×

On the stochastic Korteweg-de Vries equation. (English) Zbl 0912.60074

The authors study the following stochastic partial differential equation \[ {\partial u\over \partial t}+ {\partial^3 u\over\partial x^3} +u{\partial u\over \partial x} =f+ \Phi(u) {\partial^2 B\over \partial t\partial x}, \tag{*} \] where \(u\) is a random process defined on \((x,t)\in \mathbb{R}\times \mathbb{R}^+\), \(f\) is a deterministic forcing term, \(\Phi(u)\) is a linear operator depending on \(u\) and \(B\) is a two-parameter Brownian motion on \(\mathbb{R} \times \mathbb{R}^+\). The authors prove the existence and uniqueness of solutions in \(H^1(\mathbb{R})\) in the case of additive noise and the existence of martingale solutions in \(L^2 (\mathbb{R})\) in the case of multiplicative noise for stochastic Korteweg-de Vries equation of the form (*).

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35Q53 KdV equations (Korteweg-de Vries equations)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Akylas, T. R., On the excitation of long nonlinear water waves by a moving pressure distribution, J. Fluid Mech., 141, 455-466 (1984) · Zbl 0551.76018
[2] Akutsu, Y.; Wadati, M., Stochastic Korteweg-de Vries equation with and without damping, J. Phys. Soc. Japan, 53, 3342-3350 (1984)
[3] Artola, M., Sur un théorème d’interpolation, J. Math. Anal. Appl., 41, 148-163 (1973) · Zbl 0266.46026
[4] Bergh, J.; Lofström, J., Interpolation Spaces (1976), Springer Verlag: Springer Verlag Berlin · Zbl 0344.46071
[5] Blasco, O., Interpolation between \(H^1_{B_0}\) and \(L^p_{B_1\)
[6] Bona, J.; Scott, R., Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces, Duke Math. J., 43, 87-99 (1976) · Zbl 0335.35032
[7] Bona, J.; Smith, R., The initial value problem for the Korteweg-de Vries equation, Phil. Trans. Roy. Soc. London A, 278, 555-601 (1975) · Zbl 0306.35027
[8] Bona, J.; Zhang, B. Y., The initial value problem for the forced KdV equation, Proc. Soc. Edinburgh A, 126, 571-598 (1996) · Zbl 0865.35114
[9] Bourgain, J., Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part II, Geom. Funct. Anal., 3, 209-262 (1993) · Zbl 0787.35098
[10] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Application (1992), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0761.60052
[11] Flandoli, F.; Gatarek, D., Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Relat. Fields, 102, 367-391 (1995) · Zbl 0831.60072
[12] Gardner, C. S., Korteweg-de Vries equation and generalizations IV: The Korteweg– de Vries equation as a Hamiltonian system, J. Math. Phys., 12, 1548-1551 (1971) · Zbl 0283.35021
[13] Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes (1981), North-Holland: North-Holland Amsterdam · Zbl 0495.60005
[14] Kato, T., Quasilinear equation of evolution with applications to partial differential equations, Lect. Notes in Math. (1975), Springer: Springer Berlin, p. 27-50
[15] Kato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Stud. Appl. Math., Adv. Math. Suppl. Stud., 8, 93-128 (1993)
[16] Kato, T., On the Korteweg-de Vries equation, Manuscripta Math., 8, 89-99 (1983) · Zbl 0415.35070
[17] Kenig, C. E.; Ponce, G. P.; Vega, L., Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4, 323-347 (1991) · Zbl 0737.35102
[18] Kenig, C. E.; Ponce, G. P.; Vega, L., The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71, 1-21 (1993) · Zbl 0787.35090
[19] Kenig, C. E.; Ponce, G. P.; Vega, L., A bilinear estimate with application to the KdV equation, J. Amer. Math. Soc., 9, 573-604 (1996) · Zbl 0848.35114
[20] Korteweg, D. J.; de Vries, G., On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary waves, Phil. Mag., 39, 422-443 (1985) · JFM 26.0881.02
[21] Laurey, C., The Cauchy problem for a third order nonlinear Schrödinger equation, Nonlinear Anal., 29, 121-158 (1997) · Zbl 0879.35142
[22] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires (1969), Dunod: Dunod Paris · Zbl 0189.40603
[23] Lee, S. J.; Yates, G. T.; Wu, T. Y., Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances, J. Fluid Mech., 199, 569-593 (1989)
[24] Matsuno, Y., Forced Benjamin-Ono equations and related topics, Mathematical Problems in the Theory of Water Waves. Mathematical Problems in the Theory of Water Waves, AMS Contemporary Mathematics, 200 (1996), p. 145-156 · Zbl 0859.35111
[25] Rubio de Francia, J. L.; Ruiz, F. J.; Torrea, J. L., Caldéron-Zygmund theory for operator-valued kernels, Adv. Math., 62, 7-48 (1986) · Zbl 0627.42008
[26] Saut, J. C.; Temam, R., Remarks on the Korteweg-de Vries equation, Israel J. Math., 24, 78-87 (1976) · Zbl 0334.35062
[27] Temam, R., Sur un problème non linéaire, J. Math. Pures Appl., 48, 159-172 (1969) · Zbl 0187.03902
[28] Wadati, W., Stochastic Korteweg-de Vries equation, J. Phys. Soc. Japan, 52, 2642-2648 (1983)
[29] Wu, T. Y., Generation of upstream advancing solitons by moving disturbances, J. Fluid Mech., 184, 75-99 (1987) · Zbl 0644.76017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.