×

Well-posedness of stochastic KdV-BO equation driven by fractional Brownian motion. (English) Zbl 1335.60103

Summary: The Cauchy problem for the Korteweg-de Vries Benjamin-Ono equation driven by cylindrical fractional Brownian motion is discussed in this paper. Fractional Brownian motion is a family of processes \(B^H\). It is known that the smaller the value of Hurst parameter \(H\) is, the worse of the regularity of fBm is. Using Bourgain restriction method, we obtain the lower bound of the Hurst parameter \(H\) for the driving processes \(B^H\). With \(H > \frac{3}{8}\), we prove local existence results with initial value in classical Sobolev spaces of negative indices, i.e. \(H^s\) with \(s \geqslant - \frac{1}{8}\).

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35R60 PDEs with randomness, stochastic partial differential equations
60G22 Fractional processes, including fractional Brownian motion
35Q53 KdV equations (Korteweg-de Vries equations)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alòs, E.; Mazet, O.; Nualart, D., Stochastic calculus with respect to Gaussian processes, Ann. Probab., 29, 766-801 (2001) · Zbl 1015.60047
[2] Benjamin, T. B., A new kind of solitary waves, J. Fluid Mech., 245, 401-411 (1992) · Zbl 0779.76013
[3] Bourgain, J., Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equation, part II: the KdV equation, Geom. Funct. Anal., 2, 107-156, 209-262 (1993) · Zbl 0787.35098
[4] Caithamer, P., The stochastic wave equation driven by fractional Brownian noise and temporally correlated smooth noise, Stochastics Dyn., 5, 1, 45-64 (2005) · Zbl 1083.60053
[5] Chang, H. Y.; Lien, C.; Sukarto, S.; Raychaudhury, S.; Hill, J.; Tsikis, E. K.; Lonngren, K. E., Propagation of ion-acoustic solitons in a non-quiescent plasma, Plasma Phys. Controlled Fusion, 28, 675-681 (1986)
[6] Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions (1992), Cambridge University Press · Zbl 0761.60052
[7] de Bouard, A.; Debussche, A., On the stochastic Korteweg-de Vries equation, J. Funct. Anal., 154, 215-251 (1998) · Zbl 0912.60074
[8] de Bouard, A.; Debussche, A.; Tsutsumi, Y., White noise driven Korteweg-de Veris equation, J. Funct. Anal., 169, 532-558 (1999) · Zbl 0952.60062
[9] Duncan, T. E.; Jakubowski, J.; Pasik-Duncan, B., Stochastic integration for fractional Brownian motion in a Hilbert space, Stochastics Dyn., 6, 53-75 (2006) · Zbl 1095.60017
[10] Duncan, T. E.; Maslowski, B.; Pasik-Duncan, B., Fractional Brownian motion and stochastic equations in Hilbert spaces, Stochastics Dyn., 2, 225-250 (2002) · Zbl 1040.60054
[11] Erraoui, M.; Nualart, D.; Ouknine, Y., Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet, Stochastics Dyn., 3, 121-139 (2003) · Zbl 1040.60045
[12] Grecksch, W.; Ahn, V. V., A parabolic stochastic differential equation with fractional Brownian motion input, Stat. Probab. Lett., 41, 337-346 (1999) · Zbl 0937.60064
[13] Guo, B.; Huo, Z., The well-posedness of the Korteweg-de Vries-Benjamin-Ono equation, J. Math. Anal. Appl., 295, 444-458 (2004) · Zbl 1057.35054
[14] Herman, R. L., The stochastic, damped Korteweg-de Vries equation, J. Phys. A, 23, 1063-1084 (1990) · Zbl 0706.60068
[15] Hu, Y., Heat equation with fractional white noise potential, Appl. Math. Optim., 43, 221-243 (2001) · Zbl 0993.60065
[16] Hu, Y.; Nualart, D., Stochastic heat equation driven by fractional noise and local time, Probab. Theory Related Fields, 143, 285-328 (2009) · Zbl 1152.60331
[17] Kenig, C. E.; Ponce, G.; Vega, L., A bilinear estimate with application to the Korteweg-de Vries equation, J. Amer. Math. Soc., 9, 573-603 (1996) · Zbl 0848.35114
[18] Kenig, C. E.; Ponce, G.; 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] Kim, J. U., On the stochastic Benjamin-Ono equation, J. Differ. Equ., 228, 2, 737-768 (2006) · Zbl 1107.35105
[20] Linares, F., \(L^2\) Global well-posedness of the initial value problem associated to the Benjamin equation, J. Differ. Equ., 152, 1425-1433 (1999)
[21] Bergh, J.; Löfström, J., Interpolation Spaces (1976), Springer-Verlag · Zbl 0344.46071
[22] Matsuno, Y., Stochastic Benjamin-Ono equation and its application to the dynamics of nonlinear random waves, Phys. Rev. E, 52, 6, 6313-6322 (1996)
[23] Nualart, D., Malliavin Calculus and Related topics (1995), Springer Verlag · Zbl 0837.60050
[24] Scalerandi, M.; Romano, A.; Condat, C. A., Korteweg-de Vries solitons under additive stochastic perturbations, Phys. Rev. E, 58, 709-712 (1998)
[25] Tao, T., Multilinear weighted convolution of \(L^2\) functions, and applications to nonlinear dispersive equation, Amer. J. Math., 123, 839-908 (2001) · Zbl 0998.42005
[26] Tindel, S.; Tudor, C. A.; Viens, F., Stochastic evolution equations with fractional Brownian motion, Probab. Theory Related Fields, 127, 186-204 (2003) · Zbl 1036.60056
[27] Zähle, M., On the link between fractional and stochastic calculus, (Crauel, H.; Gundlach, M., Stochastic Dynamics (1999), Springer Verlag: Springer Verlag Cambridge), 305-325 · Zbl 0947.60060
[28] Wang, G.; Guo, B., Well-posedness of stochastic Korteweg-de Vries-Benjamin Ono equation, Front. Math. China, 5, 1, 161-177 (2010) · Zbl 1186.35251
[29] Wang, G.; Zeng, M.; Guo, B., Stochastic Burgers’ equation driven by fractional Brownian motion, J. Math. Anal. Appl., 371, 210-222 (2010) · Zbl 1197.60063
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.