The Oberbeck–Boussinesq system with non-local boundary conditions
Authors:
Anna Abbatiello and Eduard Feireisl
Journal:
Quart. Appl. Math. 81 (2023), 297-306
MSC (2020):
Primary 35Q30, 35Q35; Secondary 35K61
DOI:
https://doi.org/10.1090/qam/1635
Published electronically:
October 26, 2022
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Additional Information
Abstract: We consider the Oberbeck–Boussinesq system with non-local boundary conditions arising as a singular limit of the full Navier–Stokes–Fourier system in the regime of low Mach and low Froude numbers. The existence of strong solutions is shown on a maximal time interval $[0, T_{\mathrm {max}})$. Moreover, $T_{\mathrm {max}} = \infty$ in the two-dimensional setting.
References
- Anna Abbatiello and Eduard Feireisl, On a class of generalized solutions to equations describing incompressible viscous fluids, Ann. Mat. Pura Appl. (4) 199 (2020), no. 3, 1183–1195. MR 4102807, DOI 10.1007/s10231-019-00917-x
- P. Bella, E. Feireisl, and F. Oschmann, The incompresible limit for the Rayleigh Bénard convection problem, ArXiv Preprint, arXiv2206.14041, 2022.
- Dieter Bothe and Jan Prüss, $L_P$-theory for a class of non-Newtonian fluids, SIAM J. Math. Anal. 39 (2007), no. 2, 379–421. MR 2338412, DOI 10.1137/060663635
- Peter Constantin and Charles R. Doering, Heat transfer in convective turbulence, Nonlinearity 9 (1996), no. 4, 1049–1060. MR 1399486, DOI 10.1088/0951-7715/9/4/013
- W. A. Day, Extensions of a property of the heat equation to linear thermoelasticity and other theories, Quart. Appl. Math. 40 (1982/83), no. 3, 319–330. MR 678203, DOI 10.1090/S0033-569X-1982-0678203-2
- Robert Denk, Matthias Hieber, and Jan Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z. 257 (2007), no. 1, 193–224. MR 2318575, DOI 10.1007/s00209-007-0120-9
- Eduard Feireisl and Antonin Novotný, Mathematics of open fluid systems, Nečas Center Series, Birkhäuser/Springer, Cham, [2022] ©2022. MR 4434626, DOI 10.1007/978-3-030-94793-4
- C. Foias, O. Manley, and R. Temam, Attractors for the Bénard problem: existence and physical bounds on their fractal dimension, Nonlinear Anal. 11 (1987), no. 8, 939–967. MR 903787, DOI 10.1016/0362-546X(87)90061-7
- Avner Friedman, Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math. 44 (1986), no. 3, 401–407. MR 860893, DOI 10.1090/S0033-569X-1986-0860893-1
- Claus Gerhardt, $L^{p}$-estimates for solutions to the instationary Navier-Stokes equations in dimension two, Pacific J. Math. 79 (1978), no. 2, 375–398. MR 531326, DOI 10.2140/pjm.1978.79.375
- Yoshikazu Giga and Tetsuro Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal. 89 (1985), no. 3, 267–281. MR 786550, DOI 10.1007/BF00276875
- Jinkai Li and Edriss S. Titi, Global well-posedness of the 2D Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal. 220 (2016), no. 3, 983–1001. MR 3466839, DOI 10.1007/s00205-015-0946-y
- C. V. Pao, Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math. 88 (1998), no. 1, 225–238. Positive solutions of nonlinear problems. MR 1609090, DOI 10.1016/S0377-0427(97)00215-X
- V. A. Solonnikov, Estimates for solutions of a non-stationary linearized system of Navier-Stokes equations, Trudy Mat. Inst. Steklov. 70 (1964), 213–317 (Russian). MR 0171094
- Wolf von Wahl, Instationary Navier-Stokes equations and parabolic systems, Pacific J. Math. 72 (1977), no. 2, 557–569. MR 457965, DOI 10.2140/pjm.1977.72.557
- R. Kh. Zeytounian, Joseph Boussinesq and his approximation: a contemporary view, C. R. Mecanique 331 (2003), 575–586.
References
- Anna Abbatiello and Eduard Feireisl, On a class of generalized solutions to equations describing incompressible viscous fluids, Ann. Mat. Pura Appl. (4) 199 (2020), no. 3, 1183–1195. MR 4102807, DOI 10.1007/s10231-019-00917-x
- P. Bella, E. Feireisl, and F. Oschmann, The incompresible limit for the Rayleigh Bénard convection problem, ArXiv Preprint, arXiv2206.14041, 2022.
- Dieter Bothe and Jan Prüss, $L_P$-theory for a class of non-Newtonian fluids, SIAM J. Math. Anal. 39 (2007), no. 2, 379–421. MR 2338412, DOI 10.1137/060663635
- Peter Constantin and Charles R. Doering, Heat transfer in convective turbulence, Nonlinearity 9 (1996), no. 4, 1049–1060. MR 1399486, DOI 10.1088/0951-7715/9/4/013
- W. A. Day, Extensions of a property of the heat equation to linear thermoelasticity and other theories, Quart. Appl. Math. 40 (1982/83), no. 3, 319–330. MR 678203, DOI 10.1090/qam/678203
- Robert Denk, Matthias Hieber, and Jan Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z. 257 (2007), no. 1, 193–224. MR 2318575, DOI 10.1007/s00209-007-0120-9
- Eduard Feireisl and Antonin Novotný, Mathematics of open fluid systems, Nečas Center Series, Birkhäuser/Springer, Cham, [2022] ©2022. MR 4434626, DOI 10.1007/978-3-030-94793-4
- C. Foias, O. Manley, and R. Temam, Attractors for the Bénard problem: existence and physical bounds on their fractal dimension, Nonlinear Anal. 11 (1987), no. 8, 939–967. MR 903787, DOI 10.1016/0362-546X(87)90061-7
- Avner Friedman, Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math. 44 (1986), no. 3, 401–407. MR 860893, DOI 10.1090/qam/860893
- Claus Gerhardt, $L^{p}$-estimates for solutions to the instationary Navier-Stokes equations in dimension two, Pacific J. Math. 79 (1978), no. 2, 375–398. MR 531326
- Yoshikazu Giga and Tetsuro Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal. 89 (1985), no. 3, 267–281. MR 786550, DOI 10.1007/BF00276875
- Jinkai Li and Edriss S. Titi, Global well-posedness of the 2D Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal. 220 (2016), no. 3, 983–1001. MR 3466839, DOI 10.1007/s00205-015-0946-y
- C. V. Pao, Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math. 88 (1998), no. 1, 225–238. Positive solutions of nonlinear problems. MR 1609090, DOI 10.1016/S0377-0427(97)00215-X
- V. A. Solonnikov, Estimates for solutions of a non-stationary linearized system of Navier-Stokes equations, Trudy Mat. Inst. Steklov. 70 (1964), 213–317 (Russian). MR 0171094
- Wolf von Wahl, Instationary Navier-Stokes equations and parabolic systems, Pacific J. Math. 72 (1977), no. 2, 557–569. MR 457965
- R. Kh. Zeytounian, Joseph Boussinesq and his approximation: a contemporary view, C. R. Mecanique 331 (2003), 575–586.
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Additional Information
Anna Abbatiello
Affiliation:
Sapienza University of Rome, Department of Mathematics “G. Castelnuovo”, Piazzale Aldo Moro 5, 00185 Rome, Italy
MR Author ID:
1255355
ORCID:
0000-0001-5758-114X
Email:
anna.abbatiello@uniroma1.it
Eduard Feireisl
Affiliation:
Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, CZ-115 67 Praha 1, Czech Republic
MR Author ID:
65780
Email:
feireisl@math.cas.cz
Keywords:
Oberbeck–Boussinesq system,
non-local boundary condition,
strong solution
Received by editor(s):
June 30, 2022
Received by editor(s) in revised form:
September 16, 2022
Published electronically:
October 26, 2022
Additional Notes:
The work of the first author was supported by the ERC-STG Grant no. 759229 HiCoS “Higher Co-dimension Singularities: Minimal Surfaces and the Thin Obstacle Problem”. The work of the second author was partially supported by the Czech Sciences Foundation (GAČR), Grant Agreement 21-02411S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840.
Dedicated:
Dedicated to Constantine Dafermos on the occasion of his 80th birthday
Article copyright:
© Copyright 2022
Brown University
