June  2019, 39(6): 3215-3237. doi: 10.3934/dcds.2019133

On the well-posedness of the inviscid multi-layer quasi-geostrophic equations

Department of Mathematical Sciences, Clemson University, Clemson, SC 29631, USA

* Corresponding author: Qingshan Chen

Received  June 2018 Revised  November 2018 Published  February 2019

Fund Project: The first author is in part supported by Simons Foundation (319070)

The inviscid multi-layer quasi-geostrophic equations are considered over an arbitrary bounded domain. The no-flux but non-homogeneous boundary conditions are imposed to accommodate the free fluctuations of the top and layer interfaces. Using the barotropic and baroclinic modes in the vertical direction, the elliptic system governing the streamfunctions and the potential vorticity is decomposed into a sequence of scalar elliptic boundary value problems, where the regularity theories from the two-dimensional case can be applied. With the initial potential vorticity being essentially bounded, the multi-layer quasi-equations are then shown to be globally well-posed, and the initial and boundary conditions are satisfied in the classical sense.

Citation: Qingshan Chen. On the well-posedness of the inviscid multi-layer quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3215-3237. doi: 10.3934/dcds.2019133
References:
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G. Alberti, S. Bianchini and G. Crippa, Structure of level sets and sard-type properties of lipschitz maps.

[2]

G. AlbertiS. Bianchini and G. Crippa, A uniqueness result for the continuity equation in two dimensions, J. Eur. Math. Soc., 16 (2014), 201-234. doi: 10.4171/JEMS/431.

[3]

C. Bardos, Existence et unicité de la solution de l'équation d'Euler en dimension deux, J. Math. Anal. Appl., 40 (1972), 769-790. doi: 10.1016/0022-247X(72)90019-4.

[4]

C. Bardos and È. S. Titi, Euler equations for an ideal incompressible fluid, Russian Mathematical Surveys, 62 (2007), 5-46. doi: 10.1070/RM2007v062n03ABEH004410.

[5]

J. T. BealeT. Kato and A. Majda, Remarks on th e breakdow n of smooth solutions for the 3-D euler equations, Commun. Math. Phys., 94 (1984), 61-66. doi: 10.1007/BF01212349.

[6]

A. J. Bourgeois and J. T. Beale, Validity of the quasigeostrophic model for large-scale flow in the atmosphere and ocean, SIAM J. Math. Anal., 25 (1994), 1023-1068. doi: 10.1137/S0036141092234980.

[7]

A. Castro, D. Córdoba and J. Gómez-Serrano, Global smooth solutions for the inviscid SQG equation, arXiv: 1603.03325.

[8]

Q. Chen, The barotropic quasi-geostrophic equation under a free surface., SIAM J. Math. Anal., to appear.

[9]

P. Constantin, On the euler equations of incompressible fluids, Bull. Am. Math. Soc., 44 (2007), 603-621. doi: 10.1090/S0273-0979-07-01184-6.

[10]

P. Constantin and C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J., 42 (1993), 775-789. doi: 10.1512/iumj.1993.42.42034.

[11]

B. Desjardins and E. Grenier, Derivation of quasi-geostrophic potential vorticity equations, Adv. Differential Equations, 3 (1998), 715-752.

[12]

R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.

[13]

J. A. Dutton, The nonlinear quasi-geostrophic equation- Existence and uniqueness of solutions on a bounded domain, J. Atmos. Sci., 31 (1974), 422-433. doi: 10.1175/1520-0469(1974)031<0422:TNQGEE>2.0.CO;2.

[14]

J. A. Dutton, The Nonlinear Quasi-Geostrophic Equation. Part Ⅱ: Predictability, Recurrence and Limit Properties of Thermally-Forced and Unforced Flows, J. Atmos. Sci., 33 (1976), 1431-1453. doi: 10.1175/1520-0469(1976)033<1431:TNQGEP>2.0.CO;2.

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 1983. doi: 10.1007/978-3-642-61798-0.

[16]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.

[17]

E. N. Lorenz, Reflections on the conception, birth, and childhood of numerical weather prediction, Annu. Rev. Earth Planet. Sci., 34 (2006), 37-45. doi: 10.1146/annurev.earth.34.083105.102317.

[18]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0.

[19]

M. D. Novack, On the weak solutions to the 3D inviscid Quasi-Geostrophic system.

[20]

M. D. Novack and A. F. Vasseur, Global in time classical solutions to the 3D quasi-geostrophic system for large initial data, Communications in Mathematical Physics, 358 (2018), 237-267. doi: 10.1007/s00220-017-3049-9.

[21]

J. Pedlosky, Geophysical Fluid Dynamics, 2nd edition, Springer, 1987.

[22]

M. Puel and A. F. Vasseur, Global Weak Solutions to the Inviscid 3D Quasi-Geostrophic Equation, Commun. Math. Phys., 339 (2015), 1063-1082. doi: 10.1007/s00220-015-2428-3.

[23]

L. Talley, Descriptive Physical Oceanography, 6th edition, Elsevier, 2011.

[24]

R. Temam, Navier-Stokes equations, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.

[25] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, 2nd edition, Cambridge University Press, 2017.
[26]

V. I. Yudovich, Non-stationary flow of an ideal incompressible liquid, USSR Computational Mathematics and Mathematical Physics, 3 (1963), 1407-1456.

show all references

References:
[1]

G. Alberti, S. Bianchini and G. Crippa, Structure of level sets and sard-type properties of lipschitz maps.

[2]

G. AlbertiS. Bianchini and G. Crippa, A uniqueness result for the continuity equation in two dimensions, J. Eur. Math. Soc., 16 (2014), 201-234. doi: 10.4171/JEMS/431.

[3]

C. Bardos, Existence et unicité de la solution de l'équation d'Euler en dimension deux, J. Math. Anal. Appl., 40 (1972), 769-790. doi: 10.1016/0022-247X(72)90019-4.

[4]

C. Bardos and È. S. Titi, Euler equations for an ideal incompressible fluid, Russian Mathematical Surveys, 62 (2007), 5-46. doi: 10.1070/RM2007v062n03ABEH004410.

[5]

J. T. BealeT. Kato and A. Majda, Remarks on th e breakdow n of smooth solutions for the 3-D euler equations, Commun. Math. Phys., 94 (1984), 61-66. doi: 10.1007/BF01212349.

[6]

A. J. Bourgeois and J. T. Beale, Validity of the quasigeostrophic model for large-scale flow in the atmosphere and ocean, SIAM J. Math. Anal., 25 (1994), 1023-1068. doi: 10.1137/S0036141092234980.

[7]

A. Castro, D. Córdoba and J. Gómez-Serrano, Global smooth solutions for the inviscid SQG equation, arXiv: 1603.03325.

[8]

Q. Chen, The barotropic quasi-geostrophic equation under a free surface., SIAM J. Math. Anal., to appear.

[9]

P. Constantin, On the euler equations of incompressible fluids, Bull. Am. Math. Soc., 44 (2007), 603-621. doi: 10.1090/S0273-0979-07-01184-6.

[10]

P. Constantin and C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J., 42 (1993), 775-789. doi: 10.1512/iumj.1993.42.42034.

[11]

B. Desjardins and E. Grenier, Derivation of quasi-geostrophic potential vorticity equations, Adv. Differential Equations, 3 (1998), 715-752.

[12]

R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.

[13]

J. A. Dutton, The nonlinear quasi-geostrophic equation- Existence and uniqueness of solutions on a bounded domain, J. Atmos. Sci., 31 (1974), 422-433. doi: 10.1175/1520-0469(1974)031<0422:TNQGEE>2.0.CO;2.

[14]

J. A. Dutton, The Nonlinear Quasi-Geostrophic Equation. Part Ⅱ: Predictability, Recurrence and Limit Properties of Thermally-Forced and Unforced Flows, J. Atmos. Sci., 33 (1976), 1431-1453. doi: 10.1175/1520-0469(1976)033<1431:TNQGEP>2.0.CO;2.

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 1983. doi: 10.1007/978-3-642-61798-0.

[16]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.

[17]

E. N. Lorenz, Reflections on the conception, birth, and childhood of numerical weather prediction, Annu. Rev. Earth Planet. Sci., 34 (2006), 37-45. doi: 10.1146/annurev.earth.34.083105.102317.

[18]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0.

[19]

M. D. Novack, On the weak solutions to the 3D inviscid Quasi-Geostrophic system.

[20]

M. D. Novack and A. F. Vasseur, Global in time classical solutions to the 3D quasi-geostrophic system for large initial data, Communications in Mathematical Physics, 358 (2018), 237-267. doi: 10.1007/s00220-017-3049-9.

[21]

J. Pedlosky, Geophysical Fluid Dynamics, 2nd edition, Springer, 1987.

[22]

M. Puel and A. F. Vasseur, Global Weak Solutions to the Inviscid 3D Quasi-Geostrophic Equation, Commun. Math. Phys., 339 (2015), 1063-1082. doi: 10.1007/s00220-015-2428-3.

[23]

L. Talley, Descriptive Physical Oceanography, 6th edition, Elsevier, 2011.

[24]

R. Temam, Navier-Stokes equations, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.

[25] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, 2nd edition, Cambridge University Press, 2017.
[26]

V. I. Yudovich, Non-stationary flow of an ideal incompressible liquid, USSR Computational Mathematics and Mathematical Physics, 3 (1963), 1407-1456.

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