Entropy and the Stability of Classical Solutions

Recently, it has been shown [M. Polettini et al., 12th Joint European Thermodynamics Conference, Brescia, Italy, July 1-5, 2013] that dieomorphisms in the state space of a wide class of physical systems leave the amount Π of entropy produced per unit time inside the bulk of the system unaected. Starting from this invariance, we show that if the boundary conditions allow the system to relax towards some nal ('relaxed') state, then the necessary condition for the stability of the relaxed state against slowly evolving perturbations is the same for all the systems of this class, regardless of the detailed dynamics of the system, the amplitude of uctuations around mean values, and the possible occurrence of periodic oscillations in the relaxed state. We invoke also no Onsager symmetry, no detailed model of heat transport and production, and no approximation of local thermodynamic equilibrium. This necessary condition is the constrained minimization of suitably time-and path-ensemble-averaged Π [G. E. Crooks, Phys. Rev. E 61, 3, 2361 (2000)], the constraints being provided by the equations of motion. Even if the latter may dier, the minimum property of stable relaxed states remains the same in dierent systems. Relaxed states satisfy the equations of motion; stable relaxed states satises also the minimum condition. (Thermodynamic equilibrium corresponds to Π ≡ 0 and satises trivially this condition). Our class of physical systems includes both some mesoscopic systems described by a probability distribution function which obeys a simple Fokker-Planck equation and some macroscopic, classical uids with no net mass source and no mass ow across the boundaries; the problems of stability of these systems are dual to each other. As particular cases, we retrieve some necessary criteria for stability of relaxed states in both macroscopic and mesoscopic systems. Retrieved criteria include (among others) constrained minimization of dissipated power for plasma laments in a Dense Plasma Focus [A.

Grundlehren der mathematischen Wissenschaften, 2010

2005

We discuss the different roles of the entropy principle in modern thermodynamics. We start with the approach of rational thermodynamics in which the entropy principle becomes a selection rule for physical constitutive equations. Then we discuss the entropy principle for selecting admissible discontinuous weak solutions and to symmetrize general systems of hyperbolic balance laws. A particular attention is given on the local and global well-posedness of the relative Cauchy problem for smooth solutions. Examples are given in the case of extended thermodynamics for rarefied gases and in the case of a multi-temperature mixture of fluids.

Quarterly of Applied Mathematics, 2020

General hyperbolic systems of balance laws with inhomogeneity in space and time in all constitutive functions are studied in the context of relative entropy. A framework is developed in this setting that contributes to a measure-valued weak vs. strong uniqueness theorem, a stability theorem of viscous solutions and a convergence theorem as the viscosity parameter tends to zero. The main goal of this paper is to develop hypotheses under which the relative entropy framework can still be applied. Examples of systems with inhomogeneity that have different characteristics are presented and the hypotheses are discussed in the setting of each example.

Communications in Mathematical Sciences, 2014

We present a general framework for the approximation of systems of hyperbolic balance laws. The novelty of the analysis lies in the construction of suitable relaxation systems and the derivation of a delicate estimate on the relative entropy. We provide a direct proof of convergence in the smooth regime for a wide class of physical systems. We present results for systems arising in materials science, where the presence of source terms presents a number of additional challenges and requires delicate treatment. Our analysis is in the spirit of the framework introduced by Tzavaras [23] for systems of hyperbolic conservation laws.

Entropy, 2008

We discuss the different roles of the entropy principle in modern thermodynamics. We start with the approach of rational thermodynamics in which the entropy principle becomes a selection rule for physical constitutive equations. Then we discuss the entropy principle for selecting admissible discontinuous weak solutions and to symmetrize general systems of hyperbolic balance laws. A particular attention is given on the local and global well-posedness of the relative Cauchy problem for smooth solutions. Examples are given in the case of extended thermodynamics for rarefied gases and in the case of a multi-temperature mixture of fluids.

Archive for Rational Mechanics and Analysis, 2015

ABSTRACT We consider by a combination of analytical and numerical techniques some basic questions regarding the relations between inviscid and viscous stability and existence of a convex entropy. Specifically, for a system possessing a convex entropy, in particular for the equations of gas dynamics with a convex equation of state, we ask: (i) can inviscid instability occur? (ii) can there occur viscous instability not detected by inviscid theory? (iii) can there occur the ---necessarily viscous--- effect of Hopf bifurcation, or "galloping instability"? and, perhaps most important from a practical point of view, (iv) as shock amplitude is increased from the (stable) weak-amplitude limit, can there occur a first transition from viscous stability to instability that is not detected by inviscid theory? We show that (i) does occur for strictly hyperbolic, genuinely nonlinear gas dynamics with certain convex equations of state, while (ii) and (iii) do occur for an artifically constructed system with convex viscosity-compatible entropy. We do not know of an example for which (iv) occurs, leaving this as a key open question in viscous shock theory, related to the principal eigenvalue property of Sturm Liouville and related operators. In analogy with, and partly proceeding close to, the analysis of Smith on (non-)uniqueness of the Riemann problem, we obtain convenient criteria for shock (in)stability in the form of necessary and sufficient conditions on the equation of state.

Archive for Rational Mechanics and Analysis, 2002

We consider the Cauchy problem for n × n strictly hyperbolic systems of nonresonant balance laws

Communications in Mathematical Physics, 2022

Using methods from Convex Analysis, for each generalized pressure function we define an upper semi-continuous affine entropy-like map, establish an abstract variational principle for both countably and finitely additive probability measures and prove that equilibrium states always exist. We show that this conceptual approach imparts a new insight on dynamical systems without a measure with maximal entropy, may be used to detect second-order phase transitions, prompts the study of finitely additive ground states for non-uniformly hyperbolic transformations and grants the existence of finitely additive Lyapunov equilibrium states for singular value potentials generated by linear cocycles over continuous self-maps.

Archive for Rational Mechanics and Analysis, 2003

We propose a new notion of weak solutions (dissipative solutions) for nonisotropic, degenerate, second order, quasi-linear parabolic equations. This class of solutions is an extension of the notion of dissipative solutions for scalar conservation laws introduced by L. C. Evans. We analyze the relationship between the notions of dissipative and entropy weak solutions for non-isotropic, degenerate, second order, quasi-linear parabolic equations. As an application we prove the strong convergence of a general relaxation-type approximation for such equations.