Comptes Rendus
no. 6
Partial differential equations/Functional analysis
Nonnegative measures belonging to H−1(R2)
[Les mesures positives appartenantes à H−1(R2)]

Présenté par : Jean-Michel Coron

Grzegorz Jamróz1
1 Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland
Comptes Rendus. Mathématique, Volume 353 (2015) no. 6, pp. 529-534.

En vue d'applications en mécanique des fluides, on démontre qu'une mesure positive de Radon à support compact appartient à l'espace négatif de Sobolev H−1(R2) à condition que la fonction r↦μ(B(0,r)) soit hölderienne. En passant, on obtient un plongement d'espace des fonctions croissantes hölderiennes sur R dans l'espace de Sobolev fractionnaire H1/2(R). On discute des généralisations et des applications numériques.

Motivated by applications in fluid dynamics, we show elementarily that a nonnegative compactly supported Radon measure μ belongs to the negative Sobolev space H−1(R2) provided that function r↦μ(B(0,r)) is Hölder continuous. In passing we obtain embedding of the space of nondecreasing Hölder continuous functions on R into the fractional Sobolev space H1/2(R). We comment on possible generalizations and numerical applications.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.04.010

Grzegorz Jamróz 1

1 Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland 
@article{CRMATH_2015__353_6_529_0,
     author = {Grzegorz Jamr\'oz},
     title = {Nonnegative measures belonging to $ {H}^{-1}({\mathbb{R}}^{2})$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {529--534},
     publisher = {Elsevier},
     volume = {353},
     number = {6},
     year = {2015},
     doi = {10.1016/j.crma.2015.04.010},
     language = {en},
}
TY  - JOUR
AU  - Grzegorz Jamróz
TI  - Nonnegative measures belonging to $ {H}^{-1}({\mathbb{R}}^{2})$
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 529
EP  - 534
VL  - 353
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crma.2015.04.010
LA  - en
ID  - CRMATH_2015__353_6_529_0
ER  - 
%0 Journal Article
%A Grzegorz Jamróz
%T Nonnegative measures belonging to $ {H}^{-1}({\mathbb{R}}^{2})$
%J Comptes Rendus. Mathématique
%D 2015
%P 529-534
%V 353
%N 6
%I Elsevier
%R 10.1016/j.crma.2015.04.010
%G en
%F CRMATH_2015__353_6_529_0
Grzegorz Jamróz. Nonnegative measures belonging to $ {H}^{-1}({\mathbb{R}}^{2})$. Comptes Rendus. Mathématique, Volume 353 (2015) no. 6, pp. 529-534. doi : 10.1016/j.crma.2015.04.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.04.010/

[1] R.A. Adams; J.J.F. Fournier Sobolev Spaces, Academic Press, 2003

[2] M. Carter; B. van Brunt The Lebesgue–Stieltjes Integral. A Practical Introduction, Springer-Verlag, New York, 2000

[3] D. Chae Weak solutions of 2-D Euler incompressible Euler equations, Nonlinear Anal. TMA, Volume 23 (1994), pp. 629-638

[4] T. Cieślak; M. Szumańska A theorem on measures in dimension 2 and applications to vortex sheets, J. Funct. Anal., Volume 266 (2014), pp. 6780-6795

[5] J.-M. Delort Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc., Volume 4 (1991), pp. 553-586

[6] R. DiPerna; A. Majda Concentrations in regularizations for 2-D incompressible flow, Commun. Pure Appl. Math., Volume 40 (1987) no. 3, pp. 301-345

[7] L.C. Evans; R. Gariepy Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, USA, 1992

[8] E.A. Gorin; B.N. Kukushkin Integrals associated with the Cantor staircase, St. Petersburg Math. J., Volume 15 (2006) no. 3, pp. 449-468

[9] J. Liu; Z. Xin Convergence of vortex methods for weak solutions to the 2D Euler equations with vortex sheet data, Commun. Pure Appl. Math., Volume 48 (1995) no. 6, pp. 611-628

[10] M.C. Lopes Filho; J. Lowengrub; H.J. Nussenzveig Lopes; Y. Zheng Numerical evidence of nonuniqueness in the evolution of vortex sheets, ESAIM: Math. Model. Numer. Anal., Volume 40 (2006) no. 2, pp. 225-237

[11] M.C. Lopes Filho; H.J. Nussenzveig Lopes; E. Tadmor Approximate solutions of the incompressible Euler equations with no concentrations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 17 (2000) no. 3, pp. 371-412

[12] A. Majda Remarks on weak solutions for vortex sheets with a distinguished sign, Indiana Univ. Math. J., Volume 42 (1993) no. 3, pp. 921-939

[13] S. Schochet The point-vortex method for periodic weak solutions of the 2D Euler equations, Commun. Pure Appl. Math., Volume 49 (1996), pp. 911-965

[14] E. Tadmor On a new scale of regularity spaces with applications to Euler's equations, Nonlinearity, Volume 14 (2001) no. 3, pp. 513-532

[15] L. Tartar An Introduction to Sobolev Spaces and Interpolation Spaces, Lecture Notes of the Unione Matematica Italiana, vol. 3, Springer/UMI, Berlin/Bologna, 2007

Cité par Sources :

Commentaires - Politique