Abstract
This paper considers a family of non-diffusive active scalar equations where a viscosity type parameter enters the equations via the constitutive law that relates the drift velocity with the scalar field. The resulting operator is smooth when the viscosity is present but singular when the viscosity is zero. We obtain Gevrey-class local well-posedness results and convergence of solutions as the viscosity vanishes. We apply our results to two examples that are derived from physical systems: firstly a model for magnetostrophic turbulence in the Earth’s fluid core and secondly flow in a porous media with an “effective viscosity”.
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20 July 2021
A Correction to this paper has been published: https://doi.org/10.1007/s00021-021-00609-8
Notes
For simplicity, we sometime write and .
References
Azzam, J., Bedrossian, J.: Bounded mean oscillation and the uniqueness of active scalar equations. Trans. Am. Math. Soc. 367(5), 3095–3118 (2015)
Bahouri, H., Chemin, J., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften 343. Springer, Berlin (2011)
Bernicot, F., Keraani, S.: On the global wellposedness of the 2D Euler equation for a large class of Yudovich type data. Annales scientifiques de l’ENS 47, fascicule 3, 559–576 (2014)
Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1, 27 (1949)
Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994)
Córdoba, D., Faraco, D., Gancedo, F.: Lack of uniqueness for weak solutions of the incompressible porous media equation. Arch. Ration. Mech. Anal. 200(3), 725–746 (2011)
Córdoba, D., Gancedo, F., Orive, R.: Analytical behavior of two-dimensional incompressible flow in porous media. J. Math. Phys. 48(6), 065206 (2007)
Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. 171(3), 1903–1930 (2010)
Constantin, P., Wu, J.: Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(1), 159–180 (2009)
Friedlander, S., Gancedo, F., Sun, W., Vicol, Vlad: On a singular incompressible porous media equation. J. Math. Phys. 53, 115602 (2012)
Friedlander, S., Rusin, W., Vicol, V.: On the supercritically diffusive magnetogeostrophic equations. Nonlinearity 25(11), 3071–3097 (2012)
Friedlander, S., Rusin, W., Vicol, V.: The magnetogeostrophic equations: a survey. In: Apushkinskaya, D., Nazarov, A.I. (eds.) Proceedings of the St. Petersburg Mathematical Society, Volume XV: Advances in Mathematical Analysis of Partial Differential Equations, pp. 53–78. AMS Translations, vol. 232 (2014)
Friedlander, S., Suen, A.: Existence, uniqueness, regularity and instability results for the viscous magnetogeostrophic equation. Nonlinearity 28(9), 3193–3217 (2015)
Friedlander, S., Suen, A.: Solutions to a class of forced drift-diffusion equations with applications to the magnetogeostrophic equations. Ann. PDE 4(2), 14 (2018)
Friedlander, S., Vicol, V.: Global well-posedness for an advection-diffusion equation arising in magnetogeostrophic dynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(2), 283–301 (2011)
Friedlander, S., Vicol, V.: On the ill/wellposedness and nonlinear instability of the magnetogeostrophic equations. Nonlinearity 24(11), 3019–3042 (2011)
Kiselev, A., Nazarov, F., Volberg, A.: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math. 167, 445–453 (2007)
Kukavica, I., Vicol, V., Wang, F.: On the ill-posedness of active scalar equations with odd singular kernels. In: New Trends in Differential Equations, Control Theory and Optimization: Proceedings of the Eighth Congress of Romanian Mathematicians (2016)
Moffatt, H.K., Loper, D.E.: The magnetostrophic rise of a buoyant parcel in the earth’s core. Geophys. J. Int. 117(2), 394–402 (1994)
Paicu, M., Vicol, V.: Analyticity and Gevrey-class regularity for the second-grade fluid equations. J. Math. Fluid Mech. 13(4), 533–555 (2011)
Moffatt, H.K.: Magnetic Field Generation in Electrically Conducting Fluids. Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge (1978)
Resnick, S.: Dynamical problems in nonlinear advective partial differential equations. Ph.D. Thesis, University of Chicago (1995)
Stein, E.M.: Singular Integrals and Differentiability Properties of Function. Princeton University Press, Princeton (1970)
Ziemer, W.: Weakly Differentiable Functions. Springer, Berlin (1989)
Acknowledgements
S. Friedlander is supported by NSF DMS-1613135 and A. Suen is supported by Hong Kong Early Career Scheme (ECS) Grant Project Number 28300016.
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Friedlander, S., Suen, A. Wellposedness and Convergence of Solutions to a Class of Forced Non-diffusive Equations with Applications. J. Math. Fluid Mech. 21, 50 (2019). https://doi.org/10.1007/s00021-019-0454-1
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DOI: https://doi.org/10.1007/s00021-019-0454-1