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April  2022, 21(4): 1209-1224. doi: 10.3934/cpaa.2022016

On analyticity up to the boundary for critical quasi-geostrophic equation in the half space

Mathematical Institute, Tohoku University, Sendai, 980-8578, JAPAN

Received  September 2021 Revised  December 2021 Published  April 2022 Early access  February 2022

Fund Project: The author was supported by the Grant-in-Aid for Young Scientists (A) (No. 17H04824) from JSPS

We study the Cauchy problem for the surface quasi-geostrophic equation with the critical dissipation in the two dimensional half space under the homogeneous Dirichlet boundary condition. We show the global existence, the uniqueness and the analyticity of solutions, and the real analyticity up to the boundary is obtained. We will show a natural ways to estimate the nonlinear term for functions satisfying the Dirichlet boundary condition.

Citation: 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
References:
[1]

J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4), 14 (1981), 209–246.

[2]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math. (2), 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903.

[3]

P. Constantin, Energy Spectrum of Quasigeostrophic Turbulence, Phys. Rev. Lett., 89 (2002), 184501. 

[4]

P. Constantin, Nonlocal nonlinear advection-diffusion equations, Chin. Ann. Math. Ser. B, 38 (2017), 281-292.  doi: 10.1007/s11401-016-1071-4.

[5]

P. Constantin, D. Cordoba and J. Wu, On the critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J., 50 (2001), Special Issue, 97–107. doi: 10.1512/iumj. 2008.57.3629.

[6]

P. Constantin and M. Ignatova, Critical SQG in bounded domains, Ann. Partial Differ. Equ., 2 (2016), 42 pp. doi: 10.1007/s40818-016-0017-1.

[7]

P. Constantin and M. Ignatova, Remarks on the fractional Laplacian with Dirichlet boundary conditions and applications, Int. Math. Res. Not. IMRN, (2017), 1653-1673.  doi: 10.1093/imrn/rnw098.

[8]

P. Constantin and M. Ignatova, Estimates near the boundary for critical SQG, Ann. Partial Differ. Equ., 6 (2020), 30 pp. doi: 10.1007/s40818-020-00079-7.

[9]

P. Constantin and H. Q. Nguyen, Global weak solutions for SQG in bounded domains, Commun. Pure Appl. Math., 71 (2018), 2323-2333.  doi: 10.1002/cpa.21720.

[10]

P. Constantin and H. Q. Nguyen, Local and global strong solutions for SQG in bounded domains, Phys. D, 376/377 (2018), 195-203.  doi: 10.1016/j.physd.2017.08.008.

[11]

P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22 (2012), 1289-1321.  doi: 10.1007/s00039-012-0172-9.

[12]

P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948.  doi: 10.1137/S0036141098337333.

[13]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Commun. Math. Phys., 249 (2004), 511-528.  doi: 10.1007/s00220-004-1055-1.

[14]

M. Coti Zelati and V. Vicol, On the global regularity for the supercritical SQG equation, Indiana Univ. Math. J., 65 (2016), 535-552.  doi: 10.1512/iumj.2016.65.5807.

[15]

M. Ignatova, Construction of solutions of the critical SQG equation in bounded domains, Adv. Math., 351 (2019), 1000-1023.  doi: 10.1016/j.aim.2019.05.034.

[16]

T. Iwabuchi, Global solutions for the critical Burgers equation in the Besov spaces and the large time behavior, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 687-713.  doi: 10.1016/j.anihpc.2014.03.002.

[17]

T. Iwabuchi, The semigroup generated by the Dirichlet Laplacian of fractional order, Anal. PDE, 11 (2018), 683-703.  doi: 10.2140/apde.2018.11.683.

[18]

T. Iwabuchi, Derivatives on function spaces generated by the Dirichlet Laplacian and the Neumann Laplacian in one dimension, Commun. Math. Anal., 21 (2018), 1-8. 

[19]

T. Iwabuchi, Analyticity and large time behavior for the Burgers equation and the quasi-geostrophic equation, the both with the critical dissipation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 855-876.  doi: 10.1016/j.anihpc.2020.02.003.

[20]

T. Iwabuchi, The Leibniz rule for the Dirichlet and the Neumann Laplacian, to appear in Tohoku Math. J., arXiv: 1905.02854v2.

[21]

T. IwabuchiT. Matsuyama and K. Taniguchi, Boundedness of spectral multipliers for Schrödinger operators on open sets, Rev. Mat. Iberoam., 34 (2018), 1277-1322.  doi: 10.4171/RMI/1024.

[22]

T. IwabuchiT. Matsuyama and K. Taniguchi, Besov spaces on open sets, Bull. Sci. Math., 152 (2019), 93-149.  doi: 10.1016/j.bulsci.2019.01.008.

[23]

A. KiselevF. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453.  doi: 10.1007/s00222-006-0020-3.

[24]

L. D. Landau and E. M. Lifshitz, Fluid mechanics, Translated from the Russian by J. B. Sykes and W. H. Reid. Course of Theoretical Physics, Vol. 6, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass. 1959.

[25] E. M. Ouhabaz, Analysis of heat equations on domains, Princeton University Press, Princeton, NJ, 2005. 
[26]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag New York, 1979.

[27]

L. F. Stokols and and A. F. Vasseur, Hölder regularity up to the boundary for critical SQG on bounded domains, Arch. Ration. Mech. Anal., 236 (2020), 1543-1591.  doi: 10.1007/s00205-020-01498-3.

[28]

X. Thinh DuongE. M. Ouhabaz and and A. Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., 196 (2002), 443-485.  doi: 10.1016/S0022-1236(02)00009-5.

[29]

H. Triebel, Theory of Function Spaces, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[30]

H. Wang and and Z. Zhang, A frequency localized maximum principle applied to the 2D quasi-geostrophic equation, Commun. Math. Phys., 301 (2011), 105-129.  doi: 10.1007/s00220-010-1144-2.

show all references

References:
[1]

J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4), 14 (1981), 209–246.

[2]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math. (2), 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903.

[3]

P. Constantin, Energy Spectrum of Quasigeostrophic Turbulence, Phys. Rev. Lett., 89 (2002), 184501. 

[4]

P. Constantin, Nonlocal nonlinear advection-diffusion equations, Chin. Ann. Math. Ser. B, 38 (2017), 281-292.  doi: 10.1007/s11401-016-1071-4.

[5]

P. Constantin, D. Cordoba and J. Wu, On the critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J., 50 (2001), Special Issue, 97–107. doi: 10.1512/iumj. 2008.57.3629.

[6]

P. Constantin and M. Ignatova, Critical SQG in bounded domains, Ann. Partial Differ. Equ., 2 (2016), 42 pp. doi: 10.1007/s40818-016-0017-1.

[7]

P. Constantin and M. Ignatova, Remarks on the fractional Laplacian with Dirichlet boundary conditions and applications, Int. Math. Res. Not. IMRN, (2017), 1653-1673.  doi: 10.1093/imrn/rnw098.

[8]

P. Constantin and M. Ignatova, Estimates near the boundary for critical SQG, Ann. Partial Differ. Equ., 6 (2020), 30 pp. doi: 10.1007/s40818-020-00079-7.

[9]

P. Constantin and H. Q. Nguyen, Global weak solutions for SQG in bounded domains, Commun. Pure Appl. Math., 71 (2018), 2323-2333.  doi: 10.1002/cpa.21720.

[10]

P. Constantin and H. Q. Nguyen, Local and global strong solutions for SQG in bounded domains, Phys. D, 376/377 (2018), 195-203.  doi: 10.1016/j.physd.2017.08.008.

[11]

P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22 (2012), 1289-1321.  doi: 10.1007/s00039-012-0172-9.

[12]

P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948.  doi: 10.1137/S0036141098337333.

[13]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Commun. Math. Phys., 249 (2004), 511-528.  doi: 10.1007/s00220-004-1055-1.

[14]

M. Coti Zelati and V. Vicol, On the global regularity for the supercritical SQG equation, Indiana Univ. Math. J., 65 (2016), 535-552.  doi: 10.1512/iumj.2016.65.5807.

[15]

M. Ignatova, Construction of solutions of the critical SQG equation in bounded domains, Adv. Math., 351 (2019), 1000-1023.  doi: 10.1016/j.aim.2019.05.034.

[16]

T. Iwabuchi, Global solutions for the critical Burgers equation in the Besov spaces and the large time behavior, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 687-713.  doi: 10.1016/j.anihpc.2014.03.002.

[17]

T. Iwabuchi, The semigroup generated by the Dirichlet Laplacian of fractional order, Anal. PDE, 11 (2018), 683-703.  doi: 10.2140/apde.2018.11.683.

[18]

T. Iwabuchi, Derivatives on function spaces generated by the Dirichlet Laplacian and the Neumann Laplacian in one dimension, Commun. Math. Anal., 21 (2018), 1-8. 

[19]

T. Iwabuchi, Analyticity and large time behavior for the Burgers equation and the quasi-geostrophic equation, the both with the critical dissipation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 855-876.  doi: 10.1016/j.anihpc.2020.02.003.

[20]

T. Iwabuchi, The Leibniz rule for the Dirichlet and the Neumann Laplacian, to appear in Tohoku Math. J., arXiv: 1905.02854v2.

[21]

T. IwabuchiT. Matsuyama and K. Taniguchi, Boundedness of spectral multipliers for Schrödinger operators on open sets, Rev. Mat. Iberoam., 34 (2018), 1277-1322.  doi: 10.4171/RMI/1024.

[22]

T. IwabuchiT. Matsuyama and K. Taniguchi, Besov spaces on open sets, Bull. Sci. Math., 152 (2019), 93-149.  doi: 10.1016/j.bulsci.2019.01.008.

[23]

A. KiselevF. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453.  doi: 10.1007/s00222-006-0020-3.

[24]

L. D. Landau and E. M. Lifshitz, Fluid mechanics, Translated from the Russian by J. B. Sykes and W. H. Reid. Course of Theoretical Physics, Vol. 6, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass. 1959.

[25] E. M. Ouhabaz, Analysis of heat equations on domains, Princeton University Press, Princeton, NJ, 2005. 
[26]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag New York, 1979.

[27]

L. F. Stokols and and A. F. Vasseur, Hölder regularity up to the boundary for critical SQG on bounded domains, Arch. Ration. Mech. Anal., 236 (2020), 1543-1591.  doi: 10.1007/s00205-020-01498-3.

[28]

X. Thinh DuongE. M. Ouhabaz and and A. Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., 196 (2002), 443-485.  doi: 10.1016/S0022-1236(02)00009-5.

[29]

H. Triebel, Theory of Function Spaces, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[30]

H. Wang and and Z. Zhang, A frequency localized maximum principle applied to the 2D quasi-geostrophic equation, Commun. Math. Phys., 301 (2011), 105-129.  doi: 10.1007/s00220-010-1144-2.

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