September  2020, 15(3): 369-387. doi: 10.3934/nhm.2020023

Relative entropy method for the relaxation limit of hydrodynamic models

1. 

Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom

2. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China and, Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom

3. 

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland

* Corresponding author: José Antonio Carrillo

Received  October 2019 Revised  April 2020 Published  September 2020

We show how to obtain general nonlinear aggregation-diffusion models, including Keller-Segel type models with nonlinear diffusions, as relaxations from nonlocal compressible Euler-type hydrodynamic systems via the relative entropy method. We discuss the assumptions on the confinement and interaction potentials depending on the relative energy of the free energy functional allowing for this relaxation limit to hold. We deal with weak solutions for the nonlocal compressible Euler-type systems and strong solutions for the limiting aggregation-diffusion equations. Finally, we show the existence of weak solutions to the nonlocal compressible Euler-type systems satisfying the needed properties for completeness sake.

Citation: José Antonio Carrillo, Yingping Peng, Aneta Wróblewska-Kamińska. Relative entropy method for the relaxation limit of hydrodynamic models. Networks & Heterogeneous Media, 2020, 15 (3) : 369-387. doi: 10.3934/nhm.2020023
References:
[1]

P. Antonelli and P. Marcati, The quantum hydrodynamics system in two space dimensions, Arch. Rational Mech. Anal., 203 (2012), 499-527.  doi: 10.1007/s00205-011-0454-7.  Google Scholar

[2]

P. Antonelli and P. Marcati, On the finite energy weak solutions to a system in quantum fluid dynamics, Comm. Math. Physics, 287 (2009), 657-686.  doi: 10.1007/s00220-008-0632-0.  Google Scholar

[3]

A. BlanchetJ. A. Carrillo and P. Laurencot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168.  doi: 10.1007/s00526-008-0200-7.  Google Scholar

[4]

D. BrandonT. Lin and R. C. Rogers, Phase transitions and hysteresis in nonlocal and order-parameter models, Meccanica, 30 (1995), 541-565.  doi: 10.1007/BF01557084.  Google Scholar

[5]

V. CalvezJ. A. Carrillo and F. Hoffmann, Equilibria of homogeneous functionals in the fair-competition regime, Nonlinear Anal., 159 (2017), 85-128.  doi: 10.1016/j.na.2017.03.008.  Google Scholar

[6]

J. A. Carrillo and Y.-P. Choi, Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 925–954, arXiv: 1901.07204. doi: 10.1016/j.anihpc.2020.02.001.  Google Scholar

[7]

J. A. CarrilloY.-P. Choi and O. Tse, Convergence to equilibrium in Wasserstein distance for damped Euler equations with interaction forces, Comm. Math. Phys., 365 (2019), 329-361.  doi: 10.1007/s00220-018-3276-8.  Google Scholar

[8]

J. A. CarrilloE. FeireislP. Gwiazda and A. Świerczewska-Gwiazda, Weak solutions for Euler systems with non-local interactions, J. London Math. Soc., 95 (2017), 705-724.  doi: 10.1112/jlms.12027.  Google Scholar

[9]

J. A. Carrillo, F. Hoffmann, E. Mainini and B. Volzone, Ground states in the diffusion-dominated regime, Calc. Var. Partial Differential Equations, 57 (2018), Paper No. 127, 28 pp. doi: 10.1007/s00526-018-1402-2.  Google Scholar

[10]

J. A. CarrilloR. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263.  doi: 10.1007/s00205-005-0386-1.  Google Scholar

[11]

J. A. CarrilloR. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), 971-1018.  doi: 10.4171/RMI/376.  Google Scholar

[12]

L. ChenL. Hong and J. Wang, Parabolic elliptic type Keller-Segel system on the whole space case, Discrete Contin. Dyn. Syst., 36 (2016), 1061-1084.  doi: 10.3934/dcds.2016.36.1061.  Google Scholar

[13]

E. Chiodaroli, A counterexample to well-posedness of entropy solutions to the compressible Euler system, J. Hyperbolic Differ. Eqs., 11 (2014), 493-519.  doi: 10.1142/S0219891614500143.  Google Scholar

[14]

E. ChiodaroliE. Feireisl and O. Kreml, On the weak solutions to the equations of a compressible heat conducting gas, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 225-243.  doi: 10.1016/j.anihpc.2013.11.005.  Google Scholar

[15]

J.-F. Coulombel and T. Goudon, The strong relaxation limit of the multidimensional isothermal Euler equations, Trans. Am. Math. Soc., 359 (2007), 637-648.  doi: 10.1090/S0002-9947-06-04028-1.  Google Scholar

[16]

C. M. Dafermos, Stability of motions of thermoelastic fluids, J. Thermal Stresses, 2 (1979), 127-134.  doi: 10.1080/01495737908962394.  Google Scholar

[17]

C. M. Dafermos, The second law of thermodynamics and stability, Arch. Ration. Mech. Anal., 70 (1979), 167-179.  doi: 10.1007/BF00250353.  Google Scholar

[18]

C. De Lellis and L. Székelyhidi, On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal., 195 (2010), 225-260.  doi: 10.1007/s00205-008-0201-x.  Google Scholar

[19]

R. J. Diperna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J., 28 (1979), 137-188.  doi: 10.1512/iumj.1979.28.28011.  Google Scholar

[20]

D. DonatelliE. Feireisl and P. Marcati, Well/ill posedness for the Euler-Korteweg-Poisson system and related problems, Commun. Partial Differential Equations, 40 (2015), 1314-1335.  doi: 10.1080/03605302.2014.972517.  Google Scholar

[21]

J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal., 88 (1985), 95-133.  doi: 10.1007/BF00250907.  Google Scholar

[22]

E. Feireisl, Weak solutions to problems involving inviscid fluids, Mathematical Fluid Dynamics, Present and Future, Springer Proc. Math. Stat., Springer, Tokyo, 183 (2016), 377-399.  doi: 10.1007/978-4-431-56457-7_13.  Google Scholar

[23]

E. FeireislP. Gwiazda and A. Świerczewska-Gwiazda, On weak solutions to the 2d Savage-Hutter model of the motion of a gravity driven avalanche flow, Comm. Partial Differential Equations, 41 (2016), 759-773.  doi: 10.1080/03605302.2015.1127968.  Google Scholar

[24]

J. GiesselmannC. Lattanzio and A. E. Tzavaras, Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics, Arch. Rational Mech. Anal., 223 (2017), 1427-1484.  doi: 10.1007/s00205-016-1063-2.  Google Scholar

[25]

B. D. GoddardG. A. Pavliotis and S. Kalliadasis, The overdamped limit of dynamic density functional theory: Rigorous results, Multiscale Model. Simul., 10 (2012), 633-663.  doi: 10.1137/110844659.  Google Scholar

[26]

F. HuangP. Marcati and R. Pan, Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 176 (2005), 1-24.  doi: 10.1007/s00205-004-0349-y.  Google Scholar

[27]

F. Huang and R. Pan, Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum, J. Differ. Eqs., 220 (2006), 207–233. doi: 10.1016/j.jde.2005.03.012.  Google Scholar

[28]

F. HuangR. Pan and Z. Wang, $L^1$ convergence to the Barenblatt solution for compressible Euler equations with damping, Arch. Ration. Mech. Anal., 200 (2011), 665-689.  doi: 10.1007/s00205-010-0355-1.  Google Scholar

[29]

S. JiangQ. JuH. Li and Y. Li, Quasi-neutral limit of the two-fluid Euler-Poisson system, Commum. Pure Appl. Anal., 9 (2010), 1577-1590.  doi: 10.3934/cpaa.2010.9.1577.  Google Scholar

[30]

S. Junca and M. Rascle, Strong relaxation of the isothermal Euler system to the heat equation, Z. Angew. Math. Phys., 53 (2002), 239-264.  doi: 10.1007/s00033-002-8154-7.  Google Scholar

[31]

C. Lattanzio and A. E. Tzavaras, From gas dynamics with large friction to gradient flows describing diffusion theories, Comm. Partial Differential Equations, 42 (2017), 261–290. doi: 10.1080/03605302.2016.1269808.  Google Scholar

[32]

C. Lattanzio and A. E. Tzavaras, Relative entropy in diffusive relaxation, SIAM J. Math. Anal., 45 (2013), 1563-1584.  doi: 10.1137/120891307.  Google Scholar

[33]

T. Luo and J. Smoller, Existence and non-linear stability of rotating star solutions of the compressible Euler-Poisson equations, Arch. Ration. Mech. Anal., 191 (2009), 447-496.  doi: 10.1007/s00205-007-0108-y.  Google Scholar

[34]

P. Marcati and A. Milani, The one-dimensional Darcy's law as the limit of a compressible Euler flow, J. Differ. Eqs., 84 (1990), 129–147. doi: 10.1016/0022-0396(90)90130-H.  Google Scholar

[35]

Y.-J. Peng and Y.-G. Wang, Convergence of compressible Euler-Poisson equations to incompressible type Euler equations, Asymptotic Anal., 41 (2005), 141-160.   Google Scholar

[36]

X. Ren and L. Truskinovsky, Finite scale microstructures in nonlocal elasticity, J. Elasticity, 59 (2000), 319-355.  doi: 10.1023/A:1011003321453.  Google Scholar

[37]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenrate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.   Google Scholar

[38]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058.  Google Scholar

show all references

References:
[1]

P. Antonelli and P. Marcati, The quantum hydrodynamics system in two space dimensions, Arch. Rational Mech. Anal., 203 (2012), 499-527.  doi: 10.1007/s00205-011-0454-7.  Google Scholar

[2]

P. Antonelli and P. Marcati, On the finite energy weak solutions to a system in quantum fluid dynamics, Comm. Math. Physics, 287 (2009), 657-686.  doi: 10.1007/s00220-008-0632-0.  Google Scholar

[3]

A. BlanchetJ. A. Carrillo and P. Laurencot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168.  doi: 10.1007/s00526-008-0200-7.  Google Scholar

[4]

D. BrandonT. Lin and R. C. Rogers, Phase transitions and hysteresis in nonlocal and order-parameter models, Meccanica, 30 (1995), 541-565.  doi: 10.1007/BF01557084.  Google Scholar

[5]

V. CalvezJ. A. Carrillo and F. Hoffmann, Equilibria of homogeneous functionals in the fair-competition regime, Nonlinear Anal., 159 (2017), 85-128.  doi: 10.1016/j.na.2017.03.008.  Google Scholar

[6]

J. A. Carrillo and Y.-P. Choi, Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 925–954, arXiv: 1901.07204. doi: 10.1016/j.anihpc.2020.02.001.  Google Scholar

[7]

J. A. CarrilloY.-P. Choi and O. Tse, Convergence to equilibrium in Wasserstein distance for damped Euler equations with interaction forces, Comm. Math. Phys., 365 (2019), 329-361.  doi: 10.1007/s00220-018-3276-8.  Google Scholar

[8]

J. A. CarrilloE. FeireislP. Gwiazda and A. Świerczewska-Gwiazda, Weak solutions for Euler systems with non-local interactions, J. London Math. Soc., 95 (2017), 705-724.  doi: 10.1112/jlms.12027.  Google Scholar

[9]

J. A. Carrillo, F. Hoffmann, E. Mainini and B. Volzone, Ground states in the diffusion-dominated regime, Calc. Var. Partial Differential Equations, 57 (2018), Paper No. 127, 28 pp. doi: 10.1007/s00526-018-1402-2.  Google Scholar

[10]

J. A. CarrilloR. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263.  doi: 10.1007/s00205-005-0386-1.  Google Scholar

[11]

J. A. CarrilloR. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), 971-1018.  doi: 10.4171/RMI/376.  Google Scholar

[12]

L. ChenL. Hong and J. Wang, Parabolic elliptic type Keller-Segel system on the whole space case, Discrete Contin. Dyn. Syst., 36 (2016), 1061-1084.  doi: 10.3934/dcds.2016.36.1061.  Google Scholar

[13]

E. Chiodaroli, A counterexample to well-posedness of entropy solutions to the compressible Euler system, J. Hyperbolic Differ. Eqs., 11 (2014), 493-519.  doi: 10.1142/S0219891614500143.  Google Scholar

[14]

E. ChiodaroliE. Feireisl and O. Kreml, On the weak solutions to the equations of a compressible heat conducting gas, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 225-243.  doi: 10.1016/j.anihpc.2013.11.005.  Google Scholar

[15]

J.-F. Coulombel and T. Goudon, The strong relaxation limit of the multidimensional isothermal Euler equations, Trans. Am. Math. Soc., 359 (2007), 637-648.  doi: 10.1090/S0002-9947-06-04028-1.  Google Scholar

[16]

C. M. Dafermos, Stability of motions of thermoelastic fluids, J. Thermal Stresses, 2 (1979), 127-134.  doi: 10.1080/01495737908962394.  Google Scholar

[17]

C. M. Dafermos, The second law of thermodynamics and stability, Arch. Ration. Mech. Anal., 70 (1979), 167-179.  doi: 10.1007/BF00250353.  Google Scholar

[18]

C. De Lellis and L. Székelyhidi, On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal., 195 (2010), 225-260.  doi: 10.1007/s00205-008-0201-x.  Google Scholar

[19]

R. J. Diperna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J., 28 (1979), 137-188.  doi: 10.1512/iumj.1979.28.28011.  Google Scholar

[20]

D. DonatelliE. Feireisl and P. Marcati, Well/ill posedness for the Euler-Korteweg-Poisson system and related problems, Commun. Partial Differential Equations, 40 (2015), 1314-1335.  doi: 10.1080/03605302.2014.972517.  Google Scholar

[21]

J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal., 88 (1985), 95-133.  doi: 10.1007/BF00250907.  Google Scholar

[22]

E. Feireisl, Weak solutions to problems involving inviscid fluids, Mathematical Fluid Dynamics, Present and Future, Springer Proc. Math. Stat., Springer, Tokyo, 183 (2016), 377-399.  doi: 10.1007/978-4-431-56457-7_13.  Google Scholar

[23]

E. FeireislP. Gwiazda and A. Świerczewska-Gwiazda, On weak solutions to the 2d Savage-Hutter model of the motion of a gravity driven avalanche flow, Comm. Partial Differential Equations, 41 (2016), 759-773.  doi: 10.1080/03605302.2015.1127968.  Google Scholar

[24]

J. GiesselmannC. Lattanzio and A. E. Tzavaras, Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics, Arch. Rational Mech. Anal., 223 (2017), 1427-1484.  doi: 10.1007/s00205-016-1063-2.  Google Scholar

[25]

B. D. GoddardG. A. Pavliotis and S. Kalliadasis, The overdamped limit of dynamic density functional theory: Rigorous results, Multiscale Model. Simul., 10 (2012), 633-663.  doi: 10.1137/110844659.  Google Scholar

[26]

F. HuangP. Marcati and R. Pan, Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 176 (2005), 1-24.  doi: 10.1007/s00205-004-0349-y.  Google Scholar

[27]

F. Huang and R. Pan, Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum, J. Differ. Eqs., 220 (2006), 207–233. doi: 10.1016/j.jde.2005.03.012.  Google Scholar

[28]

F. HuangR. Pan and Z. Wang, $L^1$ convergence to the Barenblatt solution for compressible Euler equations with damping, Arch. Ration. Mech. Anal., 200 (2011), 665-689.  doi: 10.1007/s00205-010-0355-1.  Google Scholar

[29]

S. JiangQ. JuH. Li and Y. Li, Quasi-neutral limit of the two-fluid Euler-Poisson system, Commum. Pure Appl. Anal., 9 (2010), 1577-1590.  doi: 10.3934/cpaa.2010.9.1577.  Google Scholar

[30]

S. Junca and M. Rascle, Strong relaxation of the isothermal Euler system to the heat equation, Z. Angew. Math. Phys., 53 (2002), 239-264.  doi: 10.1007/s00033-002-8154-7.  Google Scholar

[31]

C. Lattanzio and A. E. Tzavaras, From gas dynamics with large friction to gradient flows describing diffusion theories, Comm. Partial Differential Equations, 42 (2017), 261–290. doi: 10.1080/03605302.2016.1269808.  Google Scholar

[32]

C. Lattanzio and A. E. Tzavaras, Relative entropy in diffusive relaxation, SIAM J. Math. Anal., 45 (2013), 1563-1584.  doi: 10.1137/120891307.  Google Scholar

[33]

T. Luo and J. Smoller, Existence and non-linear stability of rotating star solutions of the compressible Euler-Poisson equations, Arch. Ration. Mech. Anal., 191 (2009), 447-496.  doi: 10.1007/s00205-007-0108-y.  Google Scholar

[34]

P. Marcati and A. Milani, The one-dimensional Darcy's law as the limit of a compressible Euler flow, J. Differ. Eqs., 84 (1990), 129–147. doi: 10.1016/0022-0396(90)90130-H.  Google Scholar

[35]

Y.-J. Peng and Y.-G. Wang, Convergence of compressible Euler-Poisson equations to incompressible type Euler equations, Asymptotic Anal., 41 (2005), 141-160.   Google Scholar

[36]

X. Ren and L. Truskinovsky, Finite scale microstructures in nonlocal elasticity, J. Elasticity, 59 (2000), 319-355.  doi: 10.1023/A:1011003321453.  Google Scholar

[37]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenrate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.   Google Scholar

[38]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058.  Google Scholar

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