no. 4
Partial differential equations
On optimal Hölder regularity of solutions to the equation Δu + b ⋅ ∇u = 0 in two dimensions
[Sur la régularité Hölder optimale pour les solutions de l'equation Δu + b ⋅ ∇u = 0 en dimension deux]

Présenté par : Haïm Brézis

Le, Nam Q.1
1 Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
Comptes Rendus. Mathématique, Tome 355 (2017) no. 4, pp. 439-446.

Nous démontrons que, pour une dérive b∈Lloc2(R2;R2), si la norme de Hardy de div b est petite, alors les solutions faibles de Δu+b⋅∇u=0 (en dimension deux) ont la même régularité Hölder que dans le cas de la dérive incompressible, c'est-à-dire que u∈Clocα pour tout α∈(0,1).

We show that for an L2 drift b in two dimensions, if the Hardy norm of divb is small, then the weak solutions to Δu+b⋅∇u=0 have the same optimal Hölder regularity as in the case of divergence-free drift, that is, u∈Clocα for all α∈(0,1).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.02.005
Le, Nam Q. 1

1 Department of Mathematics, Indiana University, Bloomington, IN 47405, USA 
@article{CRMATH_2017__355_4_439_0,
     author = {Le, Nam Q.},
     title = {On optimal {H\"older} regularity of solutions to the equation {\ensuremath{\Delta}\protect\emph{u}\,+\,\protect\emph{b}\,\ensuremath{\cdot}\,\ensuremath{\nabla}\protect\emph{u}\,=\,0} in two dimensions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {439--446},
     publisher = {Elsevier},
     volume = {355},
     number = {4},
     year = {2017},
     doi = {10.1016/j.crma.2017.02.005},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2017.02.005/}
}
TY  - JOUR
AU  - Le, Nam Q.
TI  - On optimal Hölder regularity of solutions to the equation Δu + b ⋅ ∇u = 0 in two dimensions
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 439
EP  - 446
VL  - 355
IS  - 4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2017.02.005/
DO  - 10.1016/j.crma.2017.02.005
LA  - en
ID  - CRMATH_2017__355_4_439_0
ER  - 
%0 Journal Article
%A Le, Nam Q.
%T On optimal Hölder regularity of solutions to the equation Δu + b ⋅ ∇u = 0 in two dimensions
%J Comptes Rendus. Mathématique
%D 2017
%P 439-446
%V 355
%N 4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2017.02.005/
%R 10.1016/j.crma.2017.02.005
%G en
%F CRMATH_2017__355_4_439_0
Le, Nam Q. On optimal Hölder regularity of solutions to the equation Δu + b ⋅ ∇u = 0 in two dimensions. Comptes Rendus. Mathématique, Tome 355 (2017) no. 4, pp. 439-446. doi : 10.1016/j.crma.2017.02.005. http://www.numdam.org/articles/10.1016/j.crma.2017.02.005/

[1] Bethuel, F. Un résultat de régularité pour les solutions de l'équation de surfaces à courbure moyenne prescrite, C. R. Acad. Sci. Paris, Ser. I, Volume 314 (1992) no. 13, pp. 1003-1007

[2] Coifman, R.; Lions, P.-L.; Meyer, Y.; Semmes, S. Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9), Volume 72 (1993) no. 3, pp. 247-286

[3] Filonov, N. On the regularity of solutions to the equation −Δu+b⋅∇u=0, J. Math. Sci., Volume 195 (2013) no. 1, pp. 98-108

[4] Friedlander, S.; Vicol, V. Global well-posedness for an advection–diffusion equation arising in magneto-geostrophic dynamics, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 28 (2011) no. 2, pp. 283-301

[5] Hajłasz, P.; Strzelecki, P.; Zhong, X. A new approach to interior regularity of elliptic systems with quadratic Jacobian structure in dimension two, Manuscr. Math., Volume 127 (2008) no. 1, pp. 121-135

[6] Hélein, F. Harmonic Maps, Conservation Laws and Moving Frames, Camb. Tracts Math., vol. 150, Cambridge University Press, Cambridge, UK, 2002

[7] Rivière, T. Conservation laws for conformally invariant variational problems, Invent. Math., Volume 168 (2007) no. 1, pp. 1-22

[8] Rivière, T. The role of conservation laws in the analysis of conformally invariant problems, Topics in Modern Regularity Theory, CRM Ser., vol. 13, Edizioni della Normale, Pisa, Italy, 2012, pp. 117-167

[9] Seregin, G.; Silvestre, L.; Šverák, V.; Zlatoš, A. On divergence-free drifts, J. Differ. Equ., Volume 252 (2012) no. 1, pp. 505-540

[10] Wente, H.C. An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl., Volume 26 (1969), pp. 318-344

Cité par Sources :