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

Previous Article
Stokes and Navier-Stokes equations with perfect slip on wedge type domains
DCDS-S Home
This Issue
Next Article
Convergent finite differences for 1D viscous isentropic flow in Eulerian coordinates

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

Leonardo Kosloff ^1, and Tomas Schonbek ^2,

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

Abstract
Related Papers

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.

Keywords: Ikebe-Ramm transform., Navier-Stokes, Exterior Laplacian, quasi-geostrophic.

Mathematics Subject Classification: Primary: 35J05, 35P05, 35Q30, 35Q3.

Citation: Leonardo Kosloff, Tomas Schonbek. Existence and decay of solutions of the 2D QG equation in the presence of an obstacle. Discrete and 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), 189-227. 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.

[1]	Ludovic Godard-Cadillac. Vortex collapses for the Euler and Quasi-Geostrophic models. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3143-3168. doi: 10.3934/dcds.2022012
[2]	Yinnian He, Kaitai Li. Nonlinear Galerkin approximation of the two dimensional exterior Navier-Stokes problem. Discrete and Continuous Dynamical Systems, 1996, 2 (4) : 467-482. doi: 10.3934/dcds.1996.2.467
[3]	T. Tachim Medjo. Multi-layer quasi-geostrophic equations of the ocean with delays. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 171-196. doi: 10.3934/dcdsb.2008.10.171
[4]	Carina Geldhauser, Marco Romito. Point vortices for inviscid generalized surface quasi-geostrophic models. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2583-2606. doi: 10.3934/dcdsb.2020023
[5]	May Ramzi, Zahrouni Ezzeddine. Global existence of solutions for subcritical quasi-geostrophic equations. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1179-1191. doi: 10.3934/cpaa.2008.7.1179
[6]	Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1385-1412. doi: 10.3934/cpaa.2021025
[7]	Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5135-5148. doi: 10.3934/dcdsb.2020336
[8]	Yanhong Zhang. Global attractors of two layer baroclinic quasi-geostrophic model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6377-6385. doi: 10.3934/dcdsb.2021023
[9]	Tsukasa Iwabuchi. On analyticity up to the boundary for critical quasi-geostrophic equation in the half space. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1209-1224. doi: 10.3934/cpaa.2022016
[10]	Ben-Yu Guo, Yu-Jian Jiao. Mixed generalized Laguerre-Fourier spectral method for exterior problem of Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 315-345. doi: 10.3934/dcdsb.2009.11.315
[11]	Shuguang Shao, Shu Wang, Wen-Qing Xu, Bin Han. Global existence for the 2D Navier-Stokes flow in the exterior of a moving or rotating obstacle. Kinetic and Related Models, 2016, 9 (4) : 767-776. doi: 10.3934/krm.2016015
[12]	Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613
[13]	Yoshihiro Shibata. On the local wellposedness of free boundary problem for the Navier-Stokes equations in an exterior domain. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1681-1721. doi: 10.3934/cpaa.2018081
[14]	Takeshi Taniguchi. The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4323-4341. doi: 10.3934/dcds.2014.34.4323
[15]	Yong Zhou. Decay rate of higher order derivatives for solutions to the 2-D dissipative quasi-geostrophic flows. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 525-532. doi: 10.3934/dcds.2006.14.525
[16]	Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095
[17]	Wen Tan, Bo-Qing Dong, Zhi-Min Chen. Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152
[18]	Colin Cotter, Dan Crisan, Darryl Holm, Wei Pan, Igor Shevchenko. Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model. Foundations of Data Science, 2020, 2 (2) : 173-205. doi: 10.3934/fods.2020010
[19]	Maria Schonbek, Tomas Schonbek. Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1277-1304. doi: 10.3934/dcds.2005.13.1277
[20]	T. Tachim Medjo. Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1119-1140. doi: 10.3934/cpaa.2014.13.1119

2020 Impact Factor: 2.425

Tools

Metrics

Other articles
by authors

on AIMS
Leonardo Kosloff Tomas Schonbek
on Google Scholar
Leonardo Kosloff Tomas Schonbek

[Back to Top]