Improved a priori bounds for thermal fluid equations

Tarfulea, Andrei

doi:10.1090/tran/7529

Improved a priori bounds for thermal fluid equations
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by Andrei Tarfulea PDF

Trans. Amer. Math. Soc. 371 (2019), 2719-2737 Request permission

Abstract:

We consider two hydrodynamic model problems (one incompressible and one compressible) with three-dimensional fluid flow on the torus and temperature-dependent viscosity and conductivity. The ambient heat for the fluid is transported by the flow and fed by the local energy dissipation, modeling the transfer of kinetic energy into thermal energy through fluid friction. Both the viscosity and conductivity grow with the local temperature. We prove a strong a priori bound on the enstrophy of the velocity weighed against the temperature for initial data of arbitrary size, requiring only that the conductivity be proportionately larger than the viscosity (and, in the incompressible case, a bound on the temperature as a Muckenhoupt weight).

References

Shmuel Agmon, Lectures on elliptic boundary value problems, AMS Chelsea Publishing, Providence, RI, 2010. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr.; Revised edition of the 1965 original. MR 2589244, DOI 10.1090/chel/369
Eric Baer and Alexis Vasseur, A bound from below on the temperature for the Navier-Stokes-Fourier system, SIAM J. Math. Anal. 45 (2013), no. 4, 2046–2063. MR 3072228, DOI 10.1137/120883773
Miroslav Bulíček, Eduard Feireisl, and Josef Málek, A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients, Nonlinear Anal. Real World Appl. 10 (2009), no. 2, 992–1015. MR 2474275, DOI 10.1016/j.nonrwa.2007.11.018
Luis A. Caffarelli and Alexis F. Vasseur, The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics, Discrete Contin. Dyn. Syst. Ser. S 3 (2010), no. 3, 409–427. MR 2660718, DOI 10.3934/dcdss.2010.3.409
S. Campanato, Proprietà di hölderianità di alcune classi di funzioni, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 175–188 (Italian). MR 156188
R. R. Coifman, R. Rochberg, and Guido Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), no. 3, 611–635. MR 412721, DOI 10.2307/1970954
Bernard Ducomet and Eduard Feireisl, On the dynamics of gaseous stars, Arch. Ration. Mech. Anal. 174 (2004), no. 2, 221–266. MR 2098107, DOI 10.1007/s00205-004-0326-5
Eduard Feireisl, Dynamics of viscous compressible fluids, Oxford Lecture Series in Mathematics and its Applications, vol. 26, Oxford University Press, Oxford, 2004. MR 2040667
Eduard Feireisl, On the motion of a viscous, compressible, and heat conducting fluid, Indiana Univ. Math. J. 53 (2004), no. 6, 1705–1738. MR 2106342, DOI 10.1512/iumj.2004.53.2510
Eduard Feireisl and Antonín Novotný, Singular limits in thermodynamics of viscous fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009. MR 2499296, DOI 10.1007/978-3-7643-8843-0
Eduard Feireisl and Josef Málek, On the Navier-Stokes equations with temperature-dependent transport coefficients, Differ. Equ. Nonlinear Mech. , posted on (2006), Art. ID 90616, 14. MR 2233755, DOI 10.1155/denm/2006/90616
Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR 312139, DOI 10.1090/S0002-9947-1973-0312139-8
R. L. Johnson and C. J. Neugebauer, Properties of BMO functions whose reciprocals are also BMO, Z. Anal. Anwendungen 12 (1993), no. 1, 3–11. MR 1239426, DOI 10.4171/ZAA/583
O. A. Ladyženskaja, Uniqueness and smoothness of generalized solutions of Navier-Stokes equations, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 5 (1967), 169–185 (Russian). MR 0236541
B. Lautrup, Physics of continuous matter, CRC Press, Taylor & Francis Group, Boca Raton, FL, 2011.
Akitaka Matsumura and Takaaki Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys. 89 (1983), no. 4, 445–464. MR 713680
Antoine Mellet and Alexis Vasseur, A bound from below for the temperature in compressible Navier-Stokes equations, Monatsh. Math. 157 (2009), no. 2, 143–161. MR 2504783, DOI 10.1007/s00605-008-0021-y
Stefanie Petermichl, The sharp weighted bound for the Riesz transforms, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1237–1249. MR 2367098, DOI 10.1090/S0002-9939-07-08934-4
Luis Silvestre and Vlad Vicol, Hölder continuity for a drift-diffusion equation with pressure, Ann. Inst. H. Poincaré C Anal. Non Linéaire 29 (2012), no. 4, 637–652. MR 2948291, DOI 10.1016/j.anihpc.2012.02.003
James Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems (Proc. Sympos., Madison, Wis., 1962) Univ. Wisconsin Press, Madison, Wis., 1963, pp. 69–98. MR 0150444
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Terence Tao, Finite time blowup for an averaged three-dimensional Navier-Stokes equation, J. Amer. Math. Soc. 29 (2016), no. 3, 601–674. MR 3486169, DOI 10.1090/jams/838
Jérémie Unterberger, Global existence for strong solutions of viscous Burgers equation. (1) The bounded case, Control Cybernet. 46 (2017), no. 2, 109–136. MR 3726682