previous :: next
A sharp counterexample to the local existence of low-regularity solutions to nonlinear wave equations
Hans Lindblad
Source: Duke Math. J. Volume 72, Number 2
(1993), 503-539.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077289430
Mathematical Reviews number (MathSciNet): MR1248683
Zentralblatt MATH identifier: 0797.35123
Digital Object Identifier: doi:10.1215/S0012-7094-93-07219-5
References
[1] M. Beals and M. Bezard, Low-regularity local solutions for field equations, preprint, 1992.
Mathematical Reviews (MathSciNet): MR1373766
Zentralblatt MATH: 0852.35098
Digital Object Identifier: doi:10.1080/03605309608821176
[2] P. Brenner, On $L\sbp-L\sbp\sp\prime $ estimates for the wave-equation, Math. Z. 145 (1975), no. 3, 251–254.
Mathematical Reviews (MathSciNet): MR52:8658
Zentralblatt MATH: 0321.35052
Digital Object Identifier: doi:10.1007/BF01215290
[3] C. Fefferman, The multiplier problem for the ball, Ann. of Math. (2) 94 (1971), 330–336.
Mathematical Reviews (MathSciNet): MR45:5661
Zentralblatt MATH: 0234.42009
Digital Object Identifier: doi:10.2307/1970864
JSTOR: links.jstor.org
[4] M. Grillakis, Regularity for the wave equation with a critical nonlinearity, Comm. Pure Appl. Math. 45 (1992), no. 6, 749–774.
Mathematical Reviews (MathSciNet): MR93e:35073
Zentralblatt MATH: 0785.35065
Digital Object Identifier: doi:10.1002/cpa.3160450604
[5] J. Harmse, On Lebesgue space estimates for the wave equation, Indiana Univ. Math. J. 39 (1990), no. 1, 229–248.
Mathematical Reviews (MathSciNet): MR91j:35158
Zentralblatt MATH: 0683.35008
Digital Object Identifier: doi:10.1512/iumj.1990.39.39013
[6] L. Hörmander, Estimates for translation invariant operators in $L\spp$ spaces, Acta Math. 104 (1960), 93–140.
Mathematical Reviews (MathSciNet): MR22:12389
Zentralblatt MATH: 0093.11402
Digital Object Identifier: doi:10.1007/BF02547187
[7] L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations, Pseudodifferential operators (Oberwolfach, 1986), Lecture Notes in Math., vol. 1256, Springer, Berlin, 1987, pp. 214–280.
Mathematical Reviews (MathSciNet): MR88j:35024
Zentralblatt MATH: 0632.35045
[8] L. V. Kapitanskiĭ, Some generalizations of the Strichartz-Brenner inequality, Algebra i Analiz 1 (1989), no. 3, 127–159.
Mathematical Reviews (MathSciNet): MR90h:46063
Zentralblatt MATH: 0732.35118
[9] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907.
Mathematical Reviews (MathSciNet): MR90f:35162
Zentralblatt MATH: 0671.35066
Digital Object Identifier: doi:10.1002/cpa.3160410704
[10] C. Kenig, G. Ponce, and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527–620.
Mathematical Reviews (MathSciNet): MR94h:35229
Zentralblatt MATH: 0808.35128
Digital Object Identifier: doi:10.1002/cpa.3160460405
[11] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, preprint, 1991.
Mathematical Reviews (MathSciNet): MR1231427
Zentralblatt MATH: 0803.35095
Digital Object Identifier: doi:10.1002/cpa.3160460902
[12] H. Lindblad, Blow-up for solutions of $\square u=\vert u\vert \sp p$ with small initial data, Comm. Partial Differential Equations 15 (1990), no. 6, 757–821.
Mathematical Reviews (MathSciNet): MR91k:35168
Zentralblatt MATH: 0712.35018
Digital Object Identifier: doi:10.1080/03605309908820708
[13] H. Pecher, $L\spp$-Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen. I, Math. Z. 150 (1976), no. 2, 159–183.
Mathematical Reviews (MathSciNet): MR55:8563a
Zentralblatt MATH: 0347.35053
Digital Object Identifier: doi:10.1007/BF01215233
[14] G. Ponce and T. Sideris, Local regularity of nonlinear wave equations in three space dimensions, preprint, 1991.
Mathematical Reviews (MathSciNet): MR1211729
Zentralblatt MATH: 0803.35096
Digital Object Identifier: doi:10.1080/03605309308820925
[15] J. Rauch, Explosion for some semilinear wave equations, J. Differential Equations 74 (1988), no. 1, 29–33.
Mathematical Reviews (MathSciNet): MR89k:35022
Zentralblatt MATH: 0679.35068
Digital Object Identifier: doi:10.1016/0022-0396(88)90016-2
[16] J. Ralston, Gaussian beams and the propagation of singularities, Studies in partial differential equations, MAA Stud. Math., vol. 23, Math. Assoc. America, Washington, DC, 1982, pp. 206–248.
Mathematical Reviews (MathSciNet): MR85c:35052
Zentralblatt MATH: 0533.35062
[17] I. Segal, Space-time decay for solutions of wave equations, Advances in Math. 22 (1976), no. 3, 305–311.
Mathematical Reviews (MathSciNet): MR58:11945
Zentralblatt MATH: 0344.35058
Digital Object Identifier: doi:10.1016/0001-8708(76)90097-9
[18] J. Shatah and Shadi A. Tahvildar-Zadeh, On the Cauchy problem for equivariant wave maps, preprint, 1992.
Mathematical Reviews (MathSciNet): MR1278351
Zentralblatt MATH: 0811.58059
Digital Object Identifier: doi:10.1002/cpa.3160470507
[19] C. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, vol. 105, Cambridge University Press, Cambridge, 1993.
Mathematical Reviews (MathSciNet): MR94c:35178
Zentralblatt MATH: 0783.35001
[20] E. M. Stein, Oscillatory integrals in Fourier analysis, Beijing lectures in harmonic analysis (Beijing, 1984), Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 307–355.
Mathematical Reviews (MathSciNet): MR88g:42022
Zentralblatt MATH: 0618.42006
[21] E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482–492.
Mathematical Reviews (MathSciNet): MR18,575d
Zentralblatt MATH: 0072.32402
Digital Object Identifier: doi:10.2307/1992885
JSTOR: links.jstor.org
[22] R. S. Strichartz, Convolutions with kernels having singularities on a sphere, Trans. Amer. Math. Soc. 148 (1970), 461–471.
Mathematical Reviews (MathSciNet): MR41:876
Zentralblatt MATH: 0199.17502
Digital Object Identifier: doi:10.2307/1995383
[23] R. S. Strichartz, A priori estimates for the wave equation and some applications, J. Functional Analysis 5 (1970), 218–235.
Mathematical Reviews (MathSciNet): MR41:2231
Zentralblatt MATH: 0189.40701
Digital Object Identifier: doi:10.1016/0022-1236(70)90027-3
previous :: next
Duke Mathematical Journal
