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Optimal transport, convection, magnetic relaxation and generalized Boussinesq equations. (English) Zbl 1177.49064
Summary: We establish a connection between optimal transport theory (see Villani in Topics in optimal transportation. Graduate studies in mathematics, vol. 58, AMS, Providence, (2003; Zbl 1106.90001), for instance) and classical convection theory for geophysical flows [Pedlosky, in Geophysical fluid dynamics, Springer, New York (1979; Zbl 0429.76001)]. Our starting point is the model designed few years ago by S. Angenent, St. Haker, and A. Tannenbaum [SIAM J. Math. Anal. 35, 61–97 (2003; Zbl 1042.49040)] to solve some optimal transport problems. This model can be seen as a generalization of the Darcy-Boussinesq equations, which is a degenerate version of the Navier-Stokes-Boussinesq (NSB) equations.
In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized hydrostatic-Boussinesq equations) to various models involving optimal transport (and the related Monge-Ampère equation, Y. Brenier in Commun. Pure Appl. Math. 64, 375–417 (1991; Zbl 0768.35057); L. A. Caffarelli in Commun. Pure Appl. Math. 45, 1141–1151 (1992; Zbl 0778.35015)]. This includes the 2D semi-geostrophic equations [B. J. Hoskins in Annual review of fluid mechanics, vol. 14, 131–151 (1982; Zbl 0488.76030), Palo Alto; M. J. P. Cullen et al. in SIAM J. Appl. Math. 51, 20–31 (1991; Zbl 0713.76026), , Arch. Ration. Mech. Anal. 185:341-363; J.-D. Benamou and Y. Brenier in SIAM J. Appl. Math. 58, 1450–1461 (1998; Zbl 0915.35024); G. Loeper in SIAM J. Math. Anal. 38, 795–823 (2006; Zbl 1134.35046)] and some fully nonlinear versions of the so-called high-field limit of the Vlasov-Poisson system [J. Nieto et al. in Arch. Ration. Mech. Anal. 158, 29–59 (2001; Zbl 1038.82068)] and of the Keller-Segel for Chemotaxis [E. F. Keller and E. A. Segel in J. Theor. Biol. 30, 225–234 (1971; Zbl 1170.92307); W. Jäger and S. Luckhaus in Trans. Am. Math. Soc. 329, No. 2, 819–824 (1992; Zbl 0746.35002); F. A. C. C. Chalub et al. in Mon. Math. 142, No. 1–2, 123–141 (2004; Zbl 1052.92005)].
Mathematically speaking, we establish some existence theorems for local smooth, global smooth or global weak solutions of the different models. We also justify that the inertia terms can be rigorously neglected under appropriate scaling assumptions in the generalized Navier-Stokes-Boussinesq equations.
Finally, we show how a “stringy” generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology [see V. I. Arnold and B. A. Khesin in Topological methods in hydrodynamics. Applied mathematical sciences, vol. 125, Springer, Berlin (1998; Zbl 0902.76001); H. K. Moffatt in J. Fluid Mech. 159, 359–378 (1985; Zbl 0616.76121), Topological aspects of the dynamics of fluids and plasmas. NATO adv. sci. inst. ser. E, appl. sci., vol. 218, Kluwer, Dordrecht (1992; Zbl 0788.76097); M. E. Schonbek in Theory of the Navier-Stokes equations, Ser. adv. math. appl. sci., vol. 47, 179–184 (Zbl 0928.35136, World Sci., Singapore; V. A. Vladimirov et al. in J. Fluid Mech. 390, 127–150 (1999; Zbl 1003.76096); T. Nishiyama in Bull. Inst. Math. Acad. Sin. (N.S.) 2, 139–154 (1998; Zbl 1211.76154).

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
76W05 Magnetohydrodynamics and electrohydrodynamics
76R10 Free convection
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