June  2022, 42(6): 2945-3003. doi: 10.3934/dcds.2022005

On the analyticity of the trajectories of the particles in the planar patch problem for some active scalar equations

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

2. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona and Centre de Recerca Matemàtica, 08193 Bellaterra, Barcelona, Catalonia, Spain

*Corresponding author: Joan Mateu

Received  December 2020 Published  June 2022 Early access  January 2022

Fund Project: Both authors are partially supported by grants 2017-SGR-0395 (Generalitat de Catalunya) and MDM-2014-044 (MICINN, Spain). The second named author is also partially supported by MTM-2016-75390 and PID2020-112881GB-I00 and Severo Ochoa and Maria de Maeztu program for centers CEX2020-001084-M

We prove analyticity in time of the particle trajectories associated with the solutions of some transport equations when the initial condition is the characteristic function of a regular bounded domain. These results are obtained from a detailed study of the Beurling transform, that represents a derivative of the velocity field. The precise estimates obtained for the solutions of an equation satisfied by the Lagrangian flow, are a key point in the development.

Citation: Josep M. Burgués, Joan Mateu. On the analyticity of the trajectories of the particles in the planar patch problem for some active scalar equations. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2945-3003. doi: 10.3934/dcds.2022005
References:
[1] K. AstalaT. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Mathematical Series, 48. Princeton University Press, Princeton, NJ, 2009. 
[2]

A. L. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math. Phys., 152 (1993), 19-28.  doi: 10.1007/BF02097055.

[3]

A. BertozziJ. GarnettT. Laurent and J. Verdera, The regularity of the boundary of a multidimensional aggregation patch, SIAM J. Math. Anal., 48 (2016), 3789-3819.  doi: 10.1137/15M1033125.

[4]

A. L. BertozziT. Laurent and F. Léger, Aggregation and spreading via the Newtonian potential: The dynamics of patch solutions, Math. Models Methods Appl. Sci., 22 (2012), 39 pp. 

[5]

J. M. Burgués, Defining functions for open sets in $\Bbb R^n$, Real Anal. Exchange, 31 (2005/06), 45-53. 

[6]

A. L. Cauchy, Baron, Théorie de La Propagation des Ondes à La Surface D'un Fluide Pesant D'une Profondeur Indéfinie, Académie royale des sciences, 1816.

[7]

J.-Y. Chemin, Régularité de la trajectoire des particules d'un fluide parfait incompressible remplissant l'espace, J. Math. Pures Appl., 71 (1992), 407-417. 

[8]

J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionnels, Ann. Sci. École Norm. Sup., 26 (1993), 517-542. 

[9] J.-Y. Chemin, Perfect Incompressible Fluids, Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie. Oxford Lecture Series in Mathematics and its Applications, 14. The Clarendon Press, Oxford University Press, New York, 1998. 
[10]

A. Dold, Lectures on Algebraic Topology, Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the 1972 edition. doi: 10.1007/978-3-642-67821-9.

[11]

J. Duoandikoetxea, Fourier Analysis, Translated and revised from the 1995 Spanish original by David Cruz-Uribe. Graduate Studies in Mathematics, 29. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/029.

[12]

U. Frisch and V. Zheligovsky, A very smooth ride in a rough sea, Comm. Math. Phys., 326 (2014), 499-505.  doi: 10.1007/s00220-013-1848-1.

[13]

T. W. Gamelin, Uniform Algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969.

[14]

J. Garnett, Analytic Capacity and Measure, Lecture Notes in Mathematics, Vol. 297. Springer-Verlag, Berlin-New York, 1972.

[15]

M. Hernandez, Mechanisms of Lagrangian analyticity in fluids, Arch. Ration. Mech. Anal., 233 (2019), 513-598.  doi: 10.1007/s00205-019-01363-y.

[16]

S. Hofmann, M. Mitrea and M. Taylor, Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains, Int. Math. Res. Not. IMRN, (2010), 2567–2865. doi: 10.1093/imrn/rnp214.

[17]

L. Hörmander, An Introduction to Complex Analysis in Several Variables, 3$^{rd}$ edition, North-Holland Mathematical Library, 7. North-Holland Publishing Co., Amsterdam, 1990.

[18]

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], 26. Springer-Verlag, Berlin, 1997.

[19]

V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 1032–1066.

[20] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002. 
[21]

J. MateuJ. Orobitg and J. Verdera, Extra cancellation of even Calderón-Zygmund operators and quasiconformal mappings, J. Math. Pures Appl., 91 (2009), 402-431.  doi: 10.1016/j.matpur.2009.01.010.

[22]

P. Serfati, Vortex Patches Generalisés Dans $\mathbb R^n$ et Resultat De Regularité Stratifiée Pour Le Laplacien-ch4B, 1991.

[23]

P. Serfati, Étude Mathématique De Flammes Infiniment Minces En Combustion. Résultats De Structure Et De Régularité Pour l'équation D'Euler Incompressible, PhD thesis, 1992.

[24]

P. Serfati, Une preuve directe d'existence globale des vortex patches 2D, C. R. Acad. Sci. Paris Sér. I Math., 318 (1994), 515-518. 

[25]

P. Serfati, Équation d'Euler et holomorphies à faible régularité spatiale, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 175-180. 

[26]

P. Serfati, Structures holomorphes à faible régularité spatiale en mécanique des fluides, J. Math. Pures Appl., 74 (1995), 95-104. 

[27]

A. Shnirelman, On the analyticity of particle trajectories in the ideal incompressible fluid, arXiv preprint, arXiv: 1205.5837, 2012.

[28]

X. Tolsa, Analytic Capacity, the Cauchy Transform, and Non-Homogeneous Calderón-Zygmund Theory, Progress in Mathematics, 307. Birkhäuser/Springer, Cham, 2014. doi: 10.1007/978-3-319-00596-6.

show all references

References:
[1] K. AstalaT. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Mathematical Series, 48. Princeton University Press, Princeton, NJ, 2009. 
[2]

A. L. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math. Phys., 152 (1993), 19-28.  doi: 10.1007/BF02097055.

[3]

A. BertozziJ. GarnettT. Laurent and J. Verdera, The regularity of the boundary of a multidimensional aggregation patch, SIAM J. Math. Anal., 48 (2016), 3789-3819.  doi: 10.1137/15M1033125.

[4]

A. L. BertozziT. Laurent and F. Léger, Aggregation and spreading via the Newtonian potential: The dynamics of patch solutions, Math. Models Methods Appl. Sci., 22 (2012), 39 pp. 

[5]

J. M. Burgués, Defining functions for open sets in $\Bbb R^n$, Real Anal. Exchange, 31 (2005/06), 45-53. 

[6]

A. L. Cauchy, Baron, Théorie de La Propagation des Ondes à La Surface D'un Fluide Pesant D'une Profondeur Indéfinie, Académie royale des sciences, 1816.

[7]

J.-Y. Chemin, Régularité de la trajectoire des particules d'un fluide parfait incompressible remplissant l'espace, J. Math. Pures Appl., 71 (1992), 407-417. 

[8]

J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionnels, Ann. Sci. École Norm. Sup., 26 (1993), 517-542. 

[9] J.-Y. Chemin, Perfect Incompressible Fluids, Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie. Oxford Lecture Series in Mathematics and its Applications, 14. The Clarendon Press, Oxford University Press, New York, 1998. 
[10]

A. Dold, Lectures on Algebraic Topology, Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the 1972 edition. doi: 10.1007/978-3-642-67821-9.

[11]

J. Duoandikoetxea, Fourier Analysis, Translated and revised from the 1995 Spanish original by David Cruz-Uribe. Graduate Studies in Mathematics, 29. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/029.

[12]

U. Frisch and V. Zheligovsky, A very smooth ride in a rough sea, Comm. Math. Phys., 326 (2014), 499-505.  doi: 10.1007/s00220-013-1848-1.

[13]

T. W. Gamelin, Uniform Algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969.

[14]

J. Garnett, Analytic Capacity and Measure, Lecture Notes in Mathematics, Vol. 297. Springer-Verlag, Berlin-New York, 1972.

[15]

M. Hernandez, Mechanisms of Lagrangian analyticity in fluids, Arch. Ration. Mech. Anal., 233 (2019), 513-598.  doi: 10.1007/s00205-019-01363-y.

[16]

S. Hofmann, M. Mitrea and M. Taylor, Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains, Int. Math. Res. Not. IMRN, (2010), 2567–2865. doi: 10.1093/imrn/rnp214.

[17]

L. Hörmander, An Introduction to Complex Analysis in Several Variables, 3$^{rd}$ edition, North-Holland Mathematical Library, 7. North-Holland Publishing Co., Amsterdam, 1990.

[18]

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], 26. Springer-Verlag, Berlin, 1997.

[19]

V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 1032–1066.

[20] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002. 
[21]

J. MateuJ. Orobitg and J. Verdera, Extra cancellation of even Calderón-Zygmund operators and quasiconformal mappings, J. Math. Pures Appl., 91 (2009), 402-431.  doi: 10.1016/j.matpur.2009.01.010.

[22]

P. Serfati, Vortex Patches Generalisés Dans $\mathbb R^n$ et Resultat De Regularité Stratifiée Pour Le Laplacien-ch4B, 1991.

[23]

P. Serfati, Étude Mathématique De Flammes Infiniment Minces En Combustion. Résultats De Structure Et De Régularité Pour l'équation D'Euler Incompressible, PhD thesis, 1992.

[24]

P. Serfati, Une preuve directe d'existence globale des vortex patches 2D, C. R. Acad. Sci. Paris Sér. I Math., 318 (1994), 515-518. 

[25]

P. Serfati, Équation d'Euler et holomorphies à faible régularité spatiale, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 175-180. 

[26]

P. Serfati, Structures holomorphes à faible régularité spatiale en mécanique des fluides, J. Math. Pures Appl., 74 (1995), 95-104. 

[27]

A. Shnirelman, On the analyticity of particle trajectories in the ideal incompressible fluid, arXiv preprint, arXiv: 1205.5837, 2012.

[28]

X. Tolsa, Analytic Capacity, the Cauchy Transform, and Non-Homogeneous Calderón-Zygmund Theory, Progress in Mathematics, 307. Birkhäuser/Springer, Cham, 2014. doi: 10.1007/978-3-319-00596-6.

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