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Smoothing properties and retarded estimates for some dispersive evolution equations. (English) Zbl 0762.35008
Dispersive partial differential equations of the type \[ \partial_ tu- Lu=f \] with skew adjoint operator \(L\) acting in some Hilbert space, for instance, \(L^ 2(\mathbb{R}^ n)\), are considered. The Cauchy problem with initial data \(u(t=0)=u_ 0\) is formally solved by \[ u(t)=U(t)u_ 0+\int^ t_ 0d\tau U(t-\tau)f(\tau), \] therefore an essential role in the treatment of the problem is played by the properties of the operators \[ u_ 0\to U(t)u_ 0,\quad f\to\int^ t_ 0d\tau U(t-\tau)f(\tau). \] “Smoothing” means that less derivatives occur on \(u_ 0\) and \(f\) than on their images. “Retardation” refers to the specific form of the second operator.
In this paper, the mentioned operators are treated in terms of abstract Banach space theory. The obtained estimates are applied to the Cauchy problem for the generalized Benjamin-Ono equation that corresponds to \(L=i\Phi(-i\partial/\partial x)\), \(f=(\partial/\partial x)v'(u)\), \(\Phi(\xi)=\xi|\xi|^{2\mu}\), \(v(0)=v'(0)=0\).

MSC:
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35Q53 KdV equations (Korteweg-de Vries equations)
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[1] Abdelouhad, L., Bona, J.L., Felland, M., Saut, J.C.: Non-local models for non-linear dispersive waves. Physica D40, 360–392 (1989) · Zbl 0699.35227 · doi:10.1016/0167-2789(89)90050-X
[2] Bergh, J., Löfström, J.: Interpolation spaces. Berlin, Heidelberg, New York: Springer 1976 · Zbl 0344.46071
[3] Brenner, P.: OnL p p’ estimates for the wave equation. Math. Z.145, 251–254 (1975) · Zbl 0321.35052 · doi:10.1007/BF01215290
[4] Brenner, P.: On scattering and everywhere defined scattering operators for non-linear Klein-Gordon equations. J. Diff. Eq.56, 310–344 (1985) · Zbl 0556.35107 · doi:10.1016/0022-0396(85)90083-X
[5] Constantin, P., Saut, J.C.: Local smoothing properties of dispersive equations. J. Am. Math. Soc.1, 413–439 (1988) · Zbl 0667.35061 · doi:10.1090/S0894-0347-1988-0928265-0
[6] Ginibre, J.: Le problème de Cauchy pour les équations de Korteweg-de Vries et de Benjamin-Ono généralisées. Publ. IRMAR, Univ. Rennes (1991)
[7] Scattering theory in the energy space for a class of non-linear Schrödinger equations. J. Math. Pure Appl.64, 363–401 (1985) · Zbl 0535.35069
[8] Ginibre, J., Velo, G.: The global Cauchy problem for the non-linear Klein-Gordon equation. Math. Z.189, 487–505 (1985) · Zbl 0566.35084 · doi:10.1007/BF01168155
[9] Ginibre, J., Velo, G.: Time decay of finite energy solutions of the non-linear Klein-Gordon and Schrödinger equations. Ann. IHP (Phys. Théor.)43, 399–442 (1985) · Zbl 0595.35089
[10] Ginibre, J., Velo, G.: Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation. J. Diff. Eq.93, 150–212 (1991) · Zbl 0770.35063 · doi:10.1016/0022-0396(91)90025-5
[11] Hörmander, L.: The analysis of linear partial differential operators. I. Berlin, Heidelberg, New York: Springer 1983 · Zbl 0521.35001
[12] Kato, T.: Wave operators and similarity for some non-self adjoint operators. Math. Ann.162, 258–279 (1966) · Zbl 0139.31203 · doi:10.1007/BF01360915
[13] Kato T.: On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Studies in Applied Math. Adv. Math. Suppl. Studies18, 93–128 (1983) · Zbl 0549.34001
[14] Kato, T.: On non-linear Schrödinger equations. Ann IHP (Phys. Théor.)46, 113–129 (1987)
[15] Kenig, C.E., Ponce, G., Vega, L.: Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J.40, 33–69 (1991) · Zbl 0738.35022 · doi:10.1512/iumj.1991.40.40003
[16] Kenig, C.E., Ponce, G., Vega, L.: The initial value problem for a class of non-linear dispersive equations. Lecture Notes in Math. vol. 1450, 141–156 (1990) · Zbl 0719.35086 · doi:10.1007/BFb0084903
[17] Kenig, C.E., Ponce, G., Vega, L.: Well posedness of the initial value problem for the Korteweg-de Vries equation. J. Am. Math. Soc.4, 323–347 (1991) · Zbl 0737.35102 · doi:10.1090/S0894-0347-1991-1086966-0
[18] Marshall, B.: Mixed norm estimates for the Klein-Gordon equation. Proc. of Conf. on Harmonic analysis in honor of A. Zygmund, Wadsworth (1981), 638–649
[19] Pecher, H.:L p -Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen. I. Math. Z.150, 159–183 (1976) · Zbl 0347.35053 · doi:10.1007/BF01215233
[20] Pecher, H.: Non-linear small data scattering for the wave and Klein-Gordon equations. Math. Z.185, 245–263 (1984) · Zbl 0538.35063 · doi:10.1007/BF01181697
[21] Segal, I.E.: Space-time decay for solutions of wave equations. Adv. Math.22, 304–311 (1976) · Zbl 0344.35058 · doi:10.1016/0001-8708(76)90097-9
[22] Strichartz, R.S.: Restriction of Fourier transform to quandratic surfaces and decay of solutions of wave equations. Duke Math. J.44, 705–774 (1977) · Zbl 0372.35001 · doi:10.1215/S0012-7094-77-04430-1
[23] Thomas, P.: A restriction theorem for the Fourier transform. Bull. AMS81, 477–478 (1975) · Zbl 0298.42011 · doi:10.1090/S0002-9904-1975-13790-6
[24] Yajima, K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys.110, 415–426 (1987) · Zbl 0638.35036 · doi:10.1007/BF01212420
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