Effect of random noise on solutions to the modified two-component Camassa-Holm system on $\mathbb{T}^d$

Abstract

This paper studies the high dimensional modified two-component Camassa-Holm (MCH2) system with random perturbation on the torus T d (d ≥ 1). The MCH2 system reduces to the Euler-Poincaré equation without considering the averaged density, and to the two-component Euler-Poincaré system as the potential energy term weaken to L 2 norm in the Lagrangian. First, we establish the local well-posedness of strong pathwise solutions in the sense of Hadamard for the MCH2 system driven by general nonlinear multiplicative noise. Moreover, we prove that the data-to-solution map does not uniformly depend on the initial data, assuming that the noise coefficients can be controlled by the nonlocal terms of the system itself. Second, when the noise coefficients are in the form of polynomial, say c|f | δ f with c = 0, we find that the random noises with sufficiently large intensity parameters δ 1 ≥ 1 2 and δ 2 ≥ 1 have a regularization effect on the t-variable, which improves the local strong pathwise solutions to be global-in-time ones. Note that the global existence problem with small intensity 0 ≤ δ 1 < 1 2 and 0 ≤ δ 2 < 1 remains to be an open problem. As a partial positive answer, we prove that, in the case of δ 1 = δ 2 = 0, the stochastic MCH2 system admits a unique global strong pathwise solution with high probability for sufficiently small initial data. Nevertheless, when d = 1, we show that the solutions will break in finite time for any small initial data satisfying a proper shape condition.

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

“…A typical gradient noise example is Γ x, u, ∂ x u = σ(x)∂ x u, for some function σ, like in (1.8) or (1.1). Now most of the results in the literature concern the "stochastic forcing" case, either via additive (β = β(x)) or multiplicative (β = β(u)) noise, see the works [16,17,18,32,52,38,42,47,48,53]. For gradient noise, we refer to [1] for a local well-posedness result (up to wave-breaking) for (1.6).…”