Uniqueness results for some PDEs
Journées équations aux dérivées partielles (2003), article no. 10, 13 p.

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.

@article{JEDP_2003____A10_0,
     author = {Masmoudi, Nader},
     title = {Uniqueness results for some {PDEs}},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {10},
     publisher = {Universit\'e de Nantes},
     year = {2003},
     doi = {10.5802/jedp.624},
     zbl = {02079445},
     mrnumber = {2050596},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.624/}
}
TY  - JOUR
AU  - Masmoudi, Nader
TI  - Uniqueness results for some PDEs
JO  - Journées équations aux dérivées partielles
PY  - 2003
DA  - 2003///
PB  - Université de Nantes
UR  - http://www.numdam.org/articles/10.5802/jedp.624/
UR  - https://zbmath.org/?q=an%3A02079445
UR  - https://www.ams.org/mathscinet-getitem?mr=2050596
UR  - https://doi.org/10.5802/jedp.624
DO  - 10.5802/jedp.624
LA  - en
ID  - JEDP_2003____A10_0
ER  - 
Masmoudi, Nader. Uniqueness results for some PDEs. Journées équations aux dérivées partielles (2003), article  no. 10, 13 p. doi : 10.5802/jedp.624. http://www.numdam.org/articles/10.5802/jedp.624/

[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. | MR 1391520 | Zbl 0880.35116

[2] N. Bournaveas Local existence of energy class solutions for the Dirac-Klein-Gordon equations. Comm. Partial Differential Equations 24 (1999), no. 7-8, 1167-1193. | MR 1697486 | Zbl 0931.35134

[3] T. Cazenave and F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in H s . Nonlinear Anal. 14 (1990), no. 10, 807-836. | MR 1055532 | Zbl 0706.35127

[4] J.-Y. Chemin. Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel. Journal d'Analyse Mathématique, 77(?):27-50, 1999. | MR 1753481 | Zbl 0938.35125

[5] P.A.M. Dirac, Principles of Quantum Mechanics, Oxford University Press, 4th ed., London (1958) | Zbl 0080.22005

[6] G. Furioli, and E. Terraneo. Besov spaces and unconditional well-posedness for the nonlinear Schrödinger equations in H˙ s (R n ), to appear in Commun. Contemp. Math., 2001 | MR 1992354 | Zbl 1050.35102

[7] G. Furioli, P.-G. Lemarié-Rieusset, and E. Terraneo. Sur l’unicité dans L 3 (R 3 ) des solutions ”mild” des équations de Navier-Stokes. C. R. Acad. Sci. Paris Sér. I Math., 325(12):1253-1256, 1997. | MR 1490408 | Zbl 0894.35083

[8] G. Furioli, P.-G. Lemarié-Rieusset, and E. Terraneo. Unicité dans L 3 (R 3 ) et d’autres espaces fonctionnels limites pour Navier-Stokes. Rev. Mat. Iberoamericana, 3 (2000) 605-667. | MR 1813331 | Zbl 0970.35101

[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. | MR 1078267 | Zbl 0736.35067

[10] K. Jörgens, Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen. (German) Math. Z. 77 1961 295-308. | EuDML 169993 | MR 130462 | Zbl 0111.09105

[11] T. Kato, Strong L p -solutions of the Navier-Stokes equation in R m , with applications to weak solutions., Math. Z. 187 (1984), no. 4, 471-480. | EuDML 173504 | MR 760047 | Zbl 0545.35073

[12] T. Kato. On nonlinear Schrödinger equations. II. H s -solutions and unconditional well-posedness, J. Anal. Math., 67, 1995, 281-306, | MR 1383498 | Zbl 0848.35124

[13] S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J. 74 (1994), no. 1, 19-44. | MR 1271462 | Zbl 0818.35123

[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. | MR 1231427 | Zbl 0803.35095

[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. | EuDML 235182 | JFM 59.0402.01 | Zbl 0006.16702

[16] P.-L. Lions and N. Masmoudi. Unicité des solutions faibles de Navier-Stokes dans L N (Ω). C. R. Acad. Sci. Paris Sér. I Math., 327(5):491-496, 1998. | MR 1652574 | Zbl 0990.35114

[17] P.-L. Lions and N. Masmoudi. Uniqueness of mild solutions of the Navier-Stokes system in L N (Ω). Comm. Partial Differential Equations. 26 (2001), no. 11-12, 2211-2226. | MR 1876415 | Zbl 01717449

[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. | MR 2020223 | Zbl 1029.35199

[19] N. Masmoudi and F. Planchon, On Uniqueness for the critical wave equation, preprint, 2003.

[20] N. Masmoudi and F. Planchon, On Uniqueness for wave maps, preprint, 2003. | MR 1992026

[21] S. Monniaux. Uniqueness of mild solutions of the Navier-Stokes equation and maximal L p -regularity. C. R. Acad. Sci. Paris Sér. I Math., 328(8):663-668, 1999. | MR 1680809 | Zbl 0931.35127

[22] Andrea Nahmod, Atanas Stefanov, and Karen Uhlenbeck. On the well-posedness of the Wave Map problem in high dimensions. preprint, 2001. | MR 2016196 | Zbl 1085.58022

[23] B. Najman, The nonrelativistic limit of the nonlinear Dirac equation, Ann. Inst. Henri Poincaré, Anal. Non Lineaire 9 (1992) 3-12 | EuDML 78271 | Numdam | MR 1151464 | Zbl 0746.35036

[24] F. Planchon, On uniqueness for semilinear wave equations, to appear in Math. Zeit.. 2001 | MR 1992026 | Zbl 1023.35079

[25] J. Rauch, The u 5 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 631403 | Zbl 0473.35055

[26] Jalal Shatah and Michael Struwe. Regularity results for nonlinear wave equations. Ann. of Math. (2), 138(3):503-518, 1993. | MR 1247991 | Zbl 0836.35096

[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. | MR 1283026 | Zbl 0830.35086

[28] Jalal Shatah and Michael Struwe. The Cauchy problem for wave maps. Int. Math. Res. Not., (11):555-571, 2002. | MR 1890048 | Zbl 1024.58014

[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 | MR 1692140 | Zbl 0933.35141

[21] Y. Zhou Uniqueness of generalized solutions to nonlinear wave equations. Amer. J. Math. 122 (2000), no. 5, 939-965. | MR 1781926 | Zbl 0961.35084

Cité par Sources :