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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 |
References:
[1] |
W. Borchers and T. Miyakawa, Algebraic $L^2$ decay for Navier-Stokes flows in exterior domains, Acta Math., 165 (1990), 189-227.
doi: 10.1007/BF02391905. |
[2] |
W. Borchers and T. Miyakawa, On stability of exterior stationary Navier-Stokes flows, Acta Math., 174 (1995), 311-382.
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), 415-425.
doi: 10.1007/BF01200362. |
[4] |
J. Bergh and J. Löfström, Interpolation Spaces, Springer Verlag, Heidelberg, New York, 1976 |
[5] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. P.D.E., 32 (2007), 1245-1260.
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), 1001-1018.
doi: 10.1088/0951-7715/21/5/006. |
[7] |
M. Cannone, Ondelettes, Paraproduits et Navier-Stokes, Diderot Editeur, Arts et Sciences, Paris 1995. |
[8] |
S. Chandler-Wilde and P. Monk, Wave-number-explicit bounds in time harmonic scattering, SIAM J. Math. Analysis, 39 (2008), 1428-1455.
doi: 10.1137/060662575. |
[9] |
P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Analysis, 30 (1999), 937-948.
doi: 10.1137/S0036141098337333. |
[10] |
A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Physics, 249 (2004), 511-528.
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), 251-265.
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), 25-265.
doi: 10.1007/BF00276874. |
[13] |
Y. Giga and H. Sohr, On the Stokes operator in exterior domains, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 36 (1989), 103-130. |
[14] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Publishing Inc., Boston, MA, 1985. |
[15] |
T. Ikebe, Eigenfunction expansions associated with the Schrödinger operator and their applications to scattering theory, Arch. Rational Mech. Anal., 5 (1960), 1-34.
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), 173-200. |
[17] |
T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140. |
[18] |
A. G. Ramm, Scattering by Obstacles, D. Reidel Publishing Co., Dodrecht, Holland, 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, International Press, Cambridge, MA, 1995, 269-274. |
[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), 1277-1304.
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), 2092-2122.
doi: 10.1080/03605301003735680. |
[22] |
M. E. Taylor, Partial Differential Equations, Vol 1., Chapter 9, Springer Verlag, New York, NY 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, 2, Third Revised Edition, Elsevier, 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), 189-227.
doi: 10.1007/BF02391905. |
[2] |
W. Borchers and T. Miyakawa, On stability of exterior stationary Navier-Stokes flows, Acta Math., 174 (1995), 311-382.
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), 415-425.
doi: 10.1007/BF01200362. |
[4] |
J. Bergh and J. Löfström, Interpolation Spaces, Springer Verlag, Heidelberg, New York, 1976 |
[5] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. P.D.E., 32 (2007), 1245-1260.
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), 1001-1018.
doi: 10.1088/0951-7715/21/5/006. |
[7] |
M. Cannone, Ondelettes, Paraproduits et Navier-Stokes, Diderot Editeur, Arts et Sciences, Paris 1995. |
[8] |
S. Chandler-Wilde and P. Monk, Wave-number-explicit bounds in time harmonic scattering, SIAM J. Math. Analysis, 39 (2008), 1428-1455.
doi: 10.1137/060662575. |
[9] |
P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Analysis, 30 (1999), 937-948.
doi: 10.1137/S0036141098337333. |
[10] |
A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Physics, 249 (2004), 511-528.
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), 251-265.
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), 25-265.
doi: 10.1007/BF00276874. |
[13] |
Y. Giga and H. Sohr, On the Stokes operator in exterior domains, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 36 (1989), 103-130. |
[14] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Publishing Inc., Boston, MA, 1985. |
[15] |
T. Ikebe, Eigenfunction expansions associated with the Schrödinger operator and their applications to scattering theory, Arch. Rational Mech. Anal., 5 (1960), 1-34.
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), 173-200. |
[17] |
T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140. |
[18] |
A. G. Ramm, Scattering by Obstacles, D. Reidel Publishing Co., Dodrecht, Holland, 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, International Press, Cambridge, MA, 1995, 269-274. |
[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), 1277-1304.
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), 2092-2122.
doi: 10.1080/03605301003735680. |
[22] |
M. E. Taylor, Partial Differential Equations, Vol 1., Chapter 9, Springer Verlag, New York, NY 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, 2, Third Revised Edition, Elsevier, 1984. |
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