Global well-posedness of the viscous Camassa–Holm equation with gradient noise

Helge Holden; Kenneth H. Karlsen; Peter H. C. Pang

doi:10.3934/dcds.2022163

Article Contents

2023, Volume 43, Issue 2: 568-618. Doi: 10.3934/dcds.2022163 open access

This issue Previous Article Linear stability of the elliptic relative equilibrium with $ (1 + 7) $-gon central configuration in planar $ N $-body problem Next Article A 3D-Schrödinger operator under magnetic steps with semiclassical applications

Global well-posedness of the viscous Camassa–Holm equation with gradient noise

1.
Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
2.
Department of Mathematics, University of Oslo, NO-0316 Oslo, Norway

^*Corresponding author: Helge Holden (helge.holden@ntnu.no)
^*Corresponding author: Helge Holden (helge.holden@ntnu.no)

Received: September 2022

Revised: October 2022

Early access: November 2022

Published: February 2023

This research was partially supported by the Research Council of Norway Toppforsk project Waves and Nonlinear Phenomena (250070). The final author is also partially supported by the Research Council of Norway project INICE (301538)

Abstract Full Text(HTML) Related Papers Cited by

Abstract

We analyse a nonlinear stochastic partial differential equation that corresponds to a viscous shallow water equation (of the Camassa–Holm type) perturbed by a convective, position-dependent noise term. We establish the existence of weak solutions in $ H^m $ ($ m\in\mathbb{N} $) using Galerkin approximations and the stochastic compactness method. We derive a series of a priori estimates that combine a model-specific energy law with non-standard regularity estimates. We make systematic use of a stochastic Gronwall inequality and also stopping time techniques. The proof of convergence to a solution argues via tightness of the laws of the Galerkin solutions, and Skorokhod–Jakubowski a.s. representations of random variables in quasi-Polish spaces. The spatially dependent noise function constitutes a complication throughout the analysis, repeatedly giving rise to nonlinear terms that "balance" the martingale part of the equation against the second-order Stratonovich-to-Itô correction term. Finally, via pathwise uniqueness, we conclude that the constructed solutions are probabilistically strong. The uniqueness proof is based on a finite-dimensional Itô formula and a DiPerna–Lions type regularisation procedure, where the regularisation errors are controlled by first and second order commutators.

Keywords:

Mathematics Subject Classification: Primary: 35R60, 35G25, 60H15; Secondary: 35A01, 35A02.

Citation:

FullText(HTML)

Related Papers

Cited by

References

[1]	S. Albeverio, Z. Brzeźniak and A. Daletskii, Stochastic Camassa–Holm equation with convection type noise, J. Differential Equations, 276 (2021), 404-432. doi: 10.1016/j.jde.2020.12.013.
[2]	D. Alonso-Orán, C. Rohde and H. Tang, A local-in-time theory for singular sdes with applications to fluid models with transport noise, Journal of Nonlinear Science, 31 (2021), 98. doi: 10.1007/s00332-021-09755-9.
[3]	T. M. Bendall, C. J. Cotter and D. D. Holm, Perspectives on the formation of peakons in the stochastic Camassa–Holm equation, Proc. A., 477 (2021), Paper No. 20210224, 18 pp. doi: 10.1098/rspa.2021.0224.
[4]	A. Bensoussan, Stochastic Navier–Stokes equations, Acta Applicandae Mathematica, 38 (1995), 267-304. doi: 10.1007/BF00996149.
[5]	A. Boritchev, Decaying turbulence in the generalised Burgers equation, Arch. Ration. Mech. Anal., 214 (2014), 331-357. doi: 10.1007/s00205-014-0766-5.
[6]	D. Breit and M. Hofmanová, Stochastic Navier–Stokes equations for compressible fluids, Indiana Univ. Math. J., 65 (2016), 1183-1250. doi: 10.1512/iumj.2016.65.5832.
[7]	A. Bressan and A. Constantin, Global conservative solutions of the Camassa–Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.
[8]	A. Bressan and A. Constantin, Global dissipative solutions of the Camassa–Holm equation, Anal. Appl. (Singap.), 5 (2007), 1-27. doi: 10.1142/S0219530507000857.
[9]	Z. Brzeźniak and E. Motyl, Existence of a martingale solution of the stochastic Navier–Stokes equations in unbounded 2D and 3D domains, J. Differential Equations, 254 (2013), 1627-1685. doi: 10.1016/j.jde.2012.10.009.
[10]	Z. Brzeźniak and M. Ondreját, Weak solutions to stochastic wave equations with values in Riemannian manifolds, Comm. Partial Differential Equations, 36 (2011), 1624-1653. doi: 10.1080/03605302.2011.574243.
[11]	Z. Brzeźniak and M. Ondreját, Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces, Ann. Probab., 41 (2013), 1938-1977. doi: 10.1214/11-AOP690.
[12]	Z. Brzeźniak, M. Ondreját and J. Seidler, Invariant measures for stochastic nonlinear beam and wave equations, J. Differential Equations, 260 (2016), 4157-4179. doi: 10.1016/j.jde.2015.11.007.
[13]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.
[14]	P.-L. Chow, Stochastic Partial Differential Equations, Advances in Applied Mathematics. CRC Press, Boca Raton, FL, second edition, 2015.
[15]	Y. Chen, J. Duan and H. Gao, Global well-posedness of the stochastic Camassa–Holm equation, Commun. Math. Sci., 19 (2021), 607-627. doi: 10.4310/CMS.2021.v19.n3.a2.
[16]	Y. Chen and H. Gao, Well-posedness and large deviations of the stochastic modified Camassa–Holm equation, Potential Anal., 45 (2016), 331-354. doi: 10.1007/s11118-016-9548-z.
[17]	Y. Chen, H. Gao and B. Guo, Well-posedness for stochastic Camassa–Holm equation, J. Differential Equations, 253 (2012), 2353-2379. doi: 10.1016/j.jde.2012.06.023.
[18]	Y. Chen and L. Ran, The effect of a noise on the stochastic modified Camassa–Holm equation, J. Math. Phys., 61 (2020), 091504, 16 pp. doi: 10.1063/1.5116129.
[19]	G. M. Coclite, H. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069 (electronic). doi: 10.1137/040616711.
[20]	A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303-328, http://www.numdam.org/item/?id=ASNSP_1998_4_26_2_303_0.
[21]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.
[22]	D. Crisan and D. D. Holm, Wave breaking for the stochastic Camassa–Holm equation, Phys. D, 376/377 (2018), 138-143. doi: 10.1016/j.physd.2018.02.004.
[23]	A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144. doi: 10.1016/j.physd.2011.03.009.
[24]	A. Debussche, M. Hofmanová and J. Vovelle, Degenerate parabolic stochastic partial differential equations: Quasilinear case, Ann. Probab., 44 (2016), 1916-1955. doi: 10.1214/15-AOP1013.
[25]	R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.
[26]	R. Engelking, General Topology, Heldermann Verlag, Berlin, second edition, 1989.
[27]	F. Flandoli and D. Gątarek, Martingale and stationary solutions for stochastic Navier–Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467.
[28]	B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66. doi: 10.1016/0167-2789(81)90004-X.
[29]	N. E. Glatt-Holtz and V. C. Vicol, Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise, Ann. Probab., 42 (2014), 80-145. doi: 10.1214/12-AOP773.
[30]	I. Gyöngy and N. Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Probab. Theory Related Fields, 105 (1996), 143-158. doi: 10.1007/BF01203833.
[31]	H. Holden and X. Raynaud, Global conservative solutions of the Camassa–Holm equation—a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549. doi: 10.1080/03605300601088674.
[32]	Z. Huang, H. Tang and Z. Liu, Random attractor for a stochastic viscous coupled Camassa–Holm equation, J. Inequal. Appl., 2013 (2013), 201, 30 pp. doi: 10.1186/1029-242X-2013-201.
[33]	H. Holden, K. H. Karlsen and P. H. C. Pang, The Hunter–Saxton equation with noise, J. Differential Equations, 270 (2021), 725-786. doi: 10.1016/j.jde.2020.07.031.
[34]	D. D. Holm, Variational principles for stochastic fluid dynamics, Proc. A., 471 (2015), 20140963, 19 pp. doi: 10.1098/rspa.2014.0963.
[35]	A. Jakubowski, The almost sure Skorokhod representation for subsequences in nonmetric spaces, Theory Probab. Appl., 42 (1997), 167-174. doi: 10.4213/tvp1769.
[36]	A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4190-4.
[37]	W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext. Springer, 2015. doi: 10.1007/978-3-319-22354-4.
[38]	W. Lv, P. He and Q. Wang, Well-posedness and blow-up solution for the stochastic Dullin–Gottwald–Holm equation, J. Math. Phys., 60 (2019), 083513, 10 pp. doi: 10.1063/1.5082367.
[39]	M. Ondreját, Stochastic nonlinear wave equations in local Sobolev spaces, Electron. J. Probab., 15 (2010), 1041-1091. doi: 10.1214/EJP.v15-789.
[40]	S. Punshon-Smith and S. Smith, On the Boltzmann equation with stochastic kinetic transport: Global existence of renormalized martingale solutions, Arch. Ration. Mech. Anal., 229 (2018), 627-708. doi: 10.1007/s00205-018-1225-5.
[41]	D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, Berlin, third edition, 1999. doi: 10.1007/978-3-662-06400-9.
[42]	C. Rohde and H. Tang, On the stochastic Dullin–Gottwald–Holm equation: Global existence and wave-breaking phenomena, NoDEA Nonlinear Differential Equations Appl., 28 (2021), 34 pp. doi: 10.1007/s00030-020-00661-9.
[43]	C. Rohde and H. Tang, On a stochastic Camassa–Holm type equation with higher order nonlinearities, Journal of Dynamics and Differential Equations, 33 (2021), 1823-1852. doi: 10.1007/s10884-020-09872-1.
[44]	M. Scheutzow, A stochastic Gronwall lemma, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 16 (2013), 1350019, 4 pp. doi: 10.1142/S0219025713500197.
[45]	J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.
[46]	R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications, Springer, New York, 2005.
[47]	H. Tang, On the pathwise solutions to the Camassa–Holm equation with multiplicative noise, SIAM J. Math. Anal., 50 (2018), 1322-1366. doi: 10.1137/16M1080537.
[48]	H. Tang, Noise effects on dependence on initial data and blow-up for stochastic Euler–Poincaré equations, arXiv: 2002.08719v2, 2020.
[49]	R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia, PA, second edition, 1995. doi: 10.1137/1.9781611970050.
[50]	L. Xie and X. Zhang, Ergodicity of stochastic differential equations with jumps and singular coefficients, Ann. Inst. Henri Poincaré Probab. Stat., 56 (2020), 175-229. doi: 10.1214/19-AIHP959.
[51]	Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.
[52]	L. Zhang, Local and global pathwise solutions for a stochastically perturbed nonlinear dispersive PDE, Stochastic Process. Appl., 130 (2020), 6319-6363. doi: 10.1016/j.spa.2020.05.013.
[53]	L. Zhang, Effect of random noise on solutions to the modified two-component Camassa–Holm system on $\Bbb{T}^d$, arXiv: 2107.09603, 2021.