Contour dynamics for surface quasi-geostrophic fronts. (English) Zbl 1450.35217
Summary: We derive contour dynamics equations for the motion of infinite planar surface quasi-geostrophic fronts that can be represented as a graph. We give two different derivations with the same result: one is based on a decomposition of the front velocity field into a shear flow and a perturbation that decays away from the front; the other is based on the interpretation of the Riesz transform of the piecewise-constant \(L^\infty\)-function that jumps across the front as an element of BMO. The resulting equation is equivalent to one derived by a regularization procedure introduced previously by J. K. Hunter and J. Shu [Nonlinearity 31, No. 6, 2480–2517 (2018; Zbl 1391.35328)].
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q86 | PDEs in connection with geophysics |
| 76U60 | Geophysical flows |
| 35B65 | Smoothness and regularity of solutions to PDEs |
Citations:
Zbl 1391.35328References:
| [1] | Hunter J K and Shu J 2018 Regularized and approximate equations for sharp fronts in the surface quasi-geostrophic equation and its generalization Nonlinearity31 2480-517 · Zbl 1391.35328 · doi:10.1088/1361-6544/aab1cc |
| [2] | Fefferman C and Rodrigo J L 2011 Analytic sharp fronts for the surface quasi-geostrophic equation Commun. Math. Phys.303 261-88 · Zbl 1228.35010 · doi:10.1007/s00220-011-1190-4 |
| [3] | Rodrigo J L 2005 On the evolution of sharp fronts for the quasi-geostrophic equation Commun. Pure Appl. Math.58 821-66 · Zbl 1073.35006 · doi:10.1002/cpa.20059 |
| [4] | Córdoba D, Fefferman C and Rodrigo J L 2004 Almost sharp fronts for the surface quasi-geostrophic equation Proc. Natl Acad. Sci. USA101 2687-91 · Zbl 1063.76011 · doi:10.1073/pnas.0308154101 |
| [5] | Fefferman C, Luli G and Rodrigo J 2012 The spine of an SQG almost-sharp front Nonlinearity25 329-42 · Zbl 1235.76019 · doi:10.1088/0951-7715/25/2/329 |
| [6] | Fefferman C and Rodrigo J L 2012 Almost sharp fronts for SQG: the limit equations Commun. Math. Phys.313 131-53 · Zbl 1272.35006 · doi:10.1007/s00220-012-1486-z |
| [7] | Fefferman C and Rodrigo J L 2015 Construction of almost-sharp fronts for the surface quasi-geostrophic equation Arch. Ration. Mech. Anal.218 123-62 · Zbl 1323.35146 · doi:10.1007/s00205-015-0857-y |
| [8] | Zabusky N, Hughes M H and Roberts K V 1979 Contour dynamics for the Euler equations in two dimensions J. Comput. Phys.30 96-106 · Zbl 0405.76014 · doi:10.1016/0021-9991(79)90089-5 |
| [9] | Majda A J and Bertozzi A L 2002 Vorticity and Incompressible Flow (Cambridge: Cambridge University Press) · Zbl 0983.76001 |
| [10] | Biello J and Hunter J K 2010 Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities Commun. Pure Appl. Math.63 303-36 · Zbl 1188.35119 |
| [11] | Castro A, Córdoba D and Gómez-Serrano J 2016 Uniformly rotating analytic global patch solutions for active scalars Annals PDE2 1-34 · Zbl 1397.35020 · doi:10.1007/s40818-016-0007-3 |
| [12] | Córdoba A, Córdoba D and Gancedo F 2018 Uniqueness for SQG patch solutions Trans. Am. Math. Soc. B 5 1-31 · Zbl 1390.35244 · doi:10.1090/btran/20 |
| [13] | Gancedo F 2008 Existence for the α-patch model and the QG sharp front in Sobolev spaces Adv. Math.217 2569-98 · Zbl 1148.35099 · doi:10.1016/j.aim.2007.10.010 |
| [14] | Gancedo F and Patel N 2018 On the local existence and blow-up for generalized SQG patches (arXiv:1811.00530) |
| [15] | Gancedo F and Strain R M 2014 Absence of splash singularities for SQG sharp fronts and the muskat problem Proc. Natl Acad. Sci.111 635-9 · Zbl 1355.76065 · doi:10.1073/pnas.1320554111 |
| [16] | Gómez-Serrano J 2019 On the existence of stationary patches Adv. Math.343 110-40 · Zbl 1408.35192 · doi:10.1016/j.aim.2018.11.012 |
| [17] | Kiselev A, Ryzhik L, Yao Y and Zlatoš A 2016 Finite time singularity formation for the modified SQG patch equation Ann. Math.184 909-48 · Zbl 1360.35159 · doi:10.4007/annals.2016.184.3.7 |
| [18] | Kiselev A, Yao Y and Zlatoš A 2017 Local regularity for the modified SQG patch equation Commun. Pure Appl. Math.70 1253-315 · Zbl 1371.35220 · doi:10.1002/cpa.21677 |
| [19] | Chae D, Constantin P, Córdoba D, Gancedo F and Wu J 2012 Generalized surface quasi-geostrophic equations with singular velocities Commun. Pure Appl. Math.65 1037-66 · Zbl 1244.35108 · doi:10.1002/cpa.21390 |
| [20] | Córdoba D, Gómez-Serrano J and Ionescu A D 2019 Global solutions for the generalized SQG patch equation Arch. Ration. Mech. Anal.233 1211-51 · Zbl 1420.35424 · doi:10.1007/s00205-019-01377-6 |
| [21] | Hunter J K, Shu J and Zhang Q 2018 Global solutions of a surface quasi-geostrophic front equation (arXiv:1808.07631) |
| [22] | Hunter J K, Shu J and Zhang Q 2018 Global solutions for a family of GSQG front equations (in preparation) |
| [23] | Hunter J K, Shu J and Zhang Q 2020 Two-front solutions of the SQG equation and its generalizations Commun Math. Sci. accepted · Zbl 1464.35227 |
| [24] | Duoandikoetxea J 2001 Fourier Analysis vol 29 (Providence, RI: American Mathematical Society) · Zbl 0969.42001 |
| [25] | Fefferman C and Stein E M 1972 Hp spaces of several variables Acta Math.129 137-93 · Zbl 0257.46078 · doi:10.1007/bf02392215 |
| [26] | Stein E M 1970 Singular Integrals and Differentiability Properties of Functions(Princeton Mathematical Series vol 30) (Princeton, NJ: Princeton University Press) · Zbl 0207.13501 |
| [27] | Stein E M 1993 Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals(Princeton Mathematical Series vol 43) (Princeton, NJ: Princeton University) · Zbl 0821.42001 |
| [28] | Majda A J 2003 Introduction to PDEs and Waves for the Atmosphere and Ocean(Courant Lecture Notes in Mathematics vol 9) (Providence, RI: American Mathematical Society) · Zbl 1278.76004 · doi:10.1090/cln/009 |
| [29] | Pedlosky J 1987 Geophysical Fluid Dynamics 2nd edn (New York: Springer) · Zbl 0713.76005 · doi:10.1007/978-1-4612-4650-3 |
| [30] | Lapeyre G 2017 Surface quasi-geostrophy Fluids2 · doi:10.3390/fluids2010007 |
| [31] | Kellogg O D 1967 Foundations of Potential Theory(Die Grundlehren der Mathematischen Wissenschaften) vol 31 (Berlin: Springer) (reprint) · Zbl 0152.31301 · doi:10.1007/978-3-642-86748-4 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
