# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021169
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## 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 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 & Applied Analysis, doi: 10.3934/cpaa.2021169
##### References:
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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.  Google Scholar [30] H. Q. Nguyen, Global weak solutions for generalized SQG in bounded domains, Anal. PDE, 11 (2018), 1029-1047.   Google Scholar [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. Google Scholar [32] M. Vishik, Hydrodynamics in Besov spaces, Arch. Ration. Mech. Anal., 145 (1998), 197-214.   Google Scholar [33] H. Yu, X. 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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [3] D. Chae, P. Constantin, D. Córdoba, F. 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.  Google Scholar [4] D. Chae, P. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, Chicago, IL, 1988.   Google Scholar [8] D. Córdoba, J. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] W. Hu, I. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] A. Kiselev, L. Ryzhik, Y. 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.  Google Scholar [20] A. Kiselev, Y. 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.  Google Scholar [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.  Google Scholar [22] A. Kumar, Thesis (Ph.D.)–Indiana University, Department of Mathematics Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] A. R. Nahmod, N. Pavlović, G. 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.  Google Scholar [30] H. Q. Nguyen, Global weak solutions for generalized SQG in bounded domains, Anal. PDE, 11 (2018), 1029-1047.   Google Scholar [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. Google Scholar [32] M. Vishik, Hydrodynamics in Besov spaces, Arch. Ration. Mech. Anal., 145 (1998), 197-214.   Google Scholar [33] H. Yu, X. 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.  Google Scholar
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