Optimal estimates for far field asymptotics of solutions to the quasi-geostrophic equation

Yamamoto, Masakazu; Sugiyama, Yuusuke

doi:10.1090/proc/15305

Optimal estimates for far field asymptotics of solutions to the quasi-geostrophic equation
HTML articles powered by AMS MathViewer

by Masakazu Yamamoto and Yuusuke Sugiyama PDF

Proc. Amer. Math. Soc. 149 (2021), 1099-1110 Request permission

Abstract:

The initial value problem for the two dimensional dissipative quasi-geostrophic equation of the critical and the supercritical cases is considered. Anomalous diffusion on this equation provides slow decay of solutions as the spatial parameter tends to infinity. In this paper, uniform estimates for far field asymptotics of solutions are given.

References

R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc. 95 (1960), 263–273. MR 119247, DOI 10.1090/S0002-9947-1960-0119247-6
Lorenzo Brandolese, Space-time decay of Navier-Stokes flows invariant under rotations, Math. Ann. 329 (2004), no. 4, 685–706. MR 2076682, DOI 10.1007/s00208-004-0533-2
Lorenzo Brandolese and François Vigneron, New asymptotic profiles of nonstationary solutions of the Navier-Stokes system, J. Math. Pures Appl. (9) 88 (2007), no. 1, 64–86 (English, with English and French summaries). MR 2334773, DOI 10.1016/j.matpur.2007.04.007
Lorenzo Brandolese and Grzegorz Karch, Far field asymptotics of solutions to convection equation with anomalous diffusion, J. Evol. Equ. 8 (2008), no. 2, 307–326. MR 2407204, DOI 10.1007/s00028-008-0356-9
Luis A. Caffarelli and Alexis Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2) 171 (2010), no. 3, 1903–1930. MR 2680400, DOI 10.4007/annals.2010.171.1903
Dongho Chae and Jihoon Lee, Global well-posedness in the super-critical dissipative quasi-geostrophic equations, Comm. Math. Phys. 233 (2003), no. 2, 297–311. MR 1962043, DOI 10.1007/s00220-002-0750-z
Peter Constantin, Diego Cordoba, and Jiahong Wu, On the critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J. 50 (2001), no. Special Issue, 97–107. Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000). MR 1855665, DOI 10.1512/iumj.2008.57.3629
Peter Constantin, Andrew J. Majda, and Esteban Tabak, Formation of strong fronts in the $2$-D quasigeostrophic thermal active scalar, Nonlinearity 7 (1994), no. 6, 1495–1533. MR 1304437
Antonio Córdoba and Diego Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys. 249 (2004), no. 3, 511–528. MR 2084005, DOI 10.1007/s00220-004-1055-1
Miguel Escobedo and Enrike Zuazua, Large time behavior for convection-diffusion equations in $\textbf {R}^N$, J. Funct. Anal. 100 (1991), no. 1, 119–161. MR 1124296, DOI 10.1016/0022-1236(91)90105-E
Hichem Hajaiej, Xinwei Yu, and Zhichun Zhai, Fractional Gagliardo-Nirenberg and Hardy inequalities under Lorentz norms, J. Math. Anal. Appl. 396 (2012), no. 2, 569–577. MR 2961251, DOI 10.1016/j.jmaa.2012.06.054
Lars Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–140. MR 121655, DOI 10.1007/BF02547187
Ning Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. Math. Phys. 255 (2005), no. 1, 161–181. MR 2123380, DOI 10.1007/s00220-004-1256-7
Ning Ju, Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space, Comm. Math. Phys. 251 (2004), no. 2, 365–376. MR 2100059, DOI 10.1007/s00220-004-1062-2
Agnieszka Kałamajska and Katarzyna Pietruska-Pałuba, Gagliardo-Nirenberg inequalities in logarithmic spaces, Colloq. Math. 106 (2006), no. 1, 93–107. MR 2234739, DOI 10.4064/cm106-1-8
A. Kiselev, F. Nazarov, and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 (2007), no. 3, 445–453. MR 2276260, DOI 10.1007/s00222-006-0020-3
Ryo Kobayashi and Shuichi Kawashima, Decay estimates and large time behavior of solutions to the drift-diffusion system, Funkcial. Ekvac. 51 (2008), no. 3, 371–394. MR 2493875, DOI 10.1619/fesi.51.371
Vassili N. Kolokoltsov, Markov processes, semigroups and generators, De Gruyter Studies in Mathematics, vol. 38, Walter de Gruyter & Co., Berlin, 2011. MR 2780345
V. A. Liskevich and Yu. A. Semenov, Some problems on Markov semigroups, Schrödinger operators, Markov semigroups, wavelet analysis, operator algebras, Math. Top., vol. 11, Akademie Verlag, Berlin, 1996, pp. 163–217. MR 1409835
S. G. Mikhlin, Multidimensional singular integrals and integral equations, Pergamon Press, Oxford-New York-Paris, 1965. Translated from the Russian by W. J. A. Whyte; Translation edited by I. N. Sneddon. MR 0185399
Hideyuki Miura, Dissipative quasi-geostrophic equation for large initial data in the critical Sobolev space, Comm. Math. Phys. 267 (2006), no. 1, 141–157. MR 2238907, DOI 10.1007/s00220-006-0023-3
Tetsuro Miyakawa, On space-time decay properties of nonstationary incompressible Navier-Stokes flows in $\textbf {R}^n$, Funkcial. Ekvac. 43 (2000), no. 3, 541–557. MR 1815476
Tetsuro Miyakawa and Maria Elena Schonbek, On optimal decay rates for weak solutions to the Navier-Stokes equations in $\Bbb R^n$, Proceedings of Partial Differential Equations and Applications (Olomouc, 1999), 2001, pp. 443–455. MR 1844282
Yoshihiro Mizuta, Eiichi Nakai, Yoshihiro Sawano, and Tetsu Shimomura, Gagliardo-Nirenberg inequality for generalized Riesz potentials of functions in Musielak-Orlicz spaces, Arch. Math. (Basel) 98 (2012), no. 3, 253–263. MR 2897448, DOI 10.1007/s00013-012-0362-6
Takayoshi Ogawa and Masakazu Yamamoto, Asymptotic behavior of solutions to drift-diffusion system with generalized dissipation, Math. Models Methods Appl. Sci. 19 (2009), no. 6, 939–967. MR 2535840, DOI 10.1142/S021820250900367X
Yoshihiro Shibata and Senjo Shimizu, A decay property of the Fourier transform and its application to the Stokes problem, J. Math. Fluid Mech. 3 (2001), no. 3, 213–230. MR 1860123, DOI 10.1007/PL00000970
Luis Silvestre, Eventual regularization for the slightly supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 27 (2010), no. 2, 693–704 (English, with English and French summaries). MR 2595196, DOI 10.1016/j.anihpc.2009.11.006
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Masakazu Yamamoto and Yuusuke Sugiyama, Asymptotic expansion of solutions to the drift-diffusion equation with fractional dissipation, Nonlinear Anal. 141 (2016), 57–87. MR 3512399, DOI 10.1016/j.na.2016.03.021
Masakazu Yamamoto and Yuusuke Sugiyama, Spatial-decay of solutions to the quasi-geostrophic equation with the critical and supercritical dissipation, Nonlinearity 32 (2019), no. 7, 2467–2480. MR 3957219, DOI 10.1088/1361-6544/ab0e5a
William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3