynamic entropy unless the solution fields have sufficiently low space-time Besov regularity. A quantity measuring kinetic energy cascade will also…
It is commonly believed as a fundamental principle that energy-momentum conservation of a physical system is the result of space-time symmetry. However, for classical particle-field systems, e.g., charged particles interacting through…
The existing paradox between theory and computational experiment for weak solutions of systems of conservation laws in higher space dimensions is arguably resolved. Apparently successful computations are identified with underlying…
Let $u(t,x)$ be the solution to the Cauchy problem of a scalar conservation law in one space dimension. It is well known that even for smooth initial data the solution can become discontinuous in finite time and global entropy weak solution…
In this paper, we are concerned with the minimal regularity of weak solutions implying the law of balance for both energy and helicity in the incompressible Euler equations. In the spirit of recent works due to Berselli [5] and…
In this article we study the principle of energy conservation for the Euler-Korteweg system. We formulate an Onsager-type sufficient regularity condition for weak solutions of the Euler-Korteweg system to conserve the total energy. The…
In this paper we give elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes equations in the class of H\"older continuous functions, relaxing some of the assumptions on the time variable (both…
We present the derivation of the hydrodynamic limit under Eulerian scaling for a general class of one-dimensional interacting particle systems with two or more conservation laws. Following Yau's relative entropy method it turns out that in…
We establish energy-balance for weak solutions of the stochastically forced incompressible Euler equations, enjoying H\"older regularity $C^{\alpha}$, $\alpha>1/3$. It is well known as the Onsager's conjecture for the deterministic…
The paper deals with the problem of the energy conservation for the weak solutions to the compressible Primitive Equations (CPE) system with degenerate viscosity. The sufficient conditions on the regularity of weak solutions for the energy…
Energy conservations are studied for inhomogeneous incompressible and compressible Euler equations with general pressure law in a torus or a bounded domain. We provide sufficient conditions for a weak solution to conserve the energy. By…
When thermodynamics is understood as the science (or art) of constructing effective models of natural phenomena by choosing a minimal level of description capable of capturing the essential features of the physical reality of interest, the…
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…