January  2022, 21(1): 101-120. doi: 10.3934/cpaa.2021169

On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces

1. 

Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA

2. 

Department of Mathematics and Statistics, CUNY Hunter College, New York, NY 10065, USA

* Corresponding author

Received  May 2021 Revised  September 2021 Published  January 2022 Early access  September 2021

Fund Project: The research of M.S.J. and A.K. was supported in part by the NSF grant DMS-1818754. The research of V.R.M. was supported in part by the PSC-CUNY grant 64335-00 52

This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar $ \theta $ on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity $ u $ is of lower singularity, i.e., $ u = -\nabla^{\perp} \Lambda^{ \beta-2}p( \Lambda) \theta $, where $ p $ is a logarithmic smoothing operator and $ \beta \in [0, 1] $. We complete this study by considering the more singular regime $ \beta\in(1, 2) $. The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness.

Citation: Michael S. Jolly, Anuj Kumar, Vincent R. Martinez. On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces. Communications on Pure and Applied Analysis, 2022, 21 (1) : 101-120. doi: 10.3934/cpaa.2021169
References:
[1]

H. Bahouri, J. Y. Chemin, and R. Danchin, Fourier analysis and nonlinear partial differential equations, in Grundlehren der Mathematischen Wissenschaften, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

J. Bourgain and D. Li, Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces, Invent. Math., 201 (2015), 97-157.  doi: 10.1007/s00222-014-0548-6.

[3]

D. ChaeP. ConstantinD. CórdobaF. Gancedo and J. Wu, Generalized surface quasi-geostrophic equations with singular velocities, Commun. Pure Appl. Math., 65 (2012), 1037-1066.  doi: 10.1002/cpa.21390.

[4]

D. ChaeP. Constantin and J. Wu, Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations, Arch. Ration. Mech. Anal., 202 (2011), 35-62.  doi: 10.1007/s00205-011-0411-5.

[5]

D. Chae and J. Wu, Logarithmically regularized inviscid models in borderline Sobolev spaces, J. Math. Phys., 53 (2012), 115601-15.  doi: 10.1063/1.4725531.

[6]

J. Y. Chemin, Perfect incompressible fluids, in Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press Oxford University Press, New York, 1998.

[7] P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, Chicago, IL, 1988. 
[8]

D. CórdobaJ. Gómez-Serrano and A. D. Ionescu, Global solutions for the generalized SQG patch equation, Arch. Ration. Mech. Anal., 233 (2019), 1211-1251.  doi: 10.1007/s00205-019-01377-6.

[9]

T. M. Elgindi and I. Jeong, Ill-posedness for the incompressible Euler equations in critical {S}obolev spaces, Ann. PDE, 3 (2017), 1-19.  doi: 10.1007/s40818-017-0027-7.

[10]

T. M. Elgindi and N. Masmoudi, $L^\infty$ ill-posedness for a class of equations arising in hydrodynamics, Arch. Ration. Mech. Anal., 235 (2020), 1979-2025.  doi: 10.1007/s00205-019-01457-7.

[11]

F. Flandoli and M. Saal, mSQG equations in distributional spaces and point vortex approximation, J. Evol. Equ., 19 (2019), 1071-1090.  doi: 10.1007/s00028-019-00506-8.

[12]

F. Gancedo and N. Patel, On the local existence and blow-up for generalized sqg patches, Ann. PDE, 7 (2021), 1-63.  doi: 10.1007/s40818-021-00095-1.

[13]

C. Geldhauser and M. Romito, The point vortex model for the Euler equation, AIMS Math., 4 (2019), 534-575.  doi: 10.3934/math.2019.3.534.

[14]

C. Geldhauser and M. Romito, Point vortices for inviscid generalized surface quasi-geostrophic models, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2583-2606.  doi: 10.3934/dcdsb.2020023.

[15]

A. A. Himonas and G. Misiołek, Non-uniform dependence on initial data of solutions to the Euler equations of hydrodynamics, Commun. Math. Phys., 296 (2010), 285-301.  doi: 10.1007/s00220-010-0991-1.

[16]

W. HuI. Kukavica and M. Ziane, Sur l'existence locale pour une équation de scalaires actifs, C. R. Math. Acad. Sci. Paris, 353 (2015), 241-245.  doi: 10.1016/j.crma.2014.12.008.

[17]

M. S. Jolly, A. Kumar and V. R. Martinez, On the existence, uniqueness, and smoothing of solutions to the generalized SQG equations in critical Sobolev spaces, Commun. Math. Phys., (2021), 1–46. doi: 10.1007/s00220-021-04124-9.

[18]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, Lecture Notes in Math., 1975.

[19]

A. KiselevL. RyzhikY. Yao and A. Zlatoš, Finite time singularity for the modified SQG patch equation, Ann. Math., 184 (2016), 909-948.  doi: 10.4007/annals.2016.184.3.7.

[20]

A. KiselevY. Yao and A. Zlatoš, Local regularity for the modified SQG patch equation, Commun. Pure Appl. Math., 70 (2017), 1253-1315.  doi: 10.1002/cpa.21677.

[21]

I. Kukavica, V. Vicol and F. Wang, On the ill-posedness of active scalar equations with odd singular kernels, in New Trends in Differential Equations, Control Theory and Optimization, World Sci. Publ., Hackensack, NJ, 2016.

[22]

A. Kumar, Thesis (Ph.D.)–Indiana University, Department of Mathematics

[23]

H. Kwon, Strong ill-posedness of logarithmically regularized 2D Euler equations in the borderline Sobolev space, J. Funct. Anal., 280 (2020), 108822, 55 pp. doi: 10.1016/j.jfa.2020.108822.

[24]

O. Lazar and L. Xue, Regularity results for a class of generalized surface quasi-geostrophic equations, J. Math. Pures Appl., 130 (2019), 200-250.  doi: 10.1016/j.matpur.2019.01.009.

[25]

P. G. Lemarié-Rieusset, Recent developments in the Navier-Stokes problem, in Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.

[26]

D. Luo and M. Saal, A scaling limit for the stochastic mSQG equations with multiplicative transport noises, Stoch. Dyn., 20 (2020), 2040001, 21 pp. doi: 10.1142/S0219493720400018.

[27]

G. Misiołek and T. Yoneda, Ill-posedness examples for the quasi-geostrophic and the Euler equations, in Analysis, geometry and quantum field theory, Amer. Math. Soc., Providence, RI, 2012. doi: 10.1090/conm/584/11589.

[28]

G. Misiołek and T. Yoneda, Continuity of the solution map of the Euler equations in Hölder spaces and weak norm inflation in Besov spaces, Trans. Amer. Math. Soc., 370 (2018), 4709-4730.  doi: 10.1090/tran/7101.

[29]

A. R. NahmodN. PavlovićG. Staffilani and N. Totz, Global flows with invariant measures for the inviscid modified SQG equations, Stoch. Partial Differ. Equ. Anal. Comput., 6 (2018), 184-210.  doi: 10.1007/s40072-017-0106-5.

[30]

H. Q. Nguyen, Global weak solutions for generalized SQG in bounded domains, Anal. PDE, 11 (2018), 1029-1047. 

[31]

T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin, 1996.

[32]

M. Vishik, Hydrodynamics in Besov spaces, Arch. Ration. Mech. Anal., 145 (1998), 197-214. 

[33]

H. YuX. Zheng and Q. Jiu, Remarks on well-posedness of the generalized surface quasi-geostrophic equation, Arch. Ration. Mech. Anal., 232 (2019), 265-301.  doi: 10.1007/s00205-018-1320-7.

show all references

References:
[1]

H. Bahouri, J. Y. Chemin, and R. Danchin, Fourier analysis and nonlinear partial differential equations, in Grundlehren der Mathematischen Wissenschaften, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

J. Bourgain and D. Li, Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces, Invent. Math., 201 (2015), 97-157.  doi: 10.1007/s00222-014-0548-6.

[3]

D. ChaeP. ConstantinD. CórdobaF. Gancedo and J. Wu, Generalized surface quasi-geostrophic equations with singular velocities, Commun. Pure Appl. Math., 65 (2012), 1037-1066.  doi: 10.1002/cpa.21390.

[4]

D. ChaeP. Constantin and J. Wu, Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations, Arch. Ration. Mech. Anal., 202 (2011), 35-62.  doi: 10.1007/s00205-011-0411-5.

[5]

D. Chae and J. Wu, Logarithmically regularized inviscid models in borderline Sobolev spaces, J. Math. Phys., 53 (2012), 115601-15.  doi: 10.1063/1.4725531.

[6]

J. Y. Chemin, Perfect incompressible fluids, in Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press Oxford University Press, New York, 1998.

[7] P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, Chicago, IL, 1988. 
[8]

D. CórdobaJ. Gómez-Serrano and A. D. Ionescu, Global solutions for the generalized SQG patch equation, Arch. Ration. Mech. Anal., 233 (2019), 1211-1251.  doi: 10.1007/s00205-019-01377-6.

[9]

T. M. Elgindi and I. Jeong, Ill-posedness for the incompressible Euler equations in critical {S}obolev spaces, Ann. PDE, 3 (2017), 1-19.  doi: 10.1007/s40818-017-0027-7.

[10]

T. M. Elgindi and N. Masmoudi, $L^\infty$ ill-posedness for a class of equations arising in hydrodynamics, Arch. Ration. Mech. Anal., 235 (2020), 1979-2025.  doi: 10.1007/s00205-019-01457-7.

[11]

F. Flandoli and M. Saal, mSQG equations in distributional spaces and point vortex approximation, J. Evol. Equ., 19 (2019), 1071-1090.  doi: 10.1007/s00028-019-00506-8.

[12]

F. Gancedo and N. Patel, On the local existence and blow-up for generalized sqg patches, Ann. PDE, 7 (2021), 1-63.  doi: 10.1007/s40818-021-00095-1.

[13]

C. Geldhauser and M. Romito, The point vortex model for the Euler equation, AIMS Math., 4 (2019), 534-575.  doi: 10.3934/math.2019.3.534.

[14]

C. Geldhauser and M. Romito, Point vortices for inviscid generalized surface quasi-geostrophic models, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2583-2606.  doi: 10.3934/dcdsb.2020023.

[15]

A. A. Himonas and G. Misiołek, Non-uniform dependence on initial data of solutions to the Euler equations of hydrodynamics, Commun. Math. Phys., 296 (2010), 285-301.  doi: 10.1007/s00220-010-0991-1.

[16]

W. HuI. Kukavica and M. Ziane, Sur l'existence locale pour une équation de scalaires actifs, C. R. Math. Acad. Sci. Paris, 353 (2015), 241-245.  doi: 10.1016/j.crma.2014.12.008.

[17]

M. S. Jolly, A. Kumar and V. R. Martinez, On the existence, uniqueness, and smoothing of solutions to the generalized SQG equations in critical Sobolev spaces, Commun. Math. Phys., (2021), 1–46. doi: 10.1007/s00220-021-04124-9.

[18]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, Lecture Notes in Math., 1975.

[19]

A. KiselevL. RyzhikY. Yao and A. Zlatoš, Finite time singularity for the modified SQG patch equation, Ann. Math., 184 (2016), 909-948.  doi: 10.4007/annals.2016.184.3.7.

[20]

A. KiselevY. Yao and A. Zlatoš, Local regularity for the modified SQG patch equation, Commun. Pure Appl. Math., 70 (2017), 1253-1315.  doi: 10.1002/cpa.21677.

[21]

I. Kukavica, V. Vicol and F. Wang, On the ill-posedness of active scalar equations with odd singular kernels, in New Trends in Differential Equations, Control Theory and Optimization, World Sci. Publ., Hackensack, NJ, 2016.

[22]

A. Kumar, Thesis (Ph.D.)–Indiana University, Department of Mathematics

[23]

H. Kwon, Strong ill-posedness of logarithmically regularized 2D Euler equations in the borderline Sobolev space, J. Funct. Anal., 280 (2020), 108822, 55 pp. doi: 10.1016/j.jfa.2020.108822.

[24]

O. Lazar and L. Xue, Regularity results for a class of generalized surface quasi-geostrophic equations, J. Math. Pures Appl., 130 (2019), 200-250.  doi: 10.1016/j.matpur.2019.01.009.

[25]

P. G. Lemarié-Rieusset, Recent developments in the Navier-Stokes problem, in Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.

[26]

D. Luo and M. Saal, A scaling limit for the stochastic mSQG equations with multiplicative transport noises, Stoch. Dyn., 20 (2020), 2040001, 21 pp. doi: 10.1142/S0219493720400018.

[27]

G. Misiołek and T. Yoneda, Ill-posedness examples for the quasi-geostrophic and the Euler equations, in Analysis, geometry and quantum field theory, Amer. Math. Soc., Providence, RI, 2012. doi: 10.1090/conm/584/11589.

[28]

G. Misiołek and T. Yoneda, Continuity of the solution map of the Euler equations in Hölder spaces and weak norm inflation in Besov spaces, Trans. Amer. Math. Soc., 370 (2018), 4709-4730.  doi: 10.1090/tran/7101.

[29]

A. R. NahmodN. PavlovićG. Staffilani and N. Totz, Global flows with invariant measures for the inviscid modified SQG equations, Stoch. Partial Differ. Equ. Anal. Comput., 6 (2018), 184-210.  doi: 10.1007/s40072-017-0106-5.

[30]

H. Q. Nguyen, Global weak solutions for generalized SQG in bounded domains, Anal. PDE, 11 (2018), 1029-1047. 

[31]

T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin, 1996.

[32]

M. Vishik, Hydrodynamics in Besov spaces, Arch. Ration. Mech. Anal., 145 (1998), 197-214. 

[33]

H. YuX. Zheng and Q. Jiu, Remarks on well-posedness of the generalized surface quasi-geostrophic equation, Arch. Ration. Mech. Anal., 232 (2019), 265-301.  doi: 10.1007/s00205-018-1320-7.

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