Abstract
We establish a unique global strong solution for nonhomogeneous Bénard system with zero density at infinity on the whole two-dimensional (2D) space. In particular, the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compact support. Our method relies heavily on the structure of the system under consideration and spatial dimension.
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References
Coifman, R., Lions, P.L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72, 247–286 (1993)
Guo, B., Xie, B., Zeng, L.: Exponential decay of Bénard convection problem with surface tension. J. Differ. Equ. 267, 2261–2283 (2019)
Li, J., Liang, Z.: On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier–Stokes equations with vacuum. J. Math. Pures Appl. 102, 640–671 (2014)
Lions, P.L.: Mathematical Topics in Fluid Mechanics, vol. I: Incompressible Models. Oxford University Press, Oxford (1996)
Lü, B., Shi, X., Zhong, X.: Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier–Stokes equations with vacuum. Nonlinearity 31, 2617–2632 (2018)
Ma, T., Wang, S.: Rayleigh–Bénard convection: dynamics and structure in the physical space. Commun. Math. Sci. 5, 553–574 (2007)
Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13, 115–162 (1959)
Rabinowitz, P.H.: Existence and nonuniqueness of rectangular solutions of the Bénard problem. Arch. Rational Mech. Anal. 29, 32–57 (1968)
Stein, E.M.: Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)
Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. AMS Chelsea Publishing, Providence (2001)
Wu, G., Xue, L.: Global well-posedness for the 2D inviscid Bénard system with fractional diffusivity and Yudovich’s type data. J. Differ. Equ. 253, 100–125 (2012)
Ye, Z.: Regularity criterion of the 2D Bénard equations with critical and supercritical dissipation. Nonlinear Anal. 156, 111–143 (2017)
Zhang, Q.: Global well-posedness for the \(2\frac{1}{2}\)D Bénard system with partial viscosity terms. Appl. Math. Comput. 283, 282–289 (2016)
Zhang, R., Fan, M., Li, S.: Global well-posedness of incompressible Bénard problem with zero dissipation or zero thermal diffusivity. Appl. Math. Comput. 321, 442–449 (2018)
Zhong, X.: Local strong solutions to the nonhomogeneous Bénard system with nonnegative density. Rocky Mountain J. Math. 50(4), 1497–1516 (2020)
Zhong, X.: Global strong solution of nonhomogeneous Bénard system with large initial data and vacuum in a bounded domain, accepted by Z. Anal. Anwend. (2020)
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This research was partially supported by the Innovation Support Program for Chongqing Overseas Returnees (No. cx2020082)
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Zhong, X. Global Strong Solution to the Nonhomogeneous Bénard System with Large Initial Data and Vacuum. Results Math 76, 27 (2021). https://doi.org/10.1007/s00025-020-01338-6
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DOI: https://doi.org/10.1007/s00025-020-01338-6