# American Institute of Mathematical Sciences

October  2014, 7(5): 1025-1043. doi: 10.3934/dcdss.2014.7.1025

## Existence and decay of solutions of the 2D QG equation in the presence of an obstacle

 1 Department of Mathematics, UC Riverside, 900 University Ave, Riverside, CA 92521, United States 2 Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431

Received  March 2013 Published  May 2014

We continue the study initiated in [16] of dissipative differential equations governing fluid motion in the presence of an obstacle, in which the dissipative term is given by the Laplacian, or a fractional power of the Laplacian. Our main tools are the Ikebe-Ramm transform, and the localized version of the fractional Laplacian due to Caffarelli and Silvestre [5] as improved by Stinga and Torrea [21]. We give applications to the problem of existence of weak solutions of the two dimensional dissipative quasi-geostrophic equation and the decay of these solutions in the $L^2$-norm.
Citation: Leonardo Kosloff, Tomas Schonbek. Existence and decay of solutions of the 2D QG equation in the presence of an obstacle. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1025-1043. doi: 10.3934/dcdss.2014.7.1025
##### References:
 [1] W. Borchers and T. Miyakawa, Algebraic $L^2$ decay for Navier-Stokes flows in exterior domains,, Acta Math., 165 (1990), 189.  doi: 10.1007/BF02391905.  Google Scholar [2] W. Borchers and T. Miyakawa, On stability of exterior stationary Navier-Stokes flows,, Acta Math., 174 (1995), 311.  doi: 10.1007/BF02392469.  Google Scholar [3] W. Borchers and H. Sohr, On the semigroup of the Stokes operator for exterior domains in $L^q$ spaces,, Math. Z., 196 (1987), 415.  doi: 10.1007/BF01200362.  Google Scholar [4] J. Bergh and J. Löfström, Interpolation Spaces,, Springer Verlag, (1976).   Google Scholar [5] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. P.D.E., 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar [6] J. A. Carrillo and L. C. F. Ferreira, The asymptotic behavior of subcritical dissipative quasi-geostrophic equations,, Nonlinearity, 21 (2008), 1001.  doi: 10.1088/0951-7715/21/5/006.  Google Scholar [7] M. Cannone, Ondelettes, Paraproduits et Navier-Stokes,, Diderot Editeur, (1995).   Google Scholar [8] S. Chandler-Wilde and P. Monk, Wave-number-explicit bounds in time harmonic scattering,, SIAM J. Math. Analysis, 39 (2008), 1428.  doi: 10.1137/060662575.  Google Scholar [9] P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations,, SIAM J. Math. Analysis, 30 (1999), 937.  doi: 10.1137/S0036141098337333.  Google Scholar [10] A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations,, Comm. Math. Physics, 249 (2004), 511.  doi: 10.1007/s00220-004-1055-1.  Google Scholar [11] Y. Giga, Analyticity of the semigroup generated by the Stokes operator in $L_r$ spaces,, Math. Z., 178 (1981), 251.  doi: 10.1007/BF01214869.  Google Scholar [12] Y. Giga, Domains of fractional powers of the Stokes operator in $L_r$ spaces,, Arch. Rational Mech. Anal., 89 (1985), 25.  doi: 10.1007/BF00276874.  Google Scholar [13] Y. Giga and H. Sohr, On the Stokes operator in exterior domains,, J. Fac. Sci. Univ. Tokyo, 36 (1989), 103.   Google Scholar [14] P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman Publishing Inc., (1985).   Google Scholar [15] T. Ikebe, Eigenfunction expansions associated with the Schrödinger operator and their applications to scattering theory,, Arch. Rational Mech. Anal., 5 (1960), 1.  doi: 10.1007/BF00252896.  Google Scholar [16] L. Kosloff and T. Schonbek, On the Laplacian and fractional Laplacian in an exterior domain,, Adv. Diff. Eq., 17 (2012), 173.   Google Scholar [17] T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain,, Hiroshima Math. J., 12 (1982), 115.   Google Scholar [18] A. G. Ramm, Scattering by Obstacles,, D. Reidel Publishing Co., (1986).  doi: 10.1007/978-94-009-4544-9.  Google Scholar [19] M. E. Schonbek, The Fourier splitting method,, in Advances in Geometric Analysis and Continuum Mechanics, (1995), 269.   Google Scholar [20] M. E. Schonbek and T. Schonbek, Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows,, Discrete Contin. Dyn. Syst., 13 (2005), 1277.  doi: 10.3934/dcds.2005.13.1277.  Google Scholar [21] P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators,, Comm. P.D.E., 35 (2010), 2092.  doi: 10.1080/03605301003735680.  Google Scholar [22] M. E. Taylor, Partial Differential Equations,, Vol 1., (1996).  doi: 10.1007/978-1-4684-9320-7.  Google Scholar [23] R. Temam, Navier Stokes Equations, Theory and Numerical Analysis,, Studies in Mathematics and its Applications, (1984).   Google Scholar

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##### References:
 [1] W. Borchers and T. Miyakawa, Algebraic $L^2$ decay for Navier-Stokes flows in exterior domains,, Acta Math., 165 (1990), 189.  doi: 10.1007/BF02391905.  Google Scholar [2] W. Borchers and T. Miyakawa, On stability of exterior stationary Navier-Stokes flows,, Acta Math., 174 (1995), 311.  doi: 10.1007/BF02392469.  Google Scholar [3] W. Borchers and H. Sohr, On the semigroup of the Stokes operator for exterior domains in $L^q$ spaces,, Math. Z., 196 (1987), 415.  doi: 10.1007/BF01200362.  Google Scholar [4] J. Bergh and J. Löfström, Interpolation Spaces,, Springer Verlag, (1976).   Google Scholar [5] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. P.D.E., 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar [6] J. A. Carrillo and L. C. F. Ferreira, The asymptotic behavior of subcritical dissipative quasi-geostrophic equations,, Nonlinearity, 21 (2008), 1001.  doi: 10.1088/0951-7715/21/5/006.  Google Scholar [7] M. Cannone, Ondelettes, Paraproduits et Navier-Stokes,, Diderot Editeur, (1995).   Google Scholar [8] S. Chandler-Wilde and P. Monk, Wave-number-explicit bounds in time harmonic scattering,, SIAM J. Math. Analysis, 39 (2008), 1428.  doi: 10.1137/060662575.  Google Scholar [9] P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations,, SIAM J. Math. Analysis, 30 (1999), 937.  doi: 10.1137/S0036141098337333.  Google Scholar [10] A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations,, Comm. Math. Physics, 249 (2004), 511.  doi: 10.1007/s00220-004-1055-1.  Google Scholar [11] Y. Giga, Analyticity of the semigroup generated by the Stokes operator in $L_r$ spaces,, Math. Z., 178 (1981), 251.  doi: 10.1007/BF01214869.  Google Scholar [12] Y. Giga, Domains of fractional powers of the Stokes operator in $L_r$ spaces,, Arch. Rational Mech. Anal., 89 (1985), 25.  doi: 10.1007/BF00276874.  Google Scholar [13] Y. Giga and H. Sohr, On the Stokes operator in exterior domains,, J. Fac. Sci. Univ. Tokyo, 36 (1989), 103.   Google Scholar [14] P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman Publishing Inc., (1985).   Google Scholar [15] T. Ikebe, Eigenfunction expansions associated with the Schrödinger operator and their applications to scattering theory,, Arch. Rational Mech. Anal., 5 (1960), 1.  doi: 10.1007/BF00252896.  Google Scholar [16] L. Kosloff and T. Schonbek, On the Laplacian and fractional Laplacian in an exterior domain,, Adv. Diff. Eq., 17 (2012), 173.   Google Scholar [17] T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain,, Hiroshima Math. J., 12 (1982), 115.   Google Scholar [18] A. G. Ramm, Scattering by Obstacles,, D. Reidel Publishing Co., (1986).  doi: 10.1007/978-94-009-4544-9.  Google Scholar [19] M. E. Schonbek, The Fourier splitting method,, in Advances in Geometric Analysis and Continuum Mechanics, (1995), 269.   Google Scholar [20] M. E. Schonbek and T. Schonbek, Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows,, Discrete Contin. Dyn. Syst., 13 (2005), 1277.  doi: 10.3934/dcds.2005.13.1277.  Google Scholar [21] P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators,, Comm. P.D.E., 35 (2010), 2092.  doi: 10.1080/03605301003735680.  Google Scholar [22] M. E. Taylor, Partial Differential Equations,, Vol 1., (1996).  doi: 10.1007/978-1-4684-9320-7.  Google Scholar [23] R. Temam, Navier Stokes Equations, Theory and Numerical Analysis,, Studies in Mathematics and its Applications, (1984).   Google Scholar
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