Le equazioni di Eulero dal punto di vista delle inclusioni differenziali

Camillo De Lellis

Bollettino dell'Unione Matematica Italiana (2008)

  • Volume: 1, Issue: 3, page 873-879
  • ISSN: 0392-4041

Abstract

top
In a recent joint paper with L. Székelyhidi we have proposed a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in n with n 2 . We give a reformulation of the Euler equations as a differential inclusion, and in this way we obtain transparent proofs of several celebrated results of V. Scheffer and A. Shnirelman concerning the non-uniqueness of weak solutions and the existence of energy-decreasing solutions. Our results are stronger because they work in any dimension and yield bounded velocity and pressure.

How to cite

top

De Lellis, Camillo. "Le equazioni di Eulero dal punto di vista delle inclusioni differenziali." Bollettino dell'Unione Matematica Italiana 1.3 (2008): 873-879. <http://eudml.org/doc/290488>.

@article{DeLellis2008,
author = {De Lellis, Camillo},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {ita},
month = {10},
number = {3},
pages = {873-879},
publisher = {Unione Matematica Italiana},
title = {Le equazioni di Eulero dal punto di vista delle inclusioni differenziali},
url = {http://eudml.org/doc/290488},
volume = {1},
year = {2008},
}

TY - JOUR
AU - De Lellis, Camillo
TI - Le equazioni di Eulero dal punto di vista delle inclusioni differenziali
JO - Bollettino dell'Unione Matematica Italiana
DA - 2008/10//
PB - Unione Matematica Italiana
VL - 1
IS - 3
SP - 873
EP - 879
LA - ita
UR - http://eudml.org/doc/290488
ER -

References

top
  1. BRESSAN, A. - FLORES, F., On total differential inclusions, Rend. Sem. Mat. Univ. Padova, 92 (1994), 9-16. Zbl0821.35158MR1320474
  2. CELLINA, A., On the differential inclusion x [ - 1 , + 1 ] , Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 69 (1980), 1-2 (1981), 1-6. Zbl0922.34009MR641583
  3. CHORIN, A. J., Vorticity and turbulence, vol. 103 of Applied Mathematical Sciences, (Springer-Verlag, New York, 1994). MR1281384DOI10.1007/978-1-4419-8728-0
  4. DACOROGNA, B. - MARCELLINI, P., General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases, Acta Math., 178 (1997), 1-37. Zbl0901.49027MR1448710DOI10.1007/BF02392708
  5. DAFERMOS, C. M., Hyperbolic conservation laws in continuum physics, Vol. 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] (Springer-Verlag, Berlin, 2000). Zbl0940.35002MR1763936DOI10.1007/3-540-29089-3_14
  6. DE LELLIS, C. - SZEKELYHIDI, L., The Euler equations as a differential inclusion, Preprint. To appear in Ann. of Math. (2007). MR2600877DOI10.4007/annals.2009.170.1417
  7. DE LELLIS, C. - SZEKELYHIDI, L., On admissibility criteria for weak solutions of the Euler equations, Preprint (2008). MR2564474DOI10.1007/s00205-008-0201-x
  8. DI PERNA, R. J., Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc., 292, 2 (1985), 383-420. Zbl0606.35052MR808729DOI10.2307/2000221
  9. DI PERNA, R. J. - MAJDA, A. J., Concentrations in regularizations for 2-D incom- pressible flow, Comm. Pure Appl. Math., 40, 3 (1987), 301-345. MR882068DOI10.1002/cpa.3160400304
  10. FRISCH, U., Turbulence, Cambridge University Press, Cambridge (1995), The legacy of A. N. Kolmogorov. MR1428905
  11. KIRCHHEIM, B., Deformations with finitely many gradients and stability of quasi-convex hulls, C. R. Acad. Sci. Paris Sér. I Math., 332, 3 (2001), 289-294. Zbl0989.49013MR1817378DOI10.1016/S0764-4442(00)01792-4
  12. KIRCHHEIM, B., Rigidity and Geometry of microstructures, Habilitation Thesis, University of Leipzig (2003). Zbl1140.74303
  13. KIRCHHEIM, B. - MÜLLER, S. - ŠVERÁK, V., Studying nonlinear PDE by geometry in matrix space, in Geometric analysis and Nonlinear partial differential equations, S. Hildebrandt and H. Karcher, Eds. (Springer-Verlag, 2003), 347-395. MR2008346
  14. MAJDA, A. J. - BERTOZZI, A. L., Vorticity and incompressible flow, Vol. 27 of Cambridge Texts in Applied Mathematics, Cambridge University Press (Cambridge, 2002). MR1867882
  15. MÜLLER, S. - ŠVERÁK, V., Convex integration for Lipschitz mappings and counter-examples to regularity, Ann. of Math. (2), 157 (2003), 715-742. MR1983780DOI10.4007/annals.2003.157.715
  16. SCHEFFER, V., An inviscid flow with compact support in space-time, J. Geom. Anal., 3, 4 (1993), 343-401. Zbl0836.76017MR1231007DOI10.1007/BF02921318
  17. SHNIRELMAN, A., On the nonuniqueness of weak solution of the Euler equation, Comm. Pure Appl. Math., 50, 12 (1997), 1261-1286. Zbl0909.35109MR1476315DOI10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.3.CO;2-4
  18. SHNIRELMAN, A., Weak solutions with decreasing energy of incompressible Euler equations, Comm. Math. Phys., 210, 3 (2000), 541-603. Zbl1011.35107MR1777341DOI10.1007/s002200050791
  19. TAO, T., Nonlinear dispersive equations, vol. 106 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC (2006), Local and global analysis. MR2233925DOI10.1090/cbms/106
  20. TARTAR, L., Compensated compactness and applications to partial differential equations. In Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, vol. 39 of Res. Notes in Math., Pitman (Boston, Mass., 1979), 136-212. MR584398
  21. TARTAR, L., The compensated compactness method applied to systems of conservation laws. In Systems of nonlinear partial differential equations (Oxford, 1982), vol. 111 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Reidel (Dordrecht, 1983), 263-285. MR725524
  22. TEMAM, R., Navier-Stokes equations, third ed., vol. 2 of Studies in Mathematics and its Applications, North-Holland Publishing Co. (Amsterdam, 1984). Theory and numerical analysis, With an appendix by F. Thomasset. MR769654

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.