Approximate solutions of the incompressible Euler equations with no concentrations

 Access to full text

 Full (PDF)

1. [1] Adams R., Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, 1975. Zbl0314.46030MR450957
2. [2] Anderson C., An Introduction to Vortex Methods, Lecture Notes in Math., Vol. 1360, Springer, Berlin, 1968. Zbl0648.00012MR979555
3. [3] Bell J.B., Colella P., Glaz H.M., A Second-order projection method for the incompressible Navier-Stokes equations, JCP85 (1989) 257-283. Zbl0681.76030MR1029192
4. [4] Bennett C., Intermediate spaces and the class L log+ L, Arkiv Mat.2 (1973) 215-228. Zbl0266.46025MR352966
5. [5] Bennett C., Rudnick K., On Lorentz-Zygmund spaces, Dissert. Math.175 (1980) 1-72. Zbl0456.46028MR576995
6. [6] Bennett C., Sharpley R., Interpolation of Operators, Pure and Applied Mathematics, Vol. 129, Academic Press, 1988. Zbl0647.46057MR928802
7. [7] Chacon-Rebollo T., Hou T., A Lagrangian finite element method for the 2-D Euler equations, CPAM43 (1990) 735-767. Zbl0705.76059MR1059327
8. [8] Caffarelli L., Kohn R., Nirenberg L., Partial regularity of suitable solutions of the Navier-Stokes equations, CPAM35 (1982) 771-831. Zbl0509.35067MR673830
9. [9] Chae D., Weak solutions of 2-D Euler equations with initial vorticity in L(log L), J. Differential Equations103 (1993) 323-337. Zbl0854.35082MR1221908
10. [10] Chae D., Weak solutions of 2-D incompressible Euler equations, Nonlin. Analysis: TMA23 (1994) 629-638. Zbl0814.76022MR1297282
11. [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. [12] Chorin A., A numerical method for solving incompressible viscous flow problems, JCP2 (1967) 12-26. Zbl0149.44802
13. [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. Zbl0818.35085MR1298949
14. [14] DeVore R., Lucier B., Wavelets, Acta Numerica1 (1992) 1-56. Zbl0766.65009MR1165722
15. [15] DiPerna R., Lions P.-L., Ordinary differential equations Sobolev spaces and transport theory, Invent. Math.98 (1989) 511-547. Zbl0696.34049MR1022305
16. [16] DiPerna R., Majda A., Concentrations in regularizations for 2D incompressible flow, Comm. Pure Appl. Math.40 (1987) 301-345. Zbl0850.76730MR882068
17. [ 17] DiPerna R., Majda A., Reduced Hausdorff dimension and concentration-cancelation for 2-D incompressible flow, J. Amer. Math. Soc.1 (1988) 59-95. Zbl0707.76026MR924702
18. [18] DiPerna R., Majda A., Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys.108 (1987) 667-689. Zbl0626.35059MR877643
19. [19] Delort J.-M., Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc.4 (1991) 553-586. Zbl0780.35073MR1102579
20. [20] Donaldson T.K., Trudinger N.S., Orlicz-Sobolev spaces and imbedding theorems, J. Funct. Anal.8 (1971) 52-75. Zbl0216.15702MR301500
21. [21] E W., Liu J.-G., Finite difference schemes for incompressible flows in the velocity impulse density formulation, JCP130 (1997) 67-76. Zbl0870.76048MR1427404
22. [22] Giga Y., Miyakawa T., Navier-Stokes flows in R3 and Morrey spaces, Comm. PDE14 (1989) 577-618. Zbl0681.35072MR993821
23. [23] Henshaw W., Kreiss H.-O., Reyna L., A forth-order accurate difference approximation for the incompressible Navier-Stokes equations, Comput. Fluids23 (1994) 575-593. Zbl0801.76055MR1264751
24. [24] Hou T.Y., Wetton B.T.R., Second-order convergence of a projection scheme for the incompressible Navier-Stokes equations with boundaries, SINUM30 (3) (1993) 609-629. Zbl0776.76055MR1220643
25. [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. Zbl0920.35111MR1664898
26. [26] Krasny R., Computing vortex sheet motion, in: Proc. Inter. Congress Math. Vol. I, II, Kyoto1990, Math. Soc. Japan, 1991, pp. 1573-1583. Zbl0748.76029MR1159338
27. [27] Levy D., Tadmor E., Non-oscillatory central schemes for the incompressible 2-D Euler equations, Mathematical Research Letters4 (1997) 1-20. Zbl0883.76057MR1453063
28. [28] Lin F., A new proof of Caffarelli-Kohn-Nirenberg's theorem, Preprint. MR1488514
29. [29] Liu J.G., Xin Z., Convergence of vortex methods for weak solutions to the 2D Euler equations with vortex sheet data, CPAM48 (1995) 611-628. Zbl0829.35098MR1338471
30. [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. Zbl0866.76002MR1422251
31. [31] Majda A., Remarks on weak solutions for vortex sheets with a distinguished sign, Ind. Univ. Math. J.42 (1993) 921-939. Zbl0791.76015MR1254126
32. [32] Meyer Y., Wavelets and Operators, Cambridge Studies in Mathematics, Vol. 37, Cambridge Univ. Press, 1992. Zbl0776.42019MR1228209
33. [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. Zbl0811.76007MR1197088
34. [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. MR894077
35. [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. Zbl0882.76016MR1462801
36. [36] Onsager L., Statistical hydrodynamics, Nuovo Cimento (Supplemento)6 (1949) 279-287. MR36116
37. [37] Scheffer V., An inviscid flow with compact support in space-time, J. Geom. Anal.3 (1993) 343-401. Zbl0836.76017MR1231007
38. [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. Zbl0862.35092MR1399201
39. [39] Shnirelman A., On the non-uniqueness of weak solution of the Euler equations, Comm. Pure Appl. Math.50 (1997) 1261-1286. Zbl0909.35109MR1417742
40. [40] Tartar L., Compensated compactness and applications to partial differential equations, in: Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, IV, Pitman, London, 1979. Zbl0437.35004MR584398
41. [41] Temam R., Navier-Stokes Equations, North-Holland, Amsterdam, 1977. Zbl0383.35057MR603444
42. [42] Tian G., Xin Z., Gradient estimation on Navier-Stokes equations, Preprint. Zbl0939.35139MR1685610
43. [43] Trudinger N., On imbeddings into Orlicz spaces and some applications, J. Math. and Mechanics17 (1967) 473-483. Zbl0163.36402MR216286
44. [44] Vecchi I., Wu S., On L1-vorticity for 2-D incompressible flow, Manuscripta Math.78 (1993) 403-412. Zbl0807.35115MR1208650
45. [45] Vishik M., Hydrodynamics in Besov spaces, Anch. Rat. Mech. Anal.145 (1998) 197-214. Zbl0926.35123MR1664597
46. [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. Zbl0938.35128MR1717576
47. [47] Yudovich V.I., Non-stationary flow of an ideal incompressible liquid, USSR Comp. Math. and Math. Phys.3 (1963) 1407-1456. English transl. Zbl0147.44303
48. [48] Yudovich V.I., Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Math. Res. Letters2 (1995) 27-38. Zbl0841.35092MR1312975
49. [49] Ziemer W.P., Weakly Differentiable Functions, Graduate Texts in Mathematics, Vol. 120, Springer, 1989. Zbl0692.46022MR1014685

1. Milton C. Lopes Filho, John Lowengrub, Helena J. Nussenzveig Lopes, Yuxi Zheng, Numerical evidence of nonuniqueness in the evolution of vortex sheets