×

Stochastic Camassa-Holm equation with convection type noise. (English) Zbl 1469.60202

In this article, the authors consider a stochastic Camassa-Holm (SCH) equation of the form \[ dy(t) + F(y(t)) dt + Dy(t) \circ dw(t) = 0, \,\, t \geq 0, \,\, F(y) := A(y)y, \,\, y \in H^{1,q}, \] where \(A(v)\) is given by the formula \[ [A(v) f](x) = a(v) \partial_x f(x) + b(v) f(x), \quad x \in \mathbb{R}, \] with the operator \[ D = \xi \partial_x + \eta, \quad \xi \in C_b^4, \eta \in C_b^3, \] and \(w\) is a one-dimensional Wiener process.
In their main result, the authors show that for any \(y_0 \in H^{2,q}\), \(1 < q < \infty\), there exists a unique \(H^{2,q}\)-valued local strong solution of the SCH-equation with \(y(0) = y_0\); see Theorem 2.2.
The essential idea of the proof is to reduce the SCH-equation to a PDE with time-dependent coefficients, and then to a apply the general Kato method, which is elaborated in Section 3.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H25 Random operators and equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q86 PDEs in connection with geophysics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Acquistapace, P.; Terreni, B., An approach to Itô linear equations in Hilbert spaces by approximation of white noise with coloured noise, Stoch. Anal. Appl., 2, 2, 131-186 (1984) · Zbl 0547.60066
[2] Barbu, V.; Röckner, M.; Zhang, D., Stochastic nonlinear Schrödinger equations, Nonlinear Anal., Theory Methods Appl., 136, 168-194 (2016) · Zbl 1336.60118
[3] Barostichi, R.; Himonas, A.; Petronilho, G., Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems, J. Funct. Anal., 270, 330-358 (2016) · Zbl 1331.35299
[4] Bressan, A.; Constantin, A., Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183, 215-239 (2007) · Zbl 1105.76013
[5] Bressan, A.; Constantin, A., Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5, 1, 1-27 (2007) · Zbl 1139.35378
[6] Brzeźniak, Z.; Capiński, M.; Flandoli, F., A convergence result for stochastic partial differential equations, Stochastics, 24, 4, 423-445 (1988) · Zbl 0653.60049
[7] Brzeźniak, Z.; Capiński, M.; Flandoli, F., Stochastic partial differential equations and turbulence, Math. Models Methods Appl. Sci., 1, 41-59 (1991) · Zbl 0741.60058
[8] Brzeźniak, Z.; Manna, U.; Mukherjee, D., Wong-Zakai approximation for the stochastic Landau-Lifshitz-Gilbert equations, J. Differ. Equ., 267, 776-825 (2019) · Zbl 1447.60090
[9] Brzeźniak, Z.; Flandoli, F.; Maurelli, M., Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity, Arch. Ration. Mech. Anal., 221, 1, 107-142 (2016) · Zbl 1342.35231
[10] Brzeźniak, Z.; van Neerven, J.; Veraar, M.; Weis, L., Itô’s formula in UMD Banach spaces and regularity of solutions of the Zakai equation, J. Differ. Equ., 245, 1, 3-58 (2008) · Zbl 1154.60051
[11] Brzeźniak, Z.; Veraar, M., Is the stochastic parabolicity condition dependent on p and q?, Electron. J. Probab., 17, 56 (2012), 24 pp · Zbl 1266.60113
[12] Camassa, R.; Holm, D., An integrable shallow water equation with peaked solutions, Phys. Rev. Lett., 71, 1661-1664 (1993) · Zbl 0972.35521
[13] Chen, S.; Foias, C.; Holm, D.; Olson, E.; Titi, E.; Wynn, S., The Camassa-Holm equations and turbulence, Phys. D: Nonlinear Phenom., 133, 1-4, 49-65 (1999) · Zbl 1194.76069
[14] Chiodaroli, E.; Feireisl, E.; Flandoli, F., Ill posedness for the full Euler system driven by multiplicative white noise · Zbl 1496.35294
[15] Chen, Y.; Gao, H.; Guo, B., Well-posedness for stochastic Camassa-Holm equation, J. Differ. Equ., 253, 2353-2379 (2012) · Zbl 1248.60070
[16] Chen, Y.; Gao, H., Well-posedness and large deviations of the stochastic modified Camassa-Holm equation, Potential Anal., 45, 331-354 (2016) · Zbl 1350.60055
[17] Constantin, A.; Escher, J., Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), XXVI, 303-328 (1998) · Zbl 0918.35005
[18] Cox, S.; Jentzen, A.; Kurniawan, R.; Pušnik, P., On the mild Itô formula in Banach spaces, Discrete Contin. Dyn. Syst., Ser. B, 23, 6, 2217-2243 (2018) · Zbl 1402.60067
[19] Crisan, D.; Holm, D., Wave breaking for the stochastic Camassa-Holm equation, Physica D, 376-377, 138-143 (2018) · Zbl 1398.35305
[20] Dalecky, Yu. L.; Fomin, S. V., Measures and Differential Equations in Infinite-Dimensional Spaces (1991), Kluwer, 337 pp · Zbl 0753.46027
[21] Da Prato, G.; Tubaro, L., Fully nonlinear stochastic partial differential equations, SIAM J. Math. Anal., 27, 1, 40-55 (1996) · Zbl 0853.60052
[22] Doss, H., Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. Henri Poincaré, 13, 2, 99-125 (1977) · Zbl 0359.60087
[23] E, W., Stochastic hydrodynamics, (Current Developments in Mathematics (2001), Intl. Press: Intl. Press Somerville, MA), 109-147
[24] Flandoli, F., Random Perturbation of PDEs and Fluid Dynamic Models, École d’Été de Probabilités de Saint-Flour XL-2010 (2011), Springer · Zbl 1221.35004
[25] Fokas, A.; Fuchssteiner, B., Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4, 47-66 (1981) · Zbl 1194.37114
[26] Goldys, B.; Le, Kim-N.; Tran, T., A finite element approximation for the stochastic Landau-Lifshitz-Gilbert equation, J. Differ. Equ., 260, 2, 937-970 (2016) · Zbl 1364.35449
[27] Holm, D., Variational principles for stochastic fluid dynamics, Proc. R. Soc. A, 471, Article 20140963 pp. (2015) · Zbl 1371.35219
[28] Holm, D., Stochastic parametrization of the Richardson triple, J. Nonlinear Sci., 29, 1, 89-113 (2019) · Zbl 1411.76036
[29] Holm, D.; Ivanov, R., Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys. A, Math. Theor., 43 (2010) · Zbl 1213.37097
[30] Kallenberg, O., Foundations of Modern Probability, Probability and Its Applications (New York) (2002), Springer-Verlag: Springer-Verlag New York · Zbl 0996.60001
[31] Kato, T., Quasi-linear equations of evolution, with applications to partial differential equations, (Everitt, W. N., Spectral Theory and Differential Equations. Spectral Theory and Differential Equations, Lecture Notes in Mathematics, vol. 448 (1975), Springer) · Zbl 0315.35077
[32] Kato, T., Linear evolution equations of “hyperbolic” type, II, J. Math. Soc. Jpn., 25, 4, 648-666 (1973) · Zbl 0262.34048
[33] van Neerven, J. M.A. M.; Veraar, M. C.; Weis, L. W., Stochastic integration in UMD Banach spaces, Ann. Probab., 35, 4, 1438-1478 (2007) · Zbl 1121.60060
[34] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (2012), Springer, 282 pp · Zbl 0516.47023
[35] Stein, E., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (1993), Princeton · Zbl 0821.42001
[36] Sussman, H. J., On the gap between deterministic and stochastic ordinary differential equations, Ann. Probab., 6, 19-41 (1978) · Zbl 0391.60056
[37] Tang, H., On the pathwise solutions to the Camassa-Holm equation with multiplicative noise, SIAM J. Math. Anal., 50, 1, 1322-1366 (2018) · Zbl 1387.35537
[38] Tubaro, L., Some results on stochastic partial differential equations by the stochastic characteristics method, Stoch. Anal. Appl., 6, 2, 217-230 (1988) · Zbl 0647.60070
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.