# American Institute of Mathematical Sciences

March  2013, 12(2): 939-955. doi: 10.3934/cpaa.2013.12.939

## Weak solutions for generalized large-scale semigeostrophic equations

 1 School of Engineering and Science, Jacobs University, 28759 Bremen, Germany

Received  October 2011 Published  September 2012

We prove existence, uniqueness and continuous dependence on initial data of global weak solutions to the generalized large-scale semigeostrophic equations with periodic boundary conditions. This family of Hamiltonian balance models for rapidly rotating shallow water includes the $L_1$ model derived by R. Salmon in 1985 and its 2006 generalization by the second author. The analysis is based on the vorticity formulation of the models supplemented by a nonlinear velocity-vorticity relation. The results are fundamentally due to the conservation of potential vorticity. While classical solutions are known to exist provided the initial potential vorticity is positive---a condition which is already implicit in the formal derivation of balance models, we can assert the existence of weak solutions only under the slightly stronger assumption that the potential vorticity is bounded below by $\sqrt{5}-2$ times the equilibrium potential vorticity. The reason is that the nonlinearities in the potential vorticity inversion are felt more strongly when working in weaker function spaces. Another manifestation of this effect is that point-vortex solutions are not supported by the model even in the special case when the potential vorticity inversion gains three derivatives in spaces of classical functions.
Citation: Mahmut Çalik, Marcel Oliver. Weak solutions for generalized large-scale semigeostrophic equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 939-955. doi: 10.3934/cpaa.2013.12.939
##### References:
 [1] M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables,'', tenth edition, (1972).   Google Scholar [2] R. A. Adams and J. J. F Fournier, "Sobolev Spaces,'', second edition, (2003).   Google Scholar [3] C. Bardos, Existence et unicité de la solution de l'équation d'Euler en dimension deux,, J. Math. Anal. and Appl., 40 (1972), 769.   Google Scholar [4] Y.-Z. Chen and L.-C. Wu, "Second Order Elliptic Equations and Elliptic Systems,'', AMS, (1998).   Google Scholar [5] M. Çalık, M. Oliver and S. Vasylkevych, Global well-posedness for the generalized large-scale semigeostrophic equations,, {Arch. Ration. Mech. An., (2012).   Google Scholar [6] C. R. Doering, J. D. Gibbon and C. D. Levermore, Weak and strong solutions of the complex Ginzburg-Landau equation,, Phys. D, 71 (1994), 285.  doi: 10.1016/0167-2789(94)90150-3.  Google Scholar [7] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'', Springer-Verlag, (1983).   Google Scholar [8] D. D. Holm, Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid motion,, Phys. D, 133 (1999), 215.  doi: 10.1016/S0167-2789(99)00093-7.  Google Scholar [9] D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. in Math., 137 (1998), 1.  doi: 10.1006/aima.1998.1721.  Google Scholar [10] E. Hopf, Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus,, Sitzungsber. d. Preuss. Acad. Wiss., 19 (1927), 147.   Google Scholar [11] C. D. Levermore, M. Oliver and E. S. Titi, Global well-posedness for models of shallow water in a basin of varying bottom,, {Indiana Univ. Math. J.}, 45 (1996), 479.   Google Scholar [12] J. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations on bounded domains,, Phil. Trans R. Soc Lond. A, 359 (2001), 1449.  doi: 10.1098/rsta.2001.0852.  Google Scholar [13] M. Oliver, Variational asymptotics for rotating shallow water near geostrophy: A transformational approach,, {J. Fluid Mech.}, 551 (2006), 197.  doi: 10.1017/S0022112005008256.  Google Scholar [14] M. Oliver and S. Shkoller, The vortex blob method as a second-grade non-Newtonian fluid,, Comm. in Part. Diff. Eq., 26 (2001), 295.   Google Scholar [15] M. Oliver and S. Vasylkevych, Hamiltonian formalism for models of rotating shallow water in semigeostrophic scaling,, {Discr. Cont. Dyn. Sys.}, 31 (2011), 827.  doi: 10.3934/dcds.2011.31.827.  Google Scholar [16] M. Oliver and S. Vasylkevych, Generalized LSG models with varying Coriolis parameter,, Geophys. Astrophys. Fluid Dyn., ().   Google Scholar [17] R. Salmon, New equations for nearly geostrophic flow,, {J. Fluid Mech.}, 153 (1985), 461.  doi: 10.1017/S0022112085001343.  Google Scholar [18] V. I. Yudovich, Some bounds for solutions of elliptic equations,, Amer. Math. Soc. Transl. Ser. 2, 59 (1966), 229.   Google Scholar [19] V. I. Yudovich, Non-stationary flow of an ideal incompressible liquid,, {Zh.\ Vychisl.\ Mat.\ i Mat.\ Fiz.}, 6 (1963), 1032.   Google Scholar

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##### References:
 [1] M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables,'', tenth edition, (1972).   Google Scholar [2] R. A. Adams and J. J. F Fournier, "Sobolev Spaces,'', second edition, (2003).   Google Scholar [3] C. Bardos, Existence et unicité de la solution de l'équation d'Euler en dimension deux,, J. Math. Anal. and Appl., 40 (1972), 769.   Google Scholar [4] Y.-Z. Chen and L.-C. Wu, "Second Order Elliptic Equations and Elliptic Systems,'', AMS, (1998).   Google Scholar [5] M. Çalık, M. Oliver and S. Vasylkevych, Global well-posedness for the generalized large-scale semigeostrophic equations,, {Arch. Ration. Mech. An., (2012).   Google Scholar [6] C. R. Doering, J. D. Gibbon and C. D. Levermore, Weak and strong solutions of the complex Ginzburg-Landau equation,, Phys. D, 71 (1994), 285.  doi: 10.1016/0167-2789(94)90150-3.  Google Scholar [7] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'', Springer-Verlag, (1983).   Google Scholar [8] D. D. Holm, Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid motion,, Phys. D, 133 (1999), 215.  doi: 10.1016/S0167-2789(99)00093-7.  Google Scholar [9] D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. in Math., 137 (1998), 1.  doi: 10.1006/aima.1998.1721.  Google Scholar [10] E. Hopf, Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus,, Sitzungsber. d. Preuss. Acad. Wiss., 19 (1927), 147.   Google Scholar [11] C. D. Levermore, M. Oliver and E. S. Titi, Global well-posedness for models of shallow water in a basin of varying bottom,, {Indiana Univ. Math. J.}, 45 (1996), 479.   Google Scholar [12] J. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations on bounded domains,, Phil. Trans R. Soc Lond. A, 359 (2001), 1449.  doi: 10.1098/rsta.2001.0852.  Google Scholar [13] M. Oliver, Variational asymptotics for rotating shallow water near geostrophy: A transformational approach,, {J. Fluid Mech.}, 551 (2006), 197.  doi: 10.1017/S0022112005008256.  Google Scholar [14] M. Oliver and S. Shkoller, The vortex blob method as a second-grade non-Newtonian fluid,, Comm. in Part. Diff. Eq., 26 (2001), 295.   Google Scholar [15] M. Oliver and S. Vasylkevych, Hamiltonian formalism for models of rotating shallow water in semigeostrophic scaling,, {Discr. Cont. Dyn. Sys.}, 31 (2011), 827.  doi: 10.3934/dcds.2011.31.827.  Google Scholar [16] M. Oliver and S. Vasylkevych, Generalized LSG models with varying Coriolis parameter,, Geophys. Astrophys. Fluid Dyn., ().   Google Scholar [17] R. Salmon, New equations for nearly geostrophic flow,, {J. Fluid Mech.}, 153 (1985), 461.  doi: 10.1017/S0022112085001343.  Google Scholar [18] V. I. Yudovich, Some bounds for solutions of elliptic equations,, Amer. Math. Soc. Transl. Ser. 2, 59 (1966), 229.   Google Scholar [19] V. I. Yudovich, Non-stationary flow of an ideal incompressible liquid,, {Zh.\ Vychisl.\ Mat.\ i Mat.\ Fiz.}, 6 (1963), 1032.   Google Scholar
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