Abstract
We study whether some of the non-physical properties observed for weak solutions of the incompressible Euler equations can be ruled out by studying the vorticity formulation. Our main contribution is in developing an interior regularity method in the spirit of De Giorgi–Nash–Moser, showing that local weak solutions are exponentially integrable, uniformly in time, under minimal integrability conditions. This is a Serrin-type interior regularity result
for weak solutions in the energy space \(L_t^\infty L_x^2\), satisfying appropriate vorticity estimates. We also obtain improved integrability for the vorticity—which is to be compared with the DiPerna–Lions assumptions. The argument is completely local in nature as the result follows from the structural properties of the equation alone, while completely avoiding all sorts of boundary conditions and related gradient estimates. To the best of our knowledge, the approach we follow is new in the context of Euler equations and provides an alternative look at interior regularity issues. We also show how our method can be used to give a modified proof of the classical Serrin condition for the regularity of the Navier–Stokes equations in any dimension.
Similar content being viewed by others
References
Ambrosio, L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158(2), 227–260 (2004)
Bardos, C., Titi, E.: Euler equations for incompressible ideal fluids. Russ. Math. Surv. 62(3), 409–451 (2007)
Bouchut, F., Crippa, G.: Lagrangian flows for vector fields with gradient given by a singular integral. J. Hyperbolic Differ. Equ. 10(2), 235–282 (2013)
Buckmaster, T., De Lellis, C., Isett, P., Székelyhidi Jr., L.: Anomalous dissipation for 1/5-Hölder Euler flows. Ann. Math. 182(1), 127–172 (2015)
Constantin, P.: On the Euler equations of incompressible fluids. Bull. Am. Math. Soc. (N.S.) 44(4), 603–621 (2007)
Constantin, P., Weinan, E., Titi, E.S.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Comm. Math. Phys. 165(1), 207–209 (1994)
De Giorgi, E.: Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3, 25–43 (1957)
De Lellis, C., Székelyhidi Jr., L.: The Euler equations as a differential inclusion. Ann. Math. (2) 170(3), 1417–1436 (2009)
DiPerna, R., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)
Escauriaza, L., Seregin, G., Šverák, V.: Backward uniqueness for parabolic equations. Arch. Ration. Mech. Anal. 169(2), 147–157 (2003)
Fabes, E.B., Jones Jr., B.F., Rivière, N.M.: The initial value problem for the Navier–Stokes equations with data in \(L^{p}\). Arch. Ration. Mech. Anal. 45, 222–240 (1972)
Gianazza, U., Vespri, V.: Parabolic De Giorgi classes of order p and the Harnack inequality. Calc. Var. Partial Differ. Equ. 26(3), 379–399 (2006)
Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1950)
Isett, P: Hölder continuous Euler flows in three dimensions with compact support in time. Annals of mathematics studies, vol. 196. Princeton University Press, Princeton (2017)
Isett, P.: A proof of Onsager’s conjecture. arXiv:1608.08301 [math.AP], (2016)
Kiselev, A., Šverák, V.: Small scale creation for solutions of the incompressible two-dimensional Euler equation. Ann. Math. (2) 180(3), 1205–1220 (2014)
Kiselev, A., Zlatoš, A.: Blow up for the 2D Euler equation on some bounded domains. J. Differ. Equ. 259(7), 3490–3494 (2015)
Ladyzhenskaya, O.A.: Solution in the large to the boundary-value problem for the Navier–Stokes equations in two space variables. Sov. Phys. Dokl. 123(3), 1128–1131 (1958)
Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)
Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics 27. Cambridge University Press, Cambridge (2002)
Moser, J.: On Harnack’s theorem for elliptic differential equations. Commun. Pure Appl. Math. 14, 577–591 (1961)
Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80(4), 931–954 (1958)
Scheffer, V.: An inviscid flow with compact support in space–time. J. Geom. Anal. 3(4), 343–401 (1993)
Serrin, J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Arch. Ration. Mech. Anal. 9, 187–195 (1962)
Shnirelman, A.: On the nonuniqueness of weak solution of the Euler equation. Commun. Pure Appl. Math. 50(12), 1261–1286 (1997)
Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series 30. Princeton University Press, Princeton (1970)
Struwe, M.: On partial regularity results for the Navier–Stokes equations. Commun. Pure Appl. Math. 41(4), 437–458 (1988)
Vasseur, A.: A new proof of partial regularity of solutions to Navier–Stokes equations. NoDEA Nonlinear Differ. Equ. Appl. 14(5–6), 753–785 (2007)
Acknowledgements
This work initiated while the authors were visiting the Mittag-Leffler Institute during the program “Evolutionary Problems” and was concluded while J.S. visited the University of Coimbra; we thank both institutions for the kind hospitality. J.S. was supported by the Academy of Finland Grant 259363 and by a Väisälä Foundation travel grant. J.M.U. was partially supported by the Centre for Mathematics of the University of Coimbra— UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MCTES and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Ambrosio.
Rights and permissions
About this article
Cite this article
Siljander, J., Urbano, J.M. On the interior regularity of weak solutions to the 2-D incompressible Euler equations. Calc. Var. 56, 126 (2017). https://doi.org/10.1007/s00526-017-1231-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-017-1231-8