Advertisement

Annals of PDE

, 3:14 | Cite as

Global Regularity for 2D Boussinesq Temperature Patches with No Diffusion

  • Francisco Gancedo
  • Eduardo García-Juárez
Article

Abstract

This paper considers the temperature patch problem for the incompressible Boussinesq system with no diffusion and viscosity in the whole space \(\mathbb {R}^2\). We prove that for initial patches with \(W^{2,\infty }\) boundary the curvature remains bounded for all time. The proof explores new cancellations that allow us to bound \(\nabla ^2u\), even for those components given by time dependent singular integrals with kernels with nonzero mean on circles. In addition, we give a different proof of the \(C^{1+\gamma }\) regularity result in Danchin and Zhang (Global persistence of geometrical structures for the Boussinesq equation with no diffusion. Commun Partial Differ Equ 42(1):68–99 (2017)), \(0<\gamma <1\), using the scale of Sobolev spaces for the velocity. Furthermore, taking advantage of the new cancellations, we go beyond to show the persistence of regularity for \(C^{2+\gamma }\) patches.

Keywords

Boussinesq equations Temperature patch Global regularity Singular heat kernels 

Notes

Acknowledgements

This research was partially supported by the project P12-FQM-2466 of Junta de Andalucía, Spain, the grant MTM2014-59488-P (Spain) and by the ERC through the Starting Grant project H2020-EU.1.1.-639227. EGJ was supported by MECD FPU grant from the Spanish Government. FG acknowledges support from the Ramón y Cajal program RyC-2010-07094.

References

  1. 1.
    Abidi, H., Hmidi, T.: On the global well-posedness for Boussinesq system. J. Differ. Equ. 233(1), 199–220 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Adhikari, D., Cao, C., Wu, J.: Global regularity results for the 2D Boussinesq equations with vertical dissipation. J. Differ. Equ. 251(6), 1637–1655 (2011)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Adhikari, D., Cao, C., Shang, H., Wu, J., Xu, X., Zhuan, Z.Y.: Global regularity results for the 2D Boussinesq equations with partial dissipation. J. Differ. Equ. 260(2), 18931917 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343. Springer, London (2011)zbMATHGoogle Scholar
  5. 5.
    Bertozzi, A., Constantin, P.: Global regularity for vortex patches. Commun. Math. Phys. 152(1), 19–28 (1993)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cao, C., Wu, J.: Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation. Arch. Ration. Mech. Anal. 208(3), 985–1004 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Castro, A., Córdoba, D., Fefferman, C., Gancedo, F.: Breakdown of smoothness for the Muskat problem. Arch. Ration. Mech. Anal. 208(3), 805–909 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Castro, A., Córdoba, D., Fefferman, C., Gancedo, F., Gómez-Serrano, J.: Finite time singularities for the free boundary incompressible Euler equations. Ann. Math. (2) 178(3), 1061–1134 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Castro, A., Córdoba, D., Fefferman, C., Gancedo, F., Gómez-Serrano, J.: Splash singularities for the free boundary Navier–Stokes equations, (preprint arXiv:1504.02775) (2015)
  10. 10.
    Castro, A., Córdoba, D., Fefferman, C., Gancedo, F.: Splash singularities for the one-phase Muskat problem in stable regimes. Arch. Ration. Mech. Anal. 222(1), 213–243 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Chae, D.: Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv. Math. 203(2), 497–513 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Chemin, J.-Y.: Persistance de structures géométriques dans les fluides incompressibles bidimensionnels. Ann. Sci. École Norm. Sup. (4) 26(4), 517–542 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Chemin, J.-Y.: Théorèmes d’unicité pour le système de Navier–Stokes tridimensionnel. J. d’analyse Math. 77(1), 27–50 (1999)CrossRefzbMATHGoogle Scholar
  14. 14.
    Chemin, J.-Y., McCormick, D., Robinson, J., Rodrigo, J.: Local existence for the non-resistive MHD equations in Besov spaces. Adv. Math. 286, 1–31 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Choi, K., Kiselev, A., Yao, Y.: Finite time blow up for a 1D model of 2D Boussinesq system. Commun. Math. Phys. 334(3), 1667–1679 (2015)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7(6), 1495–1533 (1994)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Constantin, P., Doering, C.R.: Infinite Prandtl number convection. J. Stat. Phys. 94, 159–172 (1999)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Córdoba, D., Fontelos, M.A., Mancho, A.M., Rodrigo, J.L.: Evidence of singularities for a family of contour dynamics equations. Proc. Natl. Acad. Sci. USA 102(17), 5949–5952 (2005)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Coutand, D., Shkoller, S.: On the finite-time splash and splat singularities for the 3-D free-surface Euler equations. Commun. Math. Phys. 325(1), 143–183 (2014)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Coutand, D., Shkoller, S.: On the splash singularity for the free-surface of a Navier–Stokes fluid, (preprint arXiv:1505.01929v1) (2015)
  21. 21.
    Danchin, R.: Uniform estimates for transport-diffusion equations. J. Hyperbol. Differ. Equ. 4(01), 117 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Danchin, R., Paicu, M.: Les théorèmes de Leray et de Fujita–Kato pour le systme de Boussinesq partiellement visqueux. Bull. Soc. Math. France 136(2), 261–309 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Danchin, R., Paicu, M.: Global well-posedness issues for the inviscid Boussinesq system with Yudovich’s type data. Commun. Math. Phys. 290(1), 1–14 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Danchin, R., Zhang, X.: Global persistence of geometrical structures for the Boussinesq equation with no diffusion. Commun Partial Differ. Equ. 42(1), 68–99 (2017)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Fefferman, C., McCormick, D., Robinson, J., Rodrigo, J.: Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces. Arch. Ration. Mech. Anal. (2016). doi: 10.1007/s00205-016-1042-7
  26. 26.
    Gancedo, F., Strain, R.: Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem. Proc. Natl. Acad. Sci. USA 111(2), 635639 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Hmidi, T., Keraani, S.: On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity. Adv. Differ. Equ. 12(4), 461–480 (2007)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Hmidi, T., Keraani, S.: On the global well-posedness of the Boussinesq system with zero viscosity. Indiana Univ. Math. J. 58(4), 1591–1618 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Hoang, V., Orcan-Ekmekci, B., Radosz, M., Yang, H.: Blowup with vorticity control for a 2D model of the Boussinesq equations (preprint arXiv:1608.01285) (2016)
  30. 30.
    Hou, T., Li, C.: Global well-posedness of the viscous Boussinesq equations. Discrete Contin. Dyn. Syst. 12(1), 1–12 (2005)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Hu, W., Kukavica, I., Ziane, M.: Persistence of regularity for the viscous Boussinesq equations with zero diffusivity. Asymptot. Anal. 91, 111–124 (2015)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Kiselev, A., Tan, C.: Finite time blow up in the hyperbolic Boussinesq system, (preprint arXiv:1609.02468v1 [math.AP])(2016)
  33. 33.
    Krylov, N.V.: Parabolic equations in \(L_p\)-spaces with mixed norms. Algebra i Analiz. 14 91–106 (2002, Russian). English translation. St. Petersburg Math. J. 14, 603–614 (2003)Google Scholar
  34. 34.
    Luo, G., Hou, T.Y.: Towards the finite-time blowup of the 3d axisymmetric Euler equations: a numerical investigation. Multiscale Model. Simul. 12, 1722–1776 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Larios, A., Lunasin, E., Titi, E.: Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion. J. Differ. Equ. 255(9), 2636–2654 (2013)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Li, J., Titi, E.: Global well-posedness of the 2D Boussinesq equations with vertical dissipation. Arch. Ration. Mech. Anal. 220(3), 983–1001 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Majda, A.J.: Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lect. Notes Math., vol. 9, AMS/CIMS (2003)Google Scholar
  38. 38.
    Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow, Vol. 27 of Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)Google Scholar
  39. 39.
    Maz’ya, V., Shaposhnikova, T. O.: Theory of Sobolev multipliers with applications to differential and integral operators. Grundlehren der mathematischen Wissenschaften, 0072-7830 ; 337. Springer, Berlin, Heidelberg (2009)Google Scholar
  40. 40.
    Pedlosky, J.: Geophysical Fluid Dynamics. Springer, New York (1987)CrossRefzbMATHGoogle Scholar
  41. 41.
    Rodrigo, J.L.: The vortex patch problem for the surface quasi-geostrophic equation. Proc. Natl. Acad. Sci. USA 101(9), 2684–2686 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Schlag, W.: Schauder and Lp estimates for parabolic systems via Campanato spaces. Commun. Partial Differ. Equ. 21(7–8), 1141–1175 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Weinan, E., Shu, C.-W.: Small-scale structures in Boussinesq convection. Phys. Fluids 6(1), 49–58 (1994)ADSCrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Departamento de Análisis Matemático and IMUSUniversidad de SevillaSevillaSpain

Personalised recommendations