Hölder continuity for a drift-diffusion equation with pressure
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 4, pp. 637-652.

Nous abordons la question de la persistance de la continuité Hölder pour les solutions faibles de lʼéquation linéaire de dérive-diffusion avec une pression non-locale

u t +b·u-u=p,·u=0
sur [0,)× n , avec n2. On suppose que la vitesse de dérive b est au niveau critique de régularité par rapport au changement dʼéchelle de lʼéquation. La démonstration sʼappuie sur la définition des espaces Hölder de Campanato, et elle utilise un argument de principe du maximum par lequel nous contrôlons la croissance en temps de certaines moyennes locales de u. Nous fournissons une estimation qui ne dépend dʼaucune condition de petitesse locale sur le champ de vecteur b, mais seulement sur des quantités invariantes par changement dʼéchelle.

We address the persistence of Hölder continuity for weak solutions of the linear drift-diffusion equation with nonlocal pressure

u t +b·u-u=p,·u=0
on [0,)× n , with n2. The drift velocity b is assumed to be at the critical regularity level, with respect to the natural scaling of the equations. The proof draws on Campanatoʼs characterization of Hölder spaces, and uses a maximum-principle-type argument by which we control the growth in time of certain local averages of u. We provide an estimate that does not depend on any local smallness condition on the vector field b, but only on scale invariant quantities.

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     title = {H\"older continuity for a drift-diffusion equation with pressure},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Silvestre, Luis; Vicol, Vlad. Hölder continuity for a drift-diffusion equation with pressure. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 4, pp. 637-652. doi : 10.1016/j.anihpc.2012.02.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2012.02.003/

[1] D.G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Sc. Norm. Super. Pisa (3) 22 (1968), 607-694 | EuDML | Numdam | MR | Zbl

[2] S. Campanato, Propietà di hölderianità di alcune classi di funzioni, Ann. Sc. Norm. Super. Pisa (3) 17 (1963), 175-188 | EuDML | Numdam | MR | Zbl

[3] C.H. Chan, A. Vasseur, Log improvement of the Prodi–Serrin criteria for Navier–Stokes equations, Methods Appl. Anal. 14 no. 2 (2007), 197-212 | MR | Zbl

[4] Z.Q. Chen, Z. Zhao, Diffusion processes and second order elliptic operators with singular coefficients for lower order terms, Math. Ann. 302 no. 2 (1995), 323-357 | EuDML | MR | Zbl

[5] A. Cheskidov, R. Shvydkoy, The regularity of weak solutions of the 3D Navier–Stokes equations in B , -1 , Arch. Ration. Mech. Anal. 195 no. 1 (2010), 159-169 | MR | Zbl

[6] A. Cheskidov, R. Shvydkoy, Regularity problem for the 3D Navier–Stokes equations: the use of Kolmogorovʼs dissipation range, arXiv:1102.1944v1 (2011) | Zbl

[7] P. Constantin, G. Iyer, J. Wu, Global regularity for a modified critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J. 57 no. 6 (2008), 2681-2692 | MR | Zbl

[8] C. Foiaş, G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier–Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova 39 (1967), 1-34 | EuDML | Numdam | MR | Zbl

[9] S. Friedlander, V. Vicol, Global well-posedness for an advection–diffusion equation arising in magneto-geostrophic dynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 no. 2 (2011), 283-301 | Numdam | MR | Zbl

[10] L. Iskauriaza, G.A. Seregin, V. Sverak, L 3, -solutions of Navier–Stokes equations and backward uniqueness, Uspekhi Mat. Nauk 58 no. 2(350) (2003), 3-44 | MR

[11] C.E. Kenig, G.S. Koch, An alternative approach to regularity for the Navier–Stokes equations in critical spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 no. 2 (2011), 159-187 | Numdam | MR | Zbl

[12] A. Kiselev, Regularity and blow up for active scalars, Math. Model. Nat. Phenom. 5 no. 4 (2010), 225-255 | EuDML | MR | Zbl

[13] A. Kiselev, F. Nazarov, A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 no. 3 (2007), 445-453 | MR | Zbl

[14] I. Kukavica, Regularity for the Navier–Stokes equations with a solution in a Morrey space, Indiana Univ. Math. J. 57 no. 6 (2008), 2843-2860 | MR | Zbl

[15] O.A. Ladyženskaja, Uniqueness and smoothness of generalized solutions of Navier–Stokes equations, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 5 (1967), 169-185 | EuDML | MR | Zbl

[16] O.A. Ladyženskaja, V.A. Solonnikov, N.N. UralʼTseva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. vol. 23, Amer. Math. Soc., Providence, RI (1967)

[17] V. Liskevich, Q.S. Zhang, Extra regularity for parabolic equations with drift terms, Manuscripta Math. 113 no. 2 (2004), 191-209 | MR | Zbl

[18] J. Moser, On Harnackʼs theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577-591 | MR | Zbl

[19] A.I. Nazarov, N.N. UralʼTseva, The Harnack inequality and related properties of solutions of elliptic and parabolic equations with divergence-free lower-order coefficients, Algebra i Analiz 23 no. 1 (2011), 136-168 | MR

[20] G. Prodi, Un teorema di unicità per le equazioni di Navier–Stokes, Ann. Mat. Pura Appl. (4) 48 (1959), 173-182 | MR | Zbl

[21] Y.A. Semenov, Regularity theorems for parabolic equations, J. Funct. Anal. 231 no. 2 (2006), 375-417 | MR | Zbl

[22] G.A. Seregin, L. Silvestre, V. Šverák, A. Zlatoš, On divergence-free drifts, J. Differential Equations 252 no. 1 (2012), 505-540 | MR | Zbl

[23] J. Serrin, On the interior regularity of weak solutions of the Navier–Stokes equations, Arch. Ration. Mech. Anal. 9 (1962), 187-195 | MR | Zbl

[24] H. Triebel, Theory of Function Spaces. II, Monogr. Math. vol. 84, Birkhäuser-Verlag, Basel (1992) | MR | Zbl

[25] Q.S. Zhang, Gaussian bounds for the fundamental solutions of (Au)+Bu-u t =0, Manuscripta Math. 93 no. 3 (1997), 381-390 | EuDML | MR

[26] Q.S. Zhang, Local estimates on two linear parabolic equations with singular coefficients, Pacific J. Math. 223 no. 2 (2006), 367-396 | MR | Zbl

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