# The Inviscid Limit to a Contact Discontinuity for the Compressible Navier--Stokes--Fourier System Using the Relative Entropy Method

## Abstract

### Keywords

### MSC codes

## Get full access to this article

View all available purchase options and get full access to this article.

## References

*$L^p$ stability for entropy solutions of scalar conservation laws with strict convex flux*, J. Differential Equations, 256 (2014), pp. 3395--3416.

*Fluid dynamic limits of kinetic equations*, I.

*Formal derivations*, J. Statist. Phys., 63 (1991), pp. 323--344.

*Fluid dynamic limits of kinetic equations,*, Comm. Pure Appl. Math., 46 (1993), pp. 667--753.

*II*. Convergence proofs for the Boltzmann equation*From discrete velocity Boltzmann equations to gas dynamics before shocks*, J. Stat. Phys., 135 (2009), pp. 153--173.

*From kinetic equations to multidimensional isentropic gas dynamics before shocks*, SIAM J. Math. Anal., 36 (2005), pp. 1807--1835.

*Vanishing viscosity solutions of nonlinear hyperbolic systems*, Ann. of Math., 161 (2005), pp. 223--342.

*Divergence-measure fields and hyperbolic conservation laws*, Arch. Ration. Mech. Anal., 147 (1999), pp. 89--119.

*Large-time behavior of entropy solutions of conservation laws*, J. Differential Equations, 152 (1999), pp. 308--357.

*Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations*, Trans. Amer. Math. Soc., 353 (2001), pp. 1103--1117.

*Extended divergence-measure fields and the Euler equations for gas dynamics*, Comm. Math. Phys., 236 (2003), pp. 251--280.

*Uniqueness and stability of Riemann solutions with large oscillation in gas dynamics*, Comm. Math. Phys., 228 (2002), pp. 201--217.

*Short-time stability of scalar viscous shocks in the inviscid limit by the relative entropy method*, SIAM J. Math. Anal., 47 (2015), pp. 1405--1418.

*The second law of thermodynamics and stability*, Arch. Ration. Mech. Anal., 70 (1979), pp. 167--179.

*Hyperbolic Conservation Laws in Continuum Physics*, Grundlehren Math. Wiss. 325, Springer-Verlag, Berlin, 2000.

*Uniqueness of solutions to hyperbolic conservation laws*, Indiana Univ. Math. J., 28 (1979), pp. 138--188.

*Dynamics of Viscous Compressible Fluids*, Oxford Lecture Ser. Math. Appl. 26, Oxford University Press, Oxford, UK, 2004.

*The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels*, Invent. Math., 155 (2004), pp. 81--161.

*Viscous limits for piecewise smooth solutions to systems of conservation laws*, Arch. Ration. Mech. Anal., 121 (1992), pp. 235--265.

*Global solutions of the equations of one-dimensional, compressible flow with large data and forces, and with differing end states*, Z. Angew. Math. Phys., 49 (1998), pp. 774--785.

*The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data*, Indiana Univ. Math. J., 38 (1989), pp. 861--915.

*Hydrodynamic limit of the Boltzmann equation with contact discontinuities*, Comm. Math. Phys., 295 (2010), pp. 293--326.

*Fluid dynamic limit to the Riemann solutions of Euler equations:*, Kinet. Relat. Models, 3 (2010), pp. 685--728.

*I*, Superposition of rarefaction waves and contact discontinuity*Vanishing viscosity limit of the compressible Navier-Stokes equations for solutions to Riemann problem*, Arch. Ration. Mech. Anal., 203 (2012), pp. 379--413.

*The limit of the Boltzmann equation to the Euler equations*, SIAM J. Math. Anal., 45 (2013), pp. 1741--1811.

*$L^2$-Contraction for Shock Waves of Scalar Viscous Conservation Laws*, preprint.

*Criteria on Contractions for Entropic Discontinuities of Systems of Conservation Laws*, preprint.

*Unique global solution with respect to time of initial- boundary value problems for one-dimensional equations of a viscous gas*, J. Appl. Math. Mech., 41 (1977), pp. 273--282.

*Asymptotic limit to shocks for scalar balance laws using relative entropy*, Abstr. Appl. Anal. (2014), 690801.

*Asymptotic limit to a shock for BGK models using the relative entropy method*, Nonlinearity, 28 (2015), pp. 531--543.

*Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equations of one-dimensional compressible heat-conducting fluids*, SIAM J. Math. Anal., 38 (2006), pp. 368--384.

*$L^2$ stability estimates for shock solutions of scalar conservation laws using the relative entropy method*, Arch. Ration. Mech. Anal., 199 (2011), pp. 761--778.

*Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non-BV perturbations*, Arch. Ration. Mech. Anal., 201 (2011), pp. 271--302

*From the Boltzmann equations to the equations of incompressible fluid mechanics*, I, Arch. Ration. Mech. Anal., 158 (2001), pp. 173--193.

*From the Boltzmann equations to the equations of incompressible fluid mechanics*, II, Arch. Ration. Mech. Anal., 158 (2001), pp. 195--211.

*Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations*, J. Differential Equations, 248 (2010), pp. 95--110.

*From the Boltzmann equation to the Stokes-Fourier system in a bounded domain*, Comm. Pure Appl. Math., 56 (2003), pp. 1263--1293.

*$L^2$-type contraction for systems of conservation laws*, J. École Polytechnique Math., 1 (2014), pp. 1--28.

*About the Relative Entropy Method for Hyperbolic Systems of Conservation Laws*, preprint, 2015.

*Shock Waves and Reaction-Diffusion Equations*, 2nd ed., Springer-Verlag, New York, 1994.

*Recent results on hydrodynamic limits*, Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ. 4, Elsevier/North-Holland, Amsterdam, 2008, pp. 323--376.

*Zero dissipation limit of the compressible heat-conducting Navier-Stokes equations in the presence of the shock*, Acta Math. Sci., 28B (2008), pp. 727--748.

*Relative entropy and hydrodynamics of Ginzburg-Landau models*, Lett. Math. Phys., 22 (1991), pp. 63--80.

## Information & Authors

### Information

#### Published In

#### Copyright

#### History

**Submitted**: 28 May 2015

**Accepted**: 15 September 2015

**Published online**: 10 November 2015

#### Keywords

#### MSC codes

### Authors

## Metrics & Citations

### Metrics

### Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

#### Cited By

- Uniqueness of Composite Wave of Shock and Rarefaction in the Inviscid Limit of Navier–Stokes EquationsSIAM Journal on Mathematical Analysis, Vol. 56, No. 3 | 4 June 2024
- Vanishing viscosity limit to rarefaction wave with vacuum for an ionized plasmaMathematical Models and Methods in Applied Sciences, Vol. 33, No. 14 | 18 November 2023
- A review of recent applications of the relative entropy method to discontinuous solutions of conservation lawsQuarterly of Applied Mathematics, Vol. 81, No. 3 | 26 April 2023
- Well-posedness of the Riemann problem with two shocks for the isentropic Euler system in a class of vanishing physical viscosity limitsJournal of Differential Equations, Vol. 338 | 1 Nov 2022
- Uniqueness of a Planar Contact Discontinuity for 3D Compressible Euler System in a Class of Zero Dissipation Limits from Navier–Stokes–Fourier SystemCommunications in Mathematical Physics, Vol. 384, No. 3 | 8 May 2021
- Uniqueness and stability of entropy shocks to the isentropic Euler system in a class of inviscid limits from a large family of Navier–Stokes systemsInventiones mathematicae, Vol. 224, No. 1 | 15 October 2020
- Finite time stability for the Riemann problem with extremal shocks for a large class of hyperbolic systemsJournal of Differential Equations, Vol. 273 | 1 Feb 2021
- L2-type contraction for shocks of scalar viscous conservation laws with strictly convex fluxJournal de Mathématiques Pures et Appliquées, Vol. 145 | 1 Jan 2021
- Study of roof water inrush forecasting based on EM-FAHP two-factor modelMathematical Biosciences and Engineering, Vol. 18, No. 5 | 1 Jan 2021
- Contraction for large perturbations of traveling waves in a hyperbolic–parabolic system arising from a chemotaxis modelMathematical Models and Methods in Applied Sciences, Vol. 30, No. 02 | 28 January 2020
- Stability and Uniqueness for Piecewise Smooth Solutions to a Nonlocal Scalar Conservation Law with Applications to Burgers--Hilbert EquationSIAM Journal on Mathematical Analysis, Vol. 52, No. 3 | 20 May 2020
- L2-contraction of large planar shock waves for multi-dimensional scalar viscous conservation lawsJournal of Differential Equations, Vol. 267, No. 5 | 1 Aug 2019
- On uniqueness of solutions to conservation laws verifying a single entropy conditionJournal of Hyperbolic Differential Equations, Vol. 16, No. 01 | 24 May 2019
- Stability of stationary solutions of singular systems of balance lawsConfluentes Mathematici, Vol. 10, No. 2 | 3 March 2019
- The stability of contact discontinuity for compressible planar magnetohydrodynamicsKinetic & Related Models, Vol. 10, No. 4 | 1 Jan 2017

## View Options

### Get Access

**Access via your Institution**- Questions about how to access this content? Contact SIAM at
**[email protected]**.