cnrs   Université de Toulouse
With cedram.org

Table of contents for this issue | Next article
Cyril Imbert; Roman Shvydkoy; François Vigneron
Global Well-Posedness of a Non-local Burgers Equation: the periodic case
Annales de la faculté des sciences de Toulouse Sér. 6, 25 no. 4 (2016), p. 723-758, doi: 10.5802/afst.1509
Article PDF

Résumé - Abstract

This paper is concerned with the study of a non-local Burgers equation for positive bounded periodic initial data. The equation reads

$$ u_t - u |\nabla | u + |\nabla |(u^2) = 0. $$

We construct global classical solutions starting from smooth positive data, and global weak solutions starting from data in $L^\infty $. We show that any weak solution is instantaneously regularized into $C^\infty $. We also describe the long-time behavior of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations.

Bibliography

[1] Baker (G.R.), Li (X.), and Morlet (A.C.).— Analytic structure of two 1D-transport equations with nonlocal fluxes. Phys. D, 91(4), p. 349-375 (1996).  MR 1382265 |  Zbl 0899.76104
[2] Barlow (M.T.), Bass (R.F.), Chen (Z-Q), and Kassmann (M.).— Non-local Dirichlet forms and symmetric jump processes. Trans. Amer. Math. Soc., 361(4), p. 1963-1999 (2009).  MR 2465826 |  Zbl 1166.60045
[3] Benilan (P.) and Brézis (H.).— Solutions faibles d’équations d’évolution dans les espaces de Hilbert. Ann. Inst. Fourier (Grenoble), 22(2), p. 311-329 (1972). Cedram |  MR 336471 |  Zbl 0226.47034
[4] Caffarelli (L.), Chan (C.H.), and Vasseur (A.).— Regularity theory for parabolic nonlinear integral operators. J. Amer. Math. Soc., 24(3), p. 849-869 (2011).  MR 2784330 |  Zbl 1223.35098
[5] Chae (D.), Córdoba (A.), Córdoba (D.), and Fontelos (M.A.).— Finite time singularities in a 1D model of the quasi-geostrophic equation. Adv. Math., 194(1), p. 203-223 (2005).  MR 2141858 |  Zbl 1128.76372
[6] Chen (Z-Q).— Symmetric jump processes and their heat kernel estimates. Sci. China Ser. A, 52(7), p. 1423-1445 (2009).  MR 2520585 |  Zbl 1186.60073
[7] Constantin (P.) and Vicol (V.).— Nonlinear maximum principles for dissipative linear nonlocal operators and applications. Geom. Funct. Anal., 22(5), p. 1289-1321 (2012).  MR 2989434 |  Zbl 1256.35078
[8] Córdoba (A.), Córdoba (D.), and Fontelos (M.A.).— Formation of singularities for a transport equation with nonlocal velocity. Ann. of Math. (2), 162(3), p.1377-1389 (2005).  MR 2179734 |  Zbl 1101.35052
[9] Di Nezza (E.), Palatucci (G.), and Valdinoci (E.).— HitchhikerÕs guide to the fractional Sobolev spaces. Bull. Sci. Math., 136(5), p. 521-573 (2012).  MR 2944369 |  Zbl 1252.46023
[10] Felsinger (M.) and Kassmann (M.).— Local regularity for parabolic nonlocal operators. Comm. Partial Differential Equations, 38(9), p. 1539-1573 (2013).  MR 3169755 |  Zbl 1277.35090
[11] Imbert (C.), Jin (T.), Shvydkoy (R.), and Silvestre (L.).— Schauder estimates for linear integrodifferential equations with general kernels. in preparation.
[12] Imbert (C.), Monneau (R.), and Rouy (E.).— Homogenization of first order equations with $(u/\epsilon )$-periodic Hamiltonians. II. Application to dislocations dynamics. Comm. Partial Differential Equations, 33(1-3), p. 479-516 (2008).  MR 2398239 |  Zbl 1143.35005
[13] Jin (T.) and Xiong (J.).— Schauder estimates for nonlocal fully nonlinear equations. To appear in Ann. Inst. H. Poincaré Anal. Non Linéaire.  MR 3542618
[14] Jin (T.) and Xiong (J.).— Schauder estimates for solutions of linear parabolic integro-differential equations. http, p. //arxiv.org/abs/1405.0755v3.  MR 3393263
[15] Kassmann (M.).— A priori estimates for integro-differential operators with measurable kernels. Calc. Var. Partial Differential Equations, 34(1), p. 1-21 (2009).  MR 2448308 |  Zbl 1158.35019
[16] Komatsu (T.).— Uniform estimates for fundamental solutions associated with non-local Dirichlet forms. Osaka J. Math., 32(4), p. 833-860 (1995).  MR 1380729 |  Zbl 0867.35123
[17] Kraichnan (R.H.) and Montgomery (D.).— Two-dimensional turbulence. Rep. Progr. Phys., 43(5), p. 547-619 (1980).  MR 587291
[18] Lelièvre (F.).— Approximation des équations de navier-stokes préservant le changement d’échelle. PhD (2010).
[19] Lelièvre (F.).— A scaling and energy equality preserving approximation for the 3D Navier-Stokes equations in the finite energy case. Nonlinear Anal., 74(17), p. 5902-5919 (2011).  MR 2833362 |  Zbl 1262.76019
[20] Lelièvre (F.).— Un modèle scalaire analogue aux équations de Navier-Stokes. C. R. Math. Acad. Sci. Paris, 349(7-8), p. 411-416 (2011).  MR 2788379 |  Zbl 1216.35097
[21] Majda (A.J.) and Bertozzi (A.L.).— Vorticity and incompressible flow, volume 27 of Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002).  MR 1867882 |  Zbl 0983.76001
[22] Mikulevicius (R.) and Pragarauskas (H.).— On the Cauchy problem for integro-differential operators in Holder classes and the uniqueness of the martingale problem. Potential Anal., 40(4), p. 539-563 (2014).  MR 3201992 |  Zbl 1296.45009
[23] Moffatt (H.K.).— Magnetostrophic turbulence and the geodynamo. In IUTAM Symposium on Computational Physics and New Perspectives in Turbulence, p. 339-346. Springer, Dordrecht (2008).  MR 2432632 |  Zbl 1208.76148
[24] Nishida (T.).— A note on a theorem of Nirenberg. J. Differential Geom., 12(4), p. 629-633 (1978), (1977).  MR 512931 |  Zbl 0368.35007
Search for an article
Search within the site
top