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.

[2]

W. Borchers and T. Miyakawa, On stability of exterior stationary Navier-Stokes flows,, Acta Math., 174 (1995), 311. doi: 10.1007/BF02392469.

[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.

[4]

J. Bergh and J. Löfström, Interpolation Spaces,, Springer Verlag, (1976).

[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.

[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.

[7]

M. Cannone, Ondelettes, Paraproduits et Navier-Stokes,, Diderot Editeur, (1995).

[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.

[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.

[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.

[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.

[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.

[13]

Y. Giga and H. Sohr, On the Stokes operator in exterior domains,, J. Fac. Sci. Univ. Tokyo, 36 (1989), 103.

[14]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman Publishing Inc., (1985).

[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.

[16]

L. Kosloff and T. Schonbek, On the Laplacian and fractional Laplacian in an exterior domain,, Adv. Diff. Eq., 17 (2012), 173.

[17]

T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain,, Hiroshima Math. J., 12 (1982), 115.

[18]

A. G. Ramm, Scattering by Obstacles,, D. Reidel Publishing Co., (1986). doi: 10.1007/978-94-009-4544-9.

[19]

M. E. Schonbek, The Fourier splitting method,, in Advances in Geometric Analysis and Continuum Mechanics, (1995), 269.

[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.

[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.

[22]

M. E. Taylor, Partial Differential Equations,, Vol 1., (1996). doi: 10.1007/978-1-4684-9320-7.

[23]

R. Temam, Navier Stokes Equations, Theory and Numerical Analysis,, Studies in Mathematics and its Applications, (1984).

show all references

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.

[2]

W. Borchers and T. Miyakawa, On stability of exterior stationary Navier-Stokes flows,, Acta Math., 174 (1995), 311. doi: 10.1007/BF02392469.

[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.

[4]

J. Bergh and J. Löfström, Interpolation Spaces,, Springer Verlag, (1976).

[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.

[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.

[7]

M. Cannone, Ondelettes, Paraproduits et Navier-Stokes,, Diderot Editeur, (1995).

[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.

[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.

[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.

[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.

[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.

[13]

Y. Giga and H. Sohr, On the Stokes operator in exterior domains,, J. Fac. Sci. Univ. Tokyo, 36 (1989), 103.

[14]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman Publishing Inc., (1985).

[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.

[16]

L. Kosloff and T. Schonbek, On the Laplacian and fractional Laplacian in an exterior domain,, Adv. Diff. Eq., 17 (2012), 173.

[17]

T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain,, Hiroshima Math. J., 12 (1982), 115.

[18]

A. G. Ramm, Scattering by Obstacles,, D. Reidel Publishing Co., (1986). doi: 10.1007/978-94-009-4544-9.

[19]

M. E. Schonbek, The Fourier splitting method,, in Advances in Geometric Analysis and Continuum Mechanics, (1995), 269.

[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.

[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.

[22]

M. E. Taylor, Partial Differential Equations,, Vol 1., (1996). doi: 10.1007/978-1-4684-9320-7.

[23]

R. Temam, Navier Stokes Equations, Theory and Numerical Analysis,, Studies in Mathematics and its Applications, (1984).

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