Skip to main content
SpringerLink
Log in

Regularity Results for Viscous 3D Boussinesq Temperature Fronts

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

This paper is about the dynamics of non-diffusive temperature fronts evolving by the incompressible viscous Boussinesq system in \({\mathbb {R}}^3\). We provide local in time existence results for initial data of arbitrary size. Furthermore, we show global in time propagation of regularity for small initial data in critical spaces. The developed techniques allow to consider general fronts where the temperature is piecewise Hölder (not necessarily constant), which preserve their structure together with the regularity of the evolving interface.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abidi, H., Hmidi, T.: On the global well-posedness for Boussinesq system. J. Differ. Equ. 233(1), 199–220 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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), 1893–1917 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343. Springer, New York (2011)

    Book  MATH  Google Scholar 

  5. Bertozzi, A., Constantin, P.: Global regularity for vortex patches. Commun. Math. Phys. 152(1), 19–28 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Boussinesq, J.: Théorie Analytique de la Chaleur, vol. 2. Gauthier-Villars, Paris (1903)

    Google Scholar 

  7. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Castro, A., Córdoba, D., Lear, D.: On the asymptotic stability of stratified solutions for the 2D Boussinesq equations with a velocity damping term. Math. Models Methods Appl. Sci. 29(7), 1227–1277 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chae, D.: Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv. Math. 203(2), 497–513 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chemin, J.-Y.: Persistance de structures géométriques dans les fluides incompressibles bidimensionnels. Ann. Sci. École Norm. Sup. (4) 26(4), 517–542 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Choi, K., Hou, T.Y., Kiselev, A., Luo, G.G., Sverak, V., Yao, Y.: On the finite time blowup of a one-dimensional model for the three-dimensional axisymmetric Euler equations. Commun. Pure Appl. Math. 70, 2218–2243 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Constantin, P., Doering, C.R.: Infinite Prandtl number convection. J. Stat. Phys. 94, 159–172 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Córdoba, D., Gancedo, F.: Absence of squirt singularities for the multi-phase Muskat problem. Commun. Math. Phys. 299(2), 561–575 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Danchin, R., Paicu, M.: Les théorèmes de Leray et de Fujita–Kato pour le système de Boussinesq partiellement visqueux. Bull. Soc. Math. France 136(2), 261–309 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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)

    Article  MathSciNet  MATH  Google Scholar 

  18. WE, C.S.: Small-scale structures in Boussinesq convection. Phys. Fluids 6(1), 49–58 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  19. Elgindi, T.M., Jeong, I.-J.: Finite-time Singularity Formation for Strong Solutions to the Boussinesq System. arXiv:1708.02724v5, (2017)

  20. Elgindi, T.M., Widmayer, K.: Sharp decay estimates for an anisotropic linear semigroup and applications to the SQG and inviscid Boussinesq systems. SIAM J. Math. Anal. 47(6), 4672–4684 (2016)

    Article  MATH  Google Scholar 

  21. Faraco, D., Rogers, K.: The Sobolev norm of characteristic functions with applications to the Calderón inverse problem. Q. J. Math. 64(1), 133–147 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fefferman, C.: Existence and smoothness of the Navier–Stokes equation. In: The Millennium Prize Problems, 57–67, Clay Math. Inst., Cambridge, MA, (2006)

  23. Fefferman, C., Ionescu, A.D., Lie, V.: On the absence of splash singularities in the case of two-fluid interfaces. Duke Math. J. 165(3), 417–462 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. 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. 223(2), 677–691 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. 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), 635–639 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Gancedo, F., García-Juárez, E.: Global regularity for 2D Boussinesq temperature patches with no diffusion. Ann. PDE 3, 14 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gancedo, F., García-Juárez, E.: Global regularity of 2d density patches for inhomogeneous Navier–Stokes. Arch. Ration. Mech. Anal. 229(1), 339–360 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gill, A.: Atmosphere-Ocean Dynamics. International Geophysics Series, vol. 30. Academic Press, New York (1982)

    Google Scholar 

  29. 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)

    MathSciNet  MATH  Google Scholar 

  30. Hoang, V., Orcan-Ekmekci, B., Radosz, M., Yang, H.: Blowup with vorticity control for a 2D model of the Boussinesq equations. J. Differ. Equ. 264(12), 7328–7356 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Hou, T., Li, C.: Global well-posedness of the viscous Boussinesq equations. Discrete Contin. Dyn. Syst. 12(1), 1–12 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hu, W., Kukavica, I., Ziane, M.: Persistence of regularity for the viscous boussinesq equations with zero diffusivity. Asymp. Anal. 91, 111–124 (2015)

    MathSciNet  MATH  Google Scholar 

  33. Kiselev, A., Tan, C.: Finite time blow up in the hyperbolic Boussinesq system. Adv. Math. 325, 34–55 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Krylov, N.V.: Parabolic equations in \(L_p\)-spaces with mixed norms. Algebra i Analiz. 14 91-106 (2002, Russian). English translation in St. Petersburg Math. J. 14, 603–614, (2003)

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  37. Luo, G., Hou, T.Y.: Potentially singular solutions of the 3D axisymmetric Euler equations. Proc. Natl. Acad. Sci. USA 111(36), 12968–12973 (2014)

    Article  ADS  MATH  Google Scholar 

  38. Majda, A.J., Bertozzi, A.L.: A.J. Majda, A.L. Bertozzi. Vorticity and Incompressible flow, Vol. 27 of Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, (2002)

  39. Majda, A.J.: Introduction to PDEs and Waves for the Atmosphere and Ocean. In: Courant Lect. Notes Math., vol. 9, AMS/CIMS, (2003)

  40. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  41. Oberbeck, A.: Ueber die Wärmeleitung der Flüssigkeiten bei Berücksichtigung der Strömungen infolge von Temperaturdifferenzen. Ann. Phys. Chem. 243, 271–292 (1879)

    Article  ADS  MATH  Google Scholar 

  42. Qiu, H., Du, Y., Yao, Z.: A blow-up criterion for 3D Boussinesq equations in Besov spaces. Nonlinear Anal. 73(3), 806–815 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  43. Robinson, J.C., Rodrigo, J.L., Sadowski, W.: The Three-Dimensional Navier–Stokes equations. Cambridge Studies in Advanced Mathematics, vol. 157. Cambridge University Press, Cambridge (2016)

    Book  Google Scholar 

  44. Rodrigo, J.L.: The vortex patch problem for the surface quasi-geostrophic equation. Proc. Natl. Acad. Sci. USA 101(9), 2684–2686 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  46. Taylor, M.E.: Partial Differential Equations I. Basic Theory. Applied Mathematical Sciences, vol. 115, 2nd edn. Springer, New York (2011)

    Book  Google Scholar 

  47. Widmayer, K.: Convergence to stratified flow for an inviscid 3D Boussinesq system. Commun. Math. Sci. 16(6), 1713–1728 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  48. Xu, F., Zhang, Q., Zheng, X.: Regularity criteria of the 3D Boussinesq equations in the Morrey–Campanato space. Acta Appl. Math. 121(1), 231–240 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  49. Yudovich, V.I.: Eleven great problems of mathematical hydrodynamics. Moscow Math. J. 3(2), 711–737 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research was partially supported by 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.

Author information

Author notes

  1. Eduardo García-Juárez

    Present address: Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab., 209 South 33rd St., Philadelphia, PA, 19104, USA

Authors and Affiliations

  1. Departamento de Análisis Matemático and IMUS, Universidad de Sevilla, C/Tarfia s/n, Campus Reina Mercedes, 41012, Sevilla, Spain

    Francisco Gancedo & Eduardo García-Juárez

Authors
  1. Francisco Gancedo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eduardo García-Juárez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francisco Gancedo.

Additional information

Communicated by C. De Lellis.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gancedo, F., García-Juárez, E. Regularity Results for Viscous 3D Boussinesq Temperature Fronts. Commun. Math. Phys. 376, 1705–1736 (2020). https://doi.org/10.1007/s00220-020-03767-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-020-03767-4

Search

Navigation