Global Existence of Weak Martingale Solutions to the Camassa-Holm Equation with Linear Multiplicative Noise

Abstract: In this paper, we consider the global existence and properties of $H<sup>1$ martingale solution to the Camassa-Holm equation with linear multiplicative noise under periodic boundary conditions. The solution is obtained as limit of regular viscous approximate solutions to parabolic SPDEs, which are constructed using the Galerkin approximations ans the stochastic compactness method. The proof of convergence to a solution argues via tightness of the laws of the viscous approximations and Skorokhod-Jakubowski a.s. representations of random variables in quasi-Polish spaces. In particular, by means of the Girsanov-type transform for regular viscous approximations and the convergence of Skorokhod-Jakubowski representations, we are able to establish the one-sided supernorm estimate and space-time higher regularity of the first-order spatial derivative, and large-time behavior of the weak martingale solution in the stochastic framework.

• The paper constructs global weak H¹-martingale solutions for the stochastic CH equation using viscous regularization and Galerkin approximations.
• Key properties such as one-sided supernorm and enhanced large-time decay are preserved or improved under linear multiplicative noise.
• A novel application of the Skorokhod-Jakubowski theorem in quasi-Polish spaces rigorously handles the limit process, ensuring robust energy bounds and convergence.

Global Existence of Weak Martingale Solutions to the Stochastic Camassa-Holm Equation

Introduction and Problem Formulation

This paper establishes global existence and identifies key qualitative properties of weak martingale solutions for the Camassa-Holm (CH) equation on the 1D periodic torus subjected to linear multiplicative noise, formulated as a stochastic partial differential equation (SPDE): $${\rm d}u + (u \partial_x u + \partial_x P[u])\,{\rm d}t = \beta u\,{\rm d}W,$$ where $$P[u] = (1-\partial_x^2)^{-1}(u^2 + \frac{1}{2}(\partial_x u)^2)$$, and $W$ denotes a 1D Wiener process. Prior analyses for deterministic CH equations have demonstrated the existence of global weak solutions and their qualitative behaviors, such as the one-sided supernorm and large-time decay, using viscous approximations and compactness methods. The stochastic CH equation naturally generalizes this model to include random environmental effects relevant for geophysical fluid dynamics.

The main contributions are:

• The construction of global weak $H^1$-martingale solutions for the stochastic CH equation with linear multiplicative noise, with minimal assumptions on the initial data.
• Preservation and, in some aspects, enhancement of key qualitative properties familiar from the deterministic CH equation, such as the one-sided supernorm and large-time decay to zero.
• Rigorous treatment of the limit process—passage from viscous approximations to martingale solutions—using a Skorokhod-Jakubowski representation approach adapted to quasi-Polish (non-metric) spaces to accommodate weak topologies required by compactness.

Methodology

Viscous Approximation and Uniform Estimates

The analysis commences with a viscous regularization of the SPDE: $${\rm d}u_\varepsilon - \varepsilon \partial_x^2 u_\varepsilon\,{\rm d}t + (u_\varepsilon \partial_x u_\varepsilon + \partial_x P[u_\varepsilon])\,{\rm d}t = u_\varepsilon\,{\rm d}W.$$ Galerkin approximations, together with stochastic compactness and energy methods, establish the existence, uniqueness, and strong a priori estimates for regularized solutions. Uniform (in $\varepsilon$) bounds in $L^\infty([0,T];H^1)$ and $L^{2+\alpha}([0,T]\times\mathbb{S})$ for spatial derivatives and nonlinear quantities $q_\varepsilon = \partial_x u_\varepsilon$ are derived.

Skorokhod-Jakubowski Representation and Tightness

The paper exploits the Skorokhod-Jakubowski theorem for quasi-Polish spaces to pass to the limit in the laws of the viscous approximations. Tightness arguments are handled in weak $L^r$-spaces, circumventing non-metrizability obstacles. This yields, along a subsequence, random variables $$P[u] = (1-\partial_x^2)^{-1}(u^2 + \frac{1}{2}(\partial_x u)^2)$$0 converging a.s. to $$P[u] = (1-\partial_x^2)^{-1}(u^2 + \frac{1}{2}(\partial_x u)^2)$$1, where $$P[u] = (1-\partial_x^2)^{-1}(u^2 + \frac{1}{2}(\partial_x u)^2)$$2 and $$P[u] = (1-\partial_x^2)^{-1}(u^2 + \frac{1}{2}(\partial_x u)^2)$$3 almost everywhere.

Passage to the Limit and Identification of Nonlinearities

A critical technical challenge is the identification of the nonlinear limit $$P[u] = (1-\partial_x^2)^{-1}(u^2 + \frac{1}{2}(\partial_x u)^2)$$4 with $$P[u] = (1-\partial_x^2)^{-1}(u^2 + \frac{1}{2}(\partial_x u)^2)$$5. To address this, the study invokes renormalized formulations for suitable nonlinear test functions $$P[u] = (1-\partial_x^2)^{-1}(u^2 + \frac{1}{2}(\partial_x u)^2)$$6, extends DiPerna–Lions-type methods to the stochastic setting, and uses stability of weak convergences and a defect-measure analysis. The positivity of certain defect measures, combined with appropriate Gronwall-type estimates and convexity arguments, ultimately yields strong convergence $$P[u] = (1-\partial_x^2)^{-1}(u^2 + \frac{1}{2}(\partial_x u)^2)$$7 in $$P[u] = (1-\partial_x^2)^{-1}(u^2 + \frac{1}{2}(\partial_x u)^2)$$8 and thereby validates the substitution $$P[u] = (1-\partial_x^2)^{-1}(u^2 + \frac{1}{2}(\partial_x u)^2)$$9 almost everywhere.

Main Results and Key Properties

The central theorem asserts global existence of weak $W$0-martingale solutions for the stochastic CH equation for any initial law on $W$1 with a sufficiently high moment: $W$2 For any such initial distribution, there exists a weak martingale solution $W$3 satisfying:

• Energy Inequality (Stochastic): For $W$4,

$W$5

almost surely.

• One-sided Supernorm Estimate: Almost surely,

$W$6

This illustrates a preservation and even enhancement (due to the noise) of the monotonicity structure on the slope of the solution.

• Space-time Higher Integrability: For any $W$7,

$W$8

• Large-time Decay: For each $W$9, almost surely,

$H^1$0

This quantifies a significant dissipative effect induced by the linear multiplicative noise.

Contrast with Deterministic Theory: Notably, the result is attained without assuming the non-negativity of the initial momentum, i.e., without $H^1$1, unlike many prior results for the deterministic CH equation. Moreover, the presence of multiplicative noise imparts a stronger large-time decay than in the deterministic framework, in line with prior results concerning dissipative stochastic perturbations.

Theoretical and Practical Implications

• Robustness of Admissibility: The stochastic framework preserves core admissibility properties and qualitative energy controls known from deterministic weak solutions, demonstrating that the relevant solution concept is robust under random perturbations typical in modeling physical systems with uncertainty.
• Noise-induced Dissipation and Regularization: Linear multiplicative noise is shown—not only heuristically but also rigorously—to enforce additional decay, facilitating global regularity and taming certain nonlinear instabilities associated with the CH dynamics (such as wave breaking).
• Framework for Further Extensions: The compactness and limit-passage technologies utilized, particularly the Skorokhod-Jakubowski theorem in quasi-Polish spaces, are broadly applicable for other singular SPDEs where weak topology and non-metric structure are intrinsic. The approach supplies a template for non-metric, weak solution theories in stochastic fluid models.

Speculation and Future Directions

Further research could pursue:

• Pathwise Uniqueness and Selection: While the present result secures global existence, uniqueness questions and the selection of physical solutions in the "martingale" (as opposed to "pathwise") sense remain open and may demand additional regularization or entropy-type structural assumptions.
• Extension to Multi-dimensional or Non-periodic Settings: Similar frameworks can be adapted to non-periodic domains, boundary-driven dynamics, or higher dimensions.
• Inclusion of More General Noises: Extending the analysis to non-linear, spatially inhomogeneous, or non-Gaussian noises would be of mathematical and modeling interest.
• Quantitative Rates of Decay and Long-time Behavior: Precise characterization of decay rates, as well as the stochastic stability and ergodicity properties of the system, remain topics for detailed investigation.

Conclusion

This paper rigorously establishes the existence of global weak martingale solutions to the Camassa-Holm equation under linear multiplicative noise and delineates the preservation—and, in some aspects, enhancement—of key energy and regularity properties from the deterministic setting. The methodology, combining viscous regularization, uniform a priori analysis, and stochastic compactness in non-metric settings, sets a comprehensive foundation for the analysis of singular, non-linear SPDEs arising in fluid dynamics and related fields (2607.01645).