zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the (generalized) Korteweg-de Vries equation. (English) Zbl 0795.35105
The well-posedness of the initial value problem for the Korteweg-de Vries equation $u\sb t+ u\sb{xxx}+ uu\sb x =0$ and its generalized form $u\sb t+ u\sb{xxx}+ a(u) u\sb x=0$ in the classical Sobolev spaces and the regularity of their solutions in $L\sb s\sp p$ spaces are studied. A global smoothing effect of the solutions of these equations is also proved. See also a paper by {\it T. Kato} [Studies in applied mathematics, Adv. Math., Suppl. Stud., Vol. 8, 93-128 (1983; Zbl 0508.00010)].

35Q53KdV-like (Korteweg-de Vries) equations
35B65Smoothness and regularity of solutions of PDE
Full Text: DOI
[1] J. Bergh and J. Löfström, Interpolation Spaces , Springer, 1970.
[2] J. L. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation , Roy. Soc. Lond. Ser A 278 (1975), no. 1287, 555-601. JSTOR: · Zbl 0306.35027 · doi:10.1098/rsta.1975.0035 · http://links.jstor.org/sici?sici=0080-4614%2819750703%29278%3A1287%3C555%3ATIPFTK%3E2.0.CO%3B2-I&origin=euclid
[3] J. Bona and R. Scott, Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces , Duke Math. J. 43 (1976), no. 1, 87-99. · Zbl 0335.35032 · doi:10.1215/S0012-7094-76-04309-X
[4] R. R. Coifman and Y. Meyer, Nonlinear harmonic analysis, operator theory and P.D.E , Beijing lectures in harmonic analysis (Beijing, 1984), Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 3-45. · Zbl 0623.47052
[5] J. Ginibre and Y. Tsutsumi, Uniqueness for the generalized Korteweg-de Vries equations , · Zbl 0702.35224
[6] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations , Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., vol. 448, Springer-Verlag, Berlin, 1975, pp. 25-70. · Zbl 0315.35077
[7] T. Kato, On the Korteweg-de Vries equation , Manuscripta Math. 28 (1979), no. 1-3, 89-99. · Zbl 0415.35070 · doi:10.1007/BF01647967 · eudml:154631
[8] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation , Studies in Applied Math., Advances in Mathematics Supplementary Studies, vol. 8, Academic Press, New York, 1983, pp. 93-128. · Zbl 0549.34001
[9] T. Kato and G. Ponce, On nonstationary flows of viscous and ideal fluids in $L^ p_s(\mathbb R^2)$ , Duke Math. J. 55 (1987), no. 3, 487-499. · Zbl 0649.76011 · doi:10.1215/S0012-7094-87-05526-8
[10] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations , Comm. Pure Appl. Math. 41 (1988), no. 7, 891-907. · Zbl 0671.35066 · doi:10.1002/cpa.3160410704
[11] S. N. Kruzhkov and A. V. Framinskii, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation , Math. U.S.S.R. Sbornik 48 (1984), 93-138. · Zbl 0549.35104 · doi:10.1070/SM1984v048n02ABEH002682
[12] B. Marshall, Mixed norm estimates for the Klein-Gordon equation , Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 638-649. · Zbl 0516.35047
[13] H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation , Math. Z. 185 (1984), no. 2, 261-270. · Zbl 0538.35063 · doi:10.1007/BF01181697 · eudml:173400
[14] J. C. Saut and R. Temam, Remarks on the Korteweg-de Vries equation , Israel J. Math. 24 (1976), no. 1, 78-87. · Zbl 0334.35062 · doi:10.1007/BF02761431
[15] E. M. Stein, Oscillatory Integrals in Fourier Analysis , Beijing Lectures in Harmonic Analysis (Beijing, 1984), Ann. of Math. Stud., vol. 112, Princeton University Press, Princeton, NJ, 1986, pp. 307-355. · Zbl 0618.42006
[16] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces , Princeton University Press, Princeton, N.J., 1971. · Zbl 0232.42007
[17] R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations , Duke Math. J. 44 (1977), no. 3, 705-714. · Zbl 0372.35001 · doi:10.1215/S0012-7094-77-04430-1
[18] R. Temam, Sur un problème non linéaire , J. Math. Pures Appl. (9) 48 (1969), 159-172. · Zbl 0187.03902
[19] P. Tomas, A restriction theorem for the Fourier transform , Bull. Amer. Math. Soc. 81 (1975), 477-478. · Zbl 0298.42011 · doi:10.1090/S0002-9904-1975-13790-6