August  2022, 15(8): 2331-2343. doi: 10.3934/dcdss.2022066

$ \Gamma $-compactness and $ \Gamma $-stability of the flow of heat-conducting fluids

Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, 38050 Povo di (Trento), Italia

On the occasion of the 60th anniversary of Maurizio Grasselli with friendship

Received  January 2022 Published  August 2022 Early access  March 2022

The flow of a homogeneous, incompressible and heat conducting fluid is here described by coupling a quasilinear Navier-Stokes-type equation with the equation of heat diffusion, convection and buoyancy. This model is formulated variationally as a problem of null-minimization.

First we review how De Giorgi's theory of $ \Gamma $-convergence can be used to prove the compactness and the stability of evolutionary problems under nonparametric perturbations. Then we illustrate how this theory can be applied to the our problem of fluid and heat flow, and to more general coupled flows.

Citation: Augusto Visintin. $ \Gamma $-compactness and $ \Gamma $-stability of the flow of heat-conducting fluids. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2331-2343. doi: 10.3934/dcdss.2022066
References:
[1] A. Braides, Γ-Convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications, 22. Oxford University Press, Oxford, 2002.  doi: 10.1093/acprof:oso/9780198507840.001.0001.
[2]

H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175.  doi: 10.5802/aif.280.

[3]

H. Brezis and I. Ekeland, Un principe variationnel associé à certaines équations paraboliques. I. Le cas indépendant du temps and II. Le cas dépendant du temps, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), Ai, A1197–A1198.

[4]

F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Functional Analysis, 11 (1972), 251-294.  doi: 10.1016/0022-1236(72)90070-5.

[5]

M. Cessenat, Mathematical Modelling of Physical Systems, Springer, Cham, 2018. doi: 10.1007/978-3-319-94758-7.

[6]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Progress in Nonlinear Differential Equations and their Applications, 8. Birkhüser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.

[7]

E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 58 (1975), 842-850. 

[8]

G. Duvaut and J.L. Lions, Les Inéquations en Mécanique et en Physique, Travaux et Recherches Mathématiques, No. 21. Dunod, Paris, 1972.

[9]

S. Fitzpatrick, Representing monotone operators by convex functions, Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), Proc. Centre Math. Anal. Austral. Nat. Univ., Austral. Nat. Univ., Canberra, 20 (1988), 59-65. 

[10]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I. Theory, Mathematics and its Applications, 419. Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4.

[11]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.

[12] P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford University Press, New York, 1996. 
[13]

J. Málek and K. R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, Evolutionary equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2 (2005), 371-459. 

[14]

B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), A1035–A1038.

[15]

B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Second edition, Applied Mathematical Sciences, 91. Springer-Verlag, New York, 2004. doi: 10.1007/978-0-387-21740-6.

[16]

L. Tartar, An Introduction to Navier-Stokes Equation and Oceanography, Lecture Notes of the Unione Matematica Italiana, 1. Springer-Verlag, Berlin, UMI, Bologna, 2006. doi: 10.1007/3-540-36545-1.

[17]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Third edition, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co., Amsterdam, 1984.

[18]

A. Visintin, Variational formulation and structural stability of monotone equations, Calc. Var. Partial Differential Equations, 47 (2013), 273-317.  doi: 10.1007/s00526-012-0519-y.

[19]

A. Visintin, On Fitzpatrick's theory and stability of flows, Rend. Lincei Mat. Appl., 27 (2016), 151-180.  doi: 10.4171/RLM/729.

[20]

A. Visintin, $\Gamma$-compactness and $\Gamma$-stability of maximal monotone flows, J. Math. Anal. and Appl., 506 (2022), Paper No. 125602, 29 pp. doi: 10.1016/j.jmaa.2021.125602.

show all references

References:
[1] A. Braides, Γ-Convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications, 22. Oxford University Press, Oxford, 2002.  doi: 10.1093/acprof:oso/9780198507840.001.0001.
[2]

H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175.  doi: 10.5802/aif.280.

[3]

H. Brezis and I. Ekeland, Un principe variationnel associé à certaines équations paraboliques. I. Le cas indépendant du temps and II. Le cas dépendant du temps, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), Ai, A1197–A1198.

[4]

F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Functional Analysis, 11 (1972), 251-294.  doi: 10.1016/0022-1236(72)90070-5.

[5]

M. Cessenat, Mathematical Modelling of Physical Systems, Springer, Cham, 2018. doi: 10.1007/978-3-319-94758-7.

[6]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Progress in Nonlinear Differential Equations and their Applications, 8. Birkhüser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.

[7]

E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 58 (1975), 842-850. 

[8]

G. Duvaut and J.L. Lions, Les Inéquations en Mécanique et en Physique, Travaux et Recherches Mathématiques, No. 21. Dunod, Paris, 1972.

[9]

S. Fitzpatrick, Representing monotone operators by convex functions, Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), Proc. Centre Math. Anal. Austral. Nat. Univ., Austral. Nat. Univ., Canberra, 20 (1988), 59-65. 

[10]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I. Theory, Mathematics and its Applications, 419. Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4.

[11]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.

[12] P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford University Press, New York, 1996. 
[13]

J. Málek and K. R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, Evolutionary equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2 (2005), 371-459. 

[14]

B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), A1035–A1038.

[15]

B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Second edition, Applied Mathematical Sciences, 91. Springer-Verlag, New York, 2004. doi: 10.1007/978-0-387-21740-6.

[16]

L. Tartar, An Introduction to Navier-Stokes Equation and Oceanography, Lecture Notes of the Unione Matematica Italiana, 1. Springer-Verlag, Berlin, UMI, Bologna, 2006. doi: 10.1007/3-540-36545-1.

[17]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Third edition, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co., Amsterdam, 1984.

[18]

A. Visintin, Variational formulation and structural stability of monotone equations, Calc. Var. Partial Differential Equations, 47 (2013), 273-317.  doi: 10.1007/s00526-012-0519-y.

[19]

A. Visintin, On Fitzpatrick's theory and stability of flows, Rend. Lincei Mat. Appl., 27 (2016), 151-180.  doi: 10.4171/RLM/729.

[20]

A. Visintin, $\Gamma$-compactness and $\Gamma$-stability of maximal monotone flows, J. Math. Anal. and Appl., 506 (2022), Paper No. 125602, 29 pp. doi: 10.1016/j.jmaa.2021.125602.

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