Uniqueness results for some PDEs
Journées équations aux dérivées partielles (2003)
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- ISSN: 0752-0360
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topMasmoudi, Nader. "Uniqueness results for some PDEs." Journées équations aux dérivées partielles (2003): 1-13. <http://eudml.org/doc/93437>.
@article{Masmoudi2003,
abstract = {Existence of solutions to many kinds of PDEs can be proved by using a fixed point argument or an iterative argument in some Banach space. This usually yields uniqueness in the same Banach space where the fixed point is performed. We give here two methods to prove uniqueness in a more natural class. The first one is based on proving some estimates in a less regular space. The second one is based on a duality argument. In this paper, we present some results obtained in collaboration with Pierre-Louis Lions, with Kenji Nakanishi and with Fabrice Planchon.},
author = {Masmoudi, Nader},
journal = {Journées équations aux dérivées partielles},
keywords = {duality argument},
language = {eng},
pages = {1-13},
publisher = {Université de Nantes},
title = {Uniqueness results for some PDEs},
url = {http://eudml.org/doc/93437},
year = {2003},
}
TY - JOUR
AU - Masmoudi, Nader
TI - Uniqueness results for some PDEs
JO - Journées équations aux dérivées partielles
PY - 2003
PB - Université de Nantes
SP - 1
EP - 13
AB - Existence of solutions to many kinds of PDEs can be proved by using a fixed point argument or an iterative argument in some Banach space. This usually yields uniqueness in the same Banach space where the fixed point is performed. We give here two methods to prove uniqueness in a more natural class. The first one is based on proving some estimates in a less regular space. The second one is based on a duality argument. In this paper, we present some results obtained in collaboration with Pierre-Louis Lions, with Kenji Nakanishi and with Fabrice Planchon.
LA - eng
KW - duality argument
UR - http://eudml.org/doc/93437
ER -
References
top- [1] N. Bournaveas, Local existence for the Maxwell-Dirac equations in three space dimensions. Comm. Partial Differential Equations 21 (1996), no. 5-6, 693-720. Zbl0880.35116MR1391520
- [2] N. BournaveasLocal existence of energy class solutions for the Dirac-Klein-Gordon equations. Comm. Partial Differential Equations 24 (1999), no. 7-8, 1167-1193. Zbl0931.35134MR1697486
- [3] T. Cazenave and F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in . Nonlinear Anal. 14 (1990), no. 10, 807-836. Zbl0706.35127MR1055532
- [4] J.-Y. Chemin. Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel. Journal d'Analyse Mathématique, 77(?):27-50, 1999. Zbl0938.35125MR1753481
- [5] P.A.M. Dirac, Principles of Quantum Mechanics, Oxford University Press, 4th ed., London (1958) Zbl0080.22005
- [6] G. Furioli, and E. Terraneo. Besov spaces and unconditional well-posedness for the nonlinear Schrödinger equations in , to appear in Commun. Contemp. Math., 2001 Zbl1050.35102MR1992354
- [7] G. Furioli, P.-G. Lemarié-Rieusset, and E. Terraneo. Sur l’unicité dans des solutions ”mild” des équations de Navier-Stokes. C. R. Acad. Sci. Paris Sér. I Math., 325(12):1253-1256, 1997. Zbl0894.35083MR1490408
- [8] G. Furioli, P.-G. Lemarié-Rieusset, and E. Terraneo. Unicité dans et d’autres espaces fonctionnels limites pour Navier-Stokes. Rev. Mat. Iberoamericana, 3 (2000) 605-667. Zbl0970.35101MR1813331
- [9] M. G. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math. (2), 132, 1990, 3, 485-509. Zbl0736.35067MR1078267
- [10] K. Jörgens, Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen. (German) Math. Z. 77 1961 295-308. Zbl0111.09105MR130462
- [11] T. Kato, Strong -solutions of the Navier-Stokes equation in , with applications to weak solutions., Math. Z. 187 (1984), no. 4, 471-480. Zbl0545.35073MR760047
- [12] T. Kato. On nonlinear Schrödinger equations. II. -solutions and unconditional well-posedness, J. Anal. Math., 67, 1995, 281-306, Zbl0848.35124MR1383498
- [13] S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J. 74 (1994), no. 1, 19-44. Zbl0818.35123MR1271462
- [14] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), no. 9, 1221-1268. Zbl0803.35095MR1231427
- [15] J. Leray. Etude de diverses équations intégrales nonlinéaires et de quelques problèmes que pose l'hydrodynamique. J. Math. Pures Appl., 12:1-82, 1933. Zbl0006.16702JFM59.0402.01
- [16] P.-L. Lions and N. Masmoudi. Unicité des solutions faibles de Navier-Stokes dans . C. R. Acad. Sci. Paris Sér. I Math., 327(5):491-496, 1998. Zbl0990.35114MR1652574
- [17] P.-L. Lions and N. Masmoudi. Uniqueness of mild solutions of the Navier-Stokes system in . Comm. Partial Differential Equations. 26 (2001), no. 11-12, 2211-2226. Zbl1086.35077MR1876415
- [18] N. Masmoudi and K. Nakanishi, Uniqueness of finite energy solutions for Maxwell-Dirac and Maxwell-Klein-Gordon equations, to appear in Comm. Math. Physics, 2003. Zbl1029.35199MR2020223
- [19] N. Masmoudi and F. Planchon, On Uniqueness for the critical wave equation, preprint, 2003. Zbl1106.35035
- [20] N. Masmoudi and F. Planchon, On Uniqueness for wave maps, preprint, 2003. MR1992026
- [21] S. Monniaux. Uniqueness of mild solutions of the Navier-Stokes equation and maximal -regularity. C. R. Acad. Sci. Paris Sér. I Math., 328(8):663-668, 1999. Zbl0931.35127MR1680809
- [22] Andrea Nahmod, Atanas Stefanov, and Karen Uhlenbeck. On the well-posedness of the Wave Map problem in high dimensions. preprint, 2001. Zbl1085.58022MR2016196
- [23] B. Najman, The nonrelativistic limit of the nonlinear Dirac equation, Ann. Inst. Henri Poincaré, Anal. Non Lineaire 9 (1992) 3-12 Zbl0746.35036MR1151464
- [24] F. Planchon, On uniqueness for semilinear wave equations, to appear in Math. Zeit.. 2001 Zbl1023.35079MR1992026
- [25] J. Rauch, The Klein-Gordon equation. II. Anomalous singularities for semilinear wave equations. Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. I (Paris, 1978/1979), pp. 335-364, Res. Notes in Math., 53, Pitman, Boston, Mass.-London, 1981. Zbl0473.35055MR631403
- [26] Jalal Shatah and Michael Struwe. Regularity results for nonlinear wave equations. Ann. of Math. (2), 138(3):503-518, 1993. Zbl0836.35096MR1247991
- [27] Jalal Shatah and Michael Struwe. Well-posedness in the energy space for semilinear wave equations with critical growth. Internat. Math. Res. Notices, (7):303ff., approx. 7 pp. (electronic), 1994. Zbl0830.35086MR1283026
- [28] Jalal Shatah and Michael Struwe. The Cauchy problem for wave maps. Int. Math. Res. Not., (11):555-571, 2002. Zbl1024.58014MR1890048
- [20] M. Struwe. Uniqueness for critical nonlinear wave equations and wave maps via the energy inequality, Comm. Pure Appl. Math., 52 n 9, 1999, 1179-1188 Zbl0933.35141MR1692140
- [21] Y. ZhouUniqueness of generalized solutions to nonlinear wave equations. Amer. J. Math. 122 (2000), no. 5, 939-965. Zbl0961.35084MR1781926
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