## 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 $$u_ t+ u_{xxx}+ uu_ x =0$$ and its generalized form $$u_ t+ u_{xxx}+ a(u) u_ x=0$$ in the classical Sobolev spaces and the regularity of their solutions in $$L_ s^ p$$ spaces are studied. A global smoothing effect of the solutions of these equations is also proved. See also a paper by T. Kato [Studies in applied mathematics, Adv. Math., Suppl. Stud., Vol. 8, 93-128 (1983; Zbl 0508.00010)].

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 35B65 Smoothness and regularity of solutions to PDEs

Zbl 0508.00010
Full Text:

### References:

 [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 [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 [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 [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 [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 [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 [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 [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 [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 [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
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