Relative Entropy in Diffusive Relaxation

Lattanzio, Corrado; Tzavaras, Athanasios E.

doi:10.1137/120891307

Cited by 70 publications

(102 citation statements)

References 28 publications

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“…It should be compared to the well known relative entropy formulas initiated in the works Dafermos [11,12], DiPerna [15] and analogs that have been successfully used in many contexts (e.g. [27,30,20,8,28,22]). It has however a different origin from all these calculations: while the latter are based on the thermodynamical structure induced by the Clausius-Duhem inequality, the formula (1.7) is based on the abstract Hamiltonian flow structure in (1.1).…”

Section: Introductionmentioning

confidence: 99%

Relative Energy for the Korteweg Theory and Related Hamiltonian Flows in Gas Dynamics

Giesselmann

Lattanzio

Tzavaras

2016

Arch Rational Mech Anal

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103

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Abstract. We consider an Euler system with dynamics generated by a potential energy functional. We propose a form for the relative energy that exploits the variational structure and derive a relative energy identity. When applied to specific energies, this yields relative energy identities for the Euler-Korteweg, the Euler-Poisson, the Quantum Hydrodynamics system, and low order approximations of the Euler-Korteweg system. For the Euler-Korteweg system we prove a stability theorem between a weak and a strong solution and an associated weak-strong uniqueness theorem. In the second part we focus on the Navier-Stokes-Korteweg system (NSK) with non-monotone pressure laws: we prove stability for the NSK system via a modified relative energy approach. We prove continuous dependence of solutions on initial data and convergence of solutions of a low order model to solutions of the NSK system. The last two results provide physically meaningful examples of how higher order regularization terms enable the use of the relative energy framework for models with energies which are not poly-or quasi-convex, compensated by higher-order gradients.

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Section: Introductionmentioning

confidence: 99%

Relative Energy for the Korteweg Theory and Related Hamiltonian Flows in Gas Dynamics

Giesselmann

Lattanzio

Tzavaras

2016

Arch Rational Mech Anal

Self Cite

103

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“…In Lattanzio and Tzavaras (but see also Huang et al) it has been shown that the sequence of densities

ρ_{ε} > 0

solving the initial value problems for converge for

ε \to 0

towards a solution

\overset{ρ}{true}

of the IVP for the porous medium equation

\partial_{t} \overset{ρ}{true} - M^{- 1} div false(\nabla p false(\overset{ρ}{true} false) false) = 0 in {double-struckR}^{d} \times false(0, T false) .

…”

Section: Compressible Flow In Porous Mediamentioning

confidence: 99%

“…Here, we rely on the technical framework that has been established in Lattanzio and Tzavaras. We start to prove a dissipation relation (Proposition ) for so‐called relative entropies in Section and conclude the convergence estimate with the main result in Theorem of Section .…”

Section: Convergence To the Parabolic Limit Systemmentioning

confidence: 99%

“…Remark If the initial datum

false({boldr}_{0}^{ε}, {boldm}_{0}^{ε} false)

converges for

ε \to 0

false({\overset{boldr}{true}}_{0}, \overset{boldm}{true} false(\cdot, 0 false) false)

L_{l o c}^{2} false({double-struckR}^{d} false)

, the estimate implies

{‖ (r^{ε}, m^{ε}) (\cdot, t) - (\bar{r}, \bar{m}) (\cdot, t) ‖}_{L^{2} (R^{d})} \to 0

due to the compatibility relation . The condition holds especially for

λ_{i j} = 0, i, j = 1, \dots, n

what corresponds exactly to Lattanzio and Tzavaras . In gaseous mixtures, is expected to hold, see Remark . …”

Section: Convergence To the Parabolic Limit Systemmentioning

confidence: 99%

“…The condition (60) holds especially for i = 0, i, = 1, … , n what corresponds exactly to Lattanzio and Tzavaras. 17 In gaseous mixtures, (60) is expected to hold, see Remark 1.…”

Section: The Convergence Estimatementioning

confidence: 99%

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Compressible multicomponent flow in porous media with Maxwell‐Stefan diffusion

Ostrowski

Rohde

2020

Math Meth Appl Sci

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We introduce a Darcy‐scale model to describe compressible multicomponent flow in a fully saturated porous medium. In order to capture cross‐diffusive effects between the different species correctly, we make use of the Maxwell–Stefan theory in a thermodynamically consistent way. For inviscid flow, the model turns out to be a nonlinear system of hyperbolic balance laws. We show that the dissipative structure of the Maxwell‐Stefan operator permits to guarantee the existence of global classical solutions for initial data close to equilibria. Furthermore, it is proven by relative entropy techniques that solutions of the Darcy‐scale model tend in a certain long‐time regime to solutions of a parabolic limit system.

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Finite Volume Schemes for One-Dimensional Systems

Godlewski

Raviart

2020

Applied Mathematical Sciences

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Relative Entropy in Diffusive Relaxation

Cited by 70 publications

References 28 publications

Relative Energy for the Korteweg Theory and Related Hamiltonian Flows in Gas Dynamics

Relative Energy for the Korteweg Theory and Related Hamiltonian Flows in Gas Dynamics

Compressible multicomponent flow in porous media with Maxwell‐Stefan diffusion

Finite Volume Schemes for One-Dimensional Systems

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