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.
@incollection{JEDP_2003____A10_0, author = {Masmoudi, Nader}, title = {Uniqueness results for some {PDEs}}, booktitle = {}, series = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, eid = {10}, pages = {1--13}, publisher = {Universit\'e de Nantes}, year = {2003}, doi = {10.5802/jedp.624}, mrnumber = {2050596}, zbl = {02079445}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jedp.624/} }
Masmoudi, Nader. Uniqueness results for some PDEs. Journées équations aux dérivées partielles (2003), article no. 10, 13 p. doi : 10.5802/jedp.624. http://archive.numdam.org/articles/10.5802/jedp.624/
[1] Local existence for the Maxwell-Dirac equations in three space dimensions. Comm. Partial Differential Equations 21 (1996), no. 5-6, 693-720. | MR | Zbl
,[2] Local existence of energy class solutions for the Dirac-Klein-Gordon equations. Comm. Partial Differential Equations 24 (1999), no. 7-8, 1167-1193. | MR | Zbl
[3] The Cauchy problem for the critical nonlinear Schrödinger equation in . Nonlinear Anal. 14 (1990), no. 10, 807-836. | MR | Zbl
and ,[4] Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel. Journal d'Analyse Mathématique, 77(?):27-50, 1999. | MR | Zbl
.[5] Principles of Quantum Mechanics, Oxford University Press, 4th ed., London (1958) | Zbl
,[6] Besov spaces and unconditional well-posedness for the nonlinear Schrödinger equations in , to appear in Commun. Contemp. Math., 2001 | MR | Zbl
, and .[7] 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. | MR | Zbl
, , and .[8] Unicité dans et d’autres espaces fonctionnels limites pour Navier-Stokes. Rev. Mat. Iberoamericana, 3 (2000) 605-667. | MR | Zbl
, , and .[9] Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math. (2), 132, 1990, 3, 485-509. | MR | Zbl
,[10] Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen. (German) Math. Z. 77 1961 295-308. | EuDML | MR | Zbl
,[11] Strong -solutions of the Navier-Stokes equation in , with applications to weak solutions., Math. Z. 187 (1984), no. 4, 471-480. | EuDML | MR | Zbl
,[12] On nonlinear Schrödinger equations. II. -solutions and unconditional well-posedness, J. Anal. Math., 67, 1995, 281-306, | MR | Zbl
.[13] On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J. 74 (1994), no. 1, 19-44. | MR | Zbl
and ,[14] Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), no. 9, 1221-1268. | MR | Zbl
and ,[15] Etude de diverses équations intégrales nonlinéaires et de quelques problèmes que pose l'hydrodynamique. J. Math. Pures Appl., 12:1-82, 1933. | EuDML | JFM | Numdam | Zbl
.[16] Unicité des solutions faibles de Navier-Stokes dans . C. R. Acad. Sci. Paris Sér. I Math., 327(5):491-496, 1998. | MR | Zbl
and .[17] Uniqueness of mild solutions of the Navier-Stokes system in . Comm. Partial Differential Equations. 26 (2001), no. 11-12, 2211-2226. | MR | Zbl
and .[18] Uniqueness of finite energy solutions for Maxwell-Dirac and Maxwell-Klein-Gordon equations, to appear in Comm. Math. Physics, 2003. | MR | Zbl
and ,[19] On Uniqueness for the critical wave equation, preprint, 2003.
and ,[20] On Uniqueness for wave maps, preprint, 2003. | MR
and ,[21] 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. | MR | Zbl
.[22] On the well-posedness of the Wave Map problem in high dimensions. preprint, 2001. | MR | Zbl
, and .[23] The nonrelativistic limit of the nonlinear Dirac equation, Ann. Inst. Henri Poincaré, Anal. Non Lineaire 9 (1992) 3-12 | EuDML | Numdam | MR | Zbl
,[24] n uniqueness for semilinear wave equations, to appear in Math. Zeit.. 2001 | MR | Zbl
, O[25] 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. | MR | Zbl
,[26] Regularity results for nonlinear wave equations. Ann. of Math. (2), 138(3):503-518, 1993. | MR | Zbl
and .[27] Well-posedness in the energy space for semilinear wave equations with critical growth. Internat. Math. Res. Notices, (7):303ff., approx. 7 pp. (electronic), 1994. | MR | Zbl
and .[28] The Cauchy problem for wave maps. Int. Math. Res. Not., (11):555-571, 2002. | MR | Zbl
and .[20] Uniqueness for critical nonlinear wave equations and wave maps via the energy inequality, Comm. Pure Appl. Math., 52 n 9, 1999, 1179-1188 | MR | Zbl
.[21] Uniqueness of generalized solutions to nonlinear wave equations. Amer. J. Math. 122 (2000), no. 5, 939-965. | MR | Zbl
Cité par Sources :