Approximate solutions of the incompressible Euler equations with no concentrations
Lopes Filho, Milton C. ; Nussenzveig Lopes, Helena J. ; Tadmor, Eitan
Annales de l'I.H.P. Analyse non linéaire, Tome 17 (2000), p. 371-412 / Harvested from Numdam
@article{AIHPC_2000__17_3_371_0,
     author = {Lopes Filho, Milton C. and Nussenzveig Lopes, Helena J. and Tadmor, Eitan},
     title = {Approximate solutions of the incompressible Euler equations with no concentrations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {17},
     year = {2000},
     pages = {371-412},
     mrnumber = {1771138},
     zbl = {0965.35110},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2000__17_3_371_0}
}
Lopes Filho, Milton C.; Nussenzveig Lopes, Helena J.; Tadmor, Eitan. Approximate solutions of the incompressible Euler equations with no concentrations. Annales de l'I.H.P. Analyse non linéaire, Tome 17 (2000) pp. 371-412. http://gdmltest.u-ga.fr/item/AIHPC_2000__17_3_371_0/

[1] Adams R., Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, 1975. | MR 450957 | Zbl 0314.46030

[2] Anderson C., An Introduction to Vortex Methods, Lecture Notes in Math., Vol. 1360, Springer, Berlin, 1968. | MR 979555 | Zbl 0648.00012

[3] Bell J.B., Colella P., Glaz H.M., A Second-order projection method for the incompressible Navier-Stokes equations, JCP 85 (1989) 257-283. | MR 1029192 | Zbl 0681.76030

[4] Bennett C., Intermediate spaces and the class L log+ L, Arkiv Mat. 2 (1973) 215-228. | MR 352966 | Zbl 0266.46025

[5] Bennett C., Rudnick K., On Lorentz-Zygmund spaces, Dissert. Math. 175 (1980) 1-72. | MR 576995 | Zbl 0456.46028

[6] Bennett C., Sharpley R., Interpolation of Operators, Pure and Applied Mathematics, Vol. 129, Academic Press, 1988. | MR 928802 | Zbl 0647.46057

[7] Chacon-Rebollo T., Hou T., A Lagrangian finite element method for the 2-D Euler equations, CPAM 43 (1990) 735-767. | MR 1059327 | Zbl 0705.76059

[8] Caffarelli L., Kohn R., Nirenberg L., Partial regularity of suitable solutions of the Navier-Stokes equations, CPAM 35 (1982) 771-831. | MR 673830 | Zbl 0509.35067

[9] Chae D., Weak solutions of 2-D Euler equations with initial vorticity in L(log L), J. Differential Equations 103 (1993) 323-337. | MR 1221908 | Zbl 0854.35082

[10] Chae D., Weak solutions of 2-D incompressible Euler equations, Nonlin. Analysis: TMA 23 (1994) 629-638. | MR 1297282 | Zbl 0814.76022

[11] Chen G.-Q., The theory of compensated compactness and the system of isentropic gas dynamics, Preprint, MSRI-00527-91, Math. Sci. Res. Inst., Berkeley.

[12] Chorin A., A numerical method for solving incompressible viscous flow problems, JCP 2 (1967) 12-26. | Zbl 0149.44802

[13] Constantin P., E W.Titi E., Onsager's conjecture on the energy conservation for solutions of Euler's equation, Comm. Math. Phys. 165 (1994) 207-209. | MR 1298949 | Zbl 0818.35085

[14] Devore R., Lucier B., Wavelets, Acta Numerica 1 (1992) 1-56. | MR 1165722 | Zbl 0766.65009

[15] Diperna R., Lions P.-L., Ordinary differential equations Sobolev spaces and transport theory, Invent. Math. 98 (1989) 511-547. | MR 1022305 | Zbl 0696.34049

[16] Diperna R., Majda A., Concentrations in regularizations for 2D incompressible flow, Comm. Pure Appl. Math. 40 (1987) 301-345. | MR 882068 | Zbl 0850.76730

[ 17] Diperna R., Majda A., Reduced Hausdorff dimension and concentration-cancelation for 2-D incompressible flow, J. Amer. Math. Soc. 1 (1988) 59-95. | MR 924702 | Zbl 0707.76026

[18] Diperna R., Majda A., Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987) 667-689. | MR 877643 | Zbl 0626.35059

[19] Delort J.-M., Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc. 4 (1991) 553-586. | MR 1102579 | Zbl 0780.35073

[20] Donaldson T.K., Trudinger N.S., Orlicz-Sobolev spaces and imbedding theorems, J. Funct. Anal. 8 (1971) 52-75. | MR 301500 | Zbl 0216.15702

[21] E W., Liu J.-G., Finite difference schemes for incompressible flows in the velocity impulse density formulation, JCP 130 (1997) 67-76. | MR 1427404 | Zbl 0870.76048

[22] Giga Y., Miyakawa T., Navier-Stokes flows in R3 and Morrey spaces, Comm. PDE 14 (1989) 577-618. | MR 993821 | Zbl 0681.35072

[23] Henshaw W., Kreiss H.-O., Reyna L., A forth-order accurate difference approximation for the incompressible Navier-Stokes equations, Comput. Fluids 23 (1994) 575-593. | MR 1264751 | Zbl 0801.76055

[24] Hou T.Y., Wetton B.T.R., Second-order convergence of a projection scheme for the incompressible Navier-Stokes equations with boundaries, SINUM 30 (3) (1993) 609-629. | MR 1220643 | Zbl 0776.76055

[25] Hounie J., Lopes Filho M.C., Nussenzveig Lopes H.J., Schochet S., A priori temporal regularity for the streamfunction of 2D incompressible, inviscid flow, Nonlinear Analysis Theor. 35 (1999) 871-884. | MR 1664898 | Zbl 0920.35111

[26] Krasny R., Computing vortex sheet motion, in: Proc. Inter. Congress Math. Vol. I, II, Kyoto 1990, Math. Soc. Japan, 1991, pp. 1573-1583. | MR 1159338 | Zbl 0748.76029

[27] Levy D., Tadmor E., Non-oscillatory central schemes for the incompressible 2-D Euler equations, Mathematical Research Letters 4 (1997) 1-20. | MR 1453063 | Zbl 0883.76057

[28] Lin F., A new proof of Caffarelli-Kohn-Nirenberg's theorem, Preprint. | MR 1488514

[29] Liu J.G., Xin Z., Convergence of vortex methods for weak solutions to the 2D Euler equations with vortex sheet data, CPAM 48 (1995) 611-628. | MR 1338471 | Zbl 0829.35098

[30] Lions P.L., Mathematical Topics in Fluid Mechanics, Vol. 1, Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, Vol. 3, Clarendon Press, 1996. | MR 1422251 | Zbl 0866.76002

[31] Majda A., Remarks on weak solutions for vortex sheets with a distinguished sign, Ind. Univ. Math. J. 42 (1993) 921-939. | MR 1254126 | Zbl 0791.76015

[32] Meyer Y., Wavelets and Operators, Cambridge Studies in Mathematics, Vol. 37, Cambridge Univ. Press, 1992. | MR 1228209 | Zbl 0776.42019

[33] Morgulis A.B., On existence of two-dimensional nonstationary flows of an ideal incompressible liquid admitting a curl nonsummable to any power greater than 1, Siberian Math. J. 33 (1992) 934-937. | MR 1197088 | Zbl 0811.76007

[34] Murat F., A survey on compensated compactness, in: Cesari L. (Ed.), Contributions to Modern Calculus of Variations, Pitman Research Notes in Mathematics Series, Wiley, New York, 1987, pp. 145-183. | MR 894077

[35] Nussenzveig Lopes H.J., A refined estimate of the size of concentration sets for 2D incompressible inviscid flow, Ind. Univ. Math. J. 46 (1997) 165-182. | MR 1462801 | Zbl 0882.76016

[36] Onsager L., Statistical hydrodynamics, Nuovo Cimento (Supplemento) 6 (1949) 279-287. | MR 36116

[37] Scheffer V., An inviscid flow with compact support in space-time, J. Geom. Anal. 3 (1993) 343-401. | MR 1231007 | Zbl 0836.76017

[38] Schochet S., The point-vortex method for periodic weak solutions of the 2-D Euler equations, Comm. Pure Appl. Math. 49 (1996) 911-965. | MR 1399201 | Zbl 0862.35092

[39] Shnirelman A., On the non-uniqueness of weak solution of the Euler equations, Comm. Pure Appl. Math. 50 (1997) 1261-1286. | MR 1417742 | Zbl 0909.35109

[40] Tartar L., Compensated compactness and applications to partial differential equations, in: Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, IV, Pitman, London, 1979. | MR 584398 | Zbl 0437.35004

[41] Temam R., Navier-Stokes Equations, North-Holland, Amsterdam, 1977. | MR 603444 | Zbl 0383.35057

[42] Tian G., Xin Z., Gradient estimation on Navier-Stokes equations, Preprint. | MR 1685610

[43] Trudinger N., On imbeddings into Orlicz spaces and some applications, J. Math. and Mechanics 17 (1967) 473-483. | MR 216286 | Zbl 0163.36402

[44] Vecchi I., Wu S., On L1-vorticity for 2-D incompressible flow, Manuscripta Math. 78 (1993) 403-412. | MR 1208650 | Zbl 0807.35115

[45] Vishik M., Hydrodynamics in Besov spaces, Anch. Rat. Mech. Anal. 145 (1998) 197-214. | MR 1664597 | Zbl 0926.35123

[46] Vishik M., Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Ann. Sci. Ecole Norm. Sup. 32 (1999) 769-812. | Numdam | MR 1717576 | Zbl 0938.35128

[47] Yudovich V.I., Non-stationary flow of an ideal incompressible liquid, USSR Comp. Math. and Math. Phys. 3 (1963) 1407-1456. English transl. | Zbl 0147.44303

[48] Yudovich V.I., Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Math. Res. Letters 2 (1995) 27-38. | MR 1312975 | Zbl 0841.35092

[49] Ziemer W.P., Weakly Differentiable Functions, Graduate Texts in Mathematics, Vol. 120, Springer, 1989. | MR 1014685 | Zbl 0692.46022