Notes on the incompressible Euler and related equations on ℝ ^N

Abstract

The author reviews briefly some of the recent results on the blow-up problem for the incompressible Euler equations on ℝ ^N, and also presents Liouville type theorems for the incompressible and compressible fluid equations.

Keywords

Euler equations / Navier-Stokes equations / Liouville theorem

References

[1] Bardos C., Titi E. S.. Euler equations of incompressible ideal fluids. Russ. Math. Surv., 2007, 62(3): 409-451

[2] Beale J. T., Kato T., Majda A.. Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys., 1984, 94(1): 61-66

[3] da Veiga H. B.. Vorticity and smoothness in incompressible viscous flows (wave phenomena and asymptotic analysis). RIMS Kokyuroku, 2003, 1315: 37-45

[4] da Veiga H. B., Berselli L. C.. On the regularizing effect of the vorticity direction in incompressible viscous flows. Diff. Int. Eqs., 2002, 15(3): 345-356

[5] Berselli L. C., Galdi G. P.. Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations. Proc. Amer. Math. Soc., 2002, 130(12): 3585-3595

[6] Chae, D., Liouville type of theorems for the Euler and the Navier-Stokes equations, 2008, preprint. arXiv:AP0809.0743

[7] Chae, D., Liouville type of theorems with weights for the Navier-Stokes equations and the Euler equations, 2008, preprint. arXiv:AP0811.4647

[8] Chae, D., On the nonexistence of stationary weak solutions to the compressible fluid equations, 2009, preprint. arXiv:AP0812.4869

[9] Chae, D., On the nonexistence of time dependent global weak solutions to the compressible fluid equations, 2009, preprint. arXiv:AP0901.3278

[10] Chae, D., On the blow-up problem and new a priori estimates for the 3D Euler and the Navier-Stokes equations, 2007, preprint. arXiv:AP0711.1113

[11] Chae D.. On the a priori estimates for the Euler, the Navier-Stokes and the quasi-geostrophic equations. Adv. Math., 2009, 221(5): 1678-1702

[12] Chae, D., Nonexistence of self-similar singularities in the ideal magnetohydrodynamics, Arch. Rational Mech. Anal., 2008, in press. DOI:10.1007/s00205-008-0182-9

[13] Chae D.. Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity. J. Funct. Anal., 2008, 254(2): 441-453

[14] Chae D.. Incompressible Euler equations: the blow-up problem and related results. Handbook of Differential Equations: Evolutionary Equations, 2008, Amsterdam: North-Holland 1-50

[15] Chae D.. Nonexistence of asymptotically self-similar singularities in the Euler and the Navier-Stokes equations. Math. Ann., 2007, 338(2): 435-449

[16] Chae D.. Nonexistence of self-similar singularities for the 3D incompressible Euler equations. Comm. Math. Phys., 2007, 273(1): 203-215

[17] Chae D.. On the regularity conditions for the Navier-Stokes and the related equations. Rev. Mat. Iberoamericana, 2007, 23(1): 373-384

[18] Chae D.. On the continuation principles for the Euler equations and the quasi-geostrophic equation. J. Diff. Eqs., 2006, 227(2): 640-651

[19] Chae D., Lee J.. Regularity criterion in terms of pressure for the Navier-Stokes equations. Nonlinear Anal., 2001, 46(5): 727-735

[20] Constantin P.. On the Euler equations of incompressible fluids. Bull. Amer. Math. Soc., 2007, 44(4): 603-621

[21] Constantin P.. A few results and open problems regarding incompressible fluids. Notices Amer. Math. Soc., 1995, 42(6): 658-663

[22] Constantin P., Fefferman C.. Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J., 1993, 42(3): 775-789

[23] Constantin P., Fefferman C., Majda A.. Geometric constraints on potential singularity formulation in the 3-D Euler equations. Comm. Part. Diff. Eqs., 1996, 21(3–4): 559-571

[24] Córdoba D., Fefferman C.. On the collapse of tubes carried by 3D incompressible flows. Comm. Math. Phys., 2001, 222(2): 293-298

[25] Córdoba D., Fefferman C.. Potato chip singularities of 3D flows. SIAM J. Math. Anal., 2001, 33(4): 786-789

[26] Deng J., Hou T. Y., Yu X. W.. Improved geometric conditions for non-blow upof the 3D incompressible Euler equations. Comm. Part. Diff. Eqs., 2006, 31(2): 293-306

[27] Guo Z. H., Jiang S.. Self-similar solutions to the isothermal compressible Navier-Stokes equations. IMA J. Appl. Math., 2006, 71(5): 658-669

[28] Kato T.. Nonstationary flows of viscous and ideal fluids in ℝ³. J. Funct. Anal., 1972, 9(3): 296-305

[29] Koch, G., Nadirashvili, N., Seregin, G., et al, Liouville theorems for the Navier-Stokes equations and applications, Acta Math., to appear. arXiv:AP0706.3599

[30] Leray J.. Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math., 1934, 63: 193-248

[31] Majda A.. Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, 1984, New York: Springer-Verlag

[32] Majda A., Bertozzi A. L.. Vorticity and Incompressible Flow, 2002, Cambridge: Cambridge University Press

[33] Miller J. R., O’Leary M., Schonbek M.. Nonexistence of singular pseudo-self-similar solutions of the Navier-Stokes system. Math. Ann., 2001, 319(4): 809-815

[34] Nečas J., Ruzicka M., Šverák V.. On Leray’s self-similar solutions of the Navier-Stokes equations. Acta Math., 1996, 176(2): 283-294

[35] Sermange M., Temam R.. Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math., 1983, 36(5): 635-664

[36] Tsai T.-P.. On Leray’s self-similar solutions of the Navier-Stokes equations satisfying local energy estimates. Arch. Rat. Mech. Anal., 1998, 143(1): 29-51