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Generalized surface quasi-geostrophic equations with singular velocities. (English) Zbl 1244.35108
Summary: This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasigeostrophic (SQG) equation with the velocity field \(u\) related to the scalar \(\theta \) by \(u=\nabla^{\bot}\Lambda^{\beta -2}\theta\), where \(1<\beta \leq 2\) and \(\Lambda =(-\Delta)^{1/2}\) is the Zygmund operator. The borderline case \(\beta = 1\) corresponds to the SQG equation and the situation is more singular for \(\beta > 1\). We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions, and the local existence of patch-type solutions. The second family is a dissipative active scalar equation with \(u=\nabla^{\bot}(\log (I-\delta ))^{\mu}\theta\) for \(\mu > 0\), which is at least logarithmically more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. This result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani [Dissipative and ideal surface quasi-geostrophic equations. Lecture presented at ICMS, Edinburgh (2010)].

35Q35 PDEs in connection with fluid mechanics
35D30 Weak solutions to PDEs
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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[1] Abidi, On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal. 40 (1) pp 167– (2008) · Zbl 1157.76054
[2] Chae, Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations, Arch. Ration. Mech. Anal. 202 (1) pp 35– (2011) · Zbl 1266.76010
[3] Chamorro , D. Remarks on a fractional diffusion transport equation with applications to the critical dissipative quasi-geostrophic equation
[4] Chemin, Perfect incompressible fluids. Oxford Lecture Series in Mathematics and Its Applications 14 (1998) · Zbl 0927.76002
[5] Chen, A new Bernstein’s inequality and the 2D dissipative quasigeostrophic equation, Commun. Math. Phys. 271 (3) pp 821– (2007) · Zbl 1142.35069
[6] Gancedo, Existence for the {\(\alpha\)}-patch model and the QG sharp front in Sobolev spaces, Adv. Math. 217 (6) pp 2569– (2008) · Zbl 1148.35099
[7] Kiselev, Global well-posedness for the critical 2D dissipative quasigeostrophic equation, Invent. Math. 167 (3) pp 445– (2007) · Zbl 1121.35115
[8] Lemarié-Rieusset, Chapman & Hall/CRC Research Notes in Mathematics 431 (2002)
[9] Li, Existence theorems for the 2D quasi-geostrophic equation with plane wave initial conditions, Nonlinearity 22 (7) pp 1639– (2009) · Zbl 1177.35170
[10] Majda, Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics 27 (2002) · Zbl 0983.76001
[11] Ohkitani , K. Dissipative and ideal surface quasi-geostrophic equations · Zbl 1309.35091
[12] Resnick , S. Dynamical problems in nonlinear advective partial differential equations 1995
[13] Runst, de Gruyter Series in Nonlinear Analysis and Applications 3 (1996)
[14] Stefanov, Global well-posedness for the 2D quasi-geostrophic equation in a critical Besov space, Electron. J. Differential Equations 150 pp 9– (2007) · Zbl 1133.35081
[15] Wang, Local well-posedness for the 2D non-dissipative quasi-geostrophic equation in Besov spaces, Nonlinear Anal. 70 (11) pp 3791– (2009) · Zbl 1160.76053
[16] Zhou, Asymptotic behaviour of the solutions to the 2D dissipative quasi-geostrophic flows, Nonlinearity 21 (9) pp 2061– (2008) · Zbl 1186.35170
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