MR3052715
Luisa CONSIGLIERI
Luisa CONSIGLIERI
Existence of Proper Weak Solutions to the Navier-Stokes-Fourier System
Funkcialaj Ekvacioj. Serio Internacia
55
2012
347--365
http://fe.math.kobe-u.ac.jp/FE/FullPapers/55-3/55_347.pdf
http://www.ams.org/mathscinet-getitem?mr=MR3052715
The existence of proper weak solutions of the Cauchy problem constituted by the Navier-Stokes-Fourier system which characterizes the incompressible homogeneous Newtonian fluids under thermal effects is studied. We call <em>proper</em> weak solutions such weak solutions that verify some local energy inequalities in analogy with the suitable weak solutions for the Navier-Stokes equations.
Navier-Stokes-Fourier system, Joule effect, Suitable weak solutions.
76D03, 35Q30, 80A20.
55-347
2012
Existence of Proper Weak Solutions to the Navier-Stokes-Fourier System
Luisa CONSIGLIERI
Luisa CONSIGLIERI
1
Boccardo, L.; Dall'aglio, A.; Gallouet, T. ; Orsina, L.
Nonlinear parabolic equations with measure data
J. Funct. Anal.
147
1997
237-258
MR1453181
2
Boccardo, L.; Gallouet, T.
Non-linear elliptic and parabolic equations involving measure
J. Funct. Anal.
87
1989
149-169
MR1025884
3
Boling, G.; Guangwei, Y.
On the suitable weak solutions for the Cauchy problem of the Boussinesq equations
Nonlinear Anal.
26
1996
1367-1385
MR1377668
4
Bulícek, M.; Consiglieri, L.; Málek, J., Slip boundary effects on unsteady flows of incompressible viscous heat conducting fluids with a non-linear internal energy-temperature relationship. Preprint CMAFUL, Pre-2007-014
5
Caffarelli, L.; Kohn, R.; Nirenberg, L.
Partial regularity of suitable weak solutions of the Navier-Stokes equations
Comm. Pure Appl. Math.
35
1982
771-831
MR0673830
6
Consiglieri, L.
Weak solutions for a class of non-Newtonian fluids with energy transfer
J. Math. Fluid Mech.
2
2000
267-293
MR1781916
7
Consiglieri, L.
Friction boundary conditions on thermal incompressible viscous flows
Ann. Mat. Pura Appl.
187
2008
647-665
MR2413372
8
Consiglieri, L.
Regularity for the Navier-Stokes-Fourier system
Differ. Equ. Appl.
1
2009
583-604
MR2604775
9
Consiglieri, L.
Partial regularity for the Navier-Stokes-Fourier system
Acta Math. Sci. Ser. B Engl. Ed.
31
2011
1653-1670
MR2884940
10
Donatelli, D.
On the artificial compressibility method for the Navier-Stokes-Fourier system
Quart. Appl. Math.
68
2010
469-485
MR2676972
11
Galdi, G. P.
An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I. Linearized steady problems
Springer Tracts in Natural Philosophy, 38, Springer-Verlag, New York
1994
MR1284205
12
Guermond, J.-L.
Faedo-Galerkin weak solutions of the Navier-Stokes equations with Dirichlet boundary conditions are suitable
J. Math. Pures Appl. (9)
88
2007
87-106
MR2334774
13
Gustafson, S.; Kang. K.; Tsai, T.-P.
Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary
J. Differential Equations
226
2006
594-618
MR2237693
14
Hishida, T.
Existence and regularizing properties of solutions for the nonstationary convection problem
Funkcial. Ekvac.
34
1991
449-474
MR1150874
1150874
15
Ladyzhenskaya, O. A.
Mathematical problems in the dynamics of a viscous incompressible fluid, 2nd ed.
Izdat. "Nauka", Moscow, 1970: English transl. of 1st ed., The mathematical theory of viscous incompressible flow, Gordon and Breach, Science Publishers, New York-London-Paris
1969
MR0254401
16
Ladyzhenskaya, O. A.; Solonnikov, V. A.; Uraltseva, N. N.
Linear and quasilinear equations of parabolic type
Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I.
1968
MR0241822
17
Lin, F.
A new proof of the Caffarelli-Kohn-Nirenberg theorem
Comm. Pure Appl. Math.
51
1998
241-257
MR1488514
18
Lions, J. L.
Quelques méthodes de résolution des problèmes aux limites non linéaires
Dunod; Gauthier-Villars, Paris
1969
MR0259693
19
Málek, J.; Nečas, J.; Rokyta, M.; Ružička, M.
Weak and Measure-valued solutions to evolutionary PDEs
Applied Mathematics and Mathematical Computation, 13, Chapman and Hall, London
1996
MR1409366
20
Naumann, J.
On the existence of weak solutions to the equations of non-stationary motion of heat-conducting incompressible viscous fluids
Math. Methods Appl. Sci.
29
2006
1883-1906
MR2259989
21
Naumann, J.; Wolf, J.
On the interior regularity of weak solutions to the non-stationary Stokes system
J. Global Optim.
40
2008
277-288
MR2373557
22
Roubíček, T.
On non-Newtonian fluids with energy transfer
J. Math. Fluid Mech.
11
2009
110-125
MR2481862
23
Scheffer, V.
Hausdorff measure and the Navier-Stokes equations
Comm. Math. Phys.
55
1977
97-112
MR0510154
24
Scheffer, V.
The Navier-Stokes equations on a bounded domain
Comm. Math. Phys.
73
1980
1-42
MR0573611
25
Simon, J.
Compact sets in the space $L^p(0,T;B)$
Ann. Mat. Pura Appl. (4)
146
1987
65-96
MR0916688
26
Walter, W.
Ordinary differential equations
Graduate Texts in Mathematics, 182, Readings in Mathematics, Springer-Verlag, New York
1998
MR1629775