Wave breaking for the Stochastic Camassa-Holm equation

We show that wave breaking occurs with positive probability for the Stochastic Camassa-Holm (SCH) equation. This means that temporal stochasticity in the diffeomorphic flow map for SCH does not prevent the wave breaking process which leads to the formation of peakon solutions. We conjecture that the time-asymptotic solutions of SCH will consist of emergent wave trains of peakons moving along stochastic space-time paths.


The deterministic Camassa-Holm (CH) equation
The deterministic CH equation, derived in [2], is a nonlinear shallow water wave equation for a fluid velocity solution whose profile u(x, t) and its gradient both decay to zero at spatial infinity, |x| → ∞, on the real line R. Namely, where subscripts t (resp. x) denote partial derivatives in time (resp. space). This nonlinear, nonlocal, completely integrable PDE may be written in Hamiltonian form for a momentum density m := u − u xx undergoing coadjoint motion, as [2]  Here, K * m := K(x, y) m(y, t)dy denotes convolution of the momentum density m with the Green's function of the Helmholtz operator L = 1 − ∂ 2 x , so that δh δm = K * m = u with K(x − y) = 1 2 exp(−|x − y|) . (1.5) Alternatively, the CH equation (1.1) may be written in advective form as  are offset in the vertical to show the evolution. The peakon wave train eventually wraps around the periodic domain, thereby allowing the faster peakons which emerge earlier to overtake slower peakons emerging later from behind in collisions that conserve momentum and preserve the peakon shape but cause phase shifts in the positions of the peaks, as discussed in [2].
Remark 1. The peakon-train solutions of CH represent an emergent phenomenon. A wave train of peakons emerges in solving the initial-value problem for the CH equation (1.1) for essentially any spatially confined initial condition. An example of the emergence of a wave train of peakons from a Gaussian initial condition is shown in Figure 1.1.

Remark 2.
By equation (1.5), the momentum density corresponding to the peakon wave train (1.7) in velocity is given by a sum over delta functions in momentum density, representing the singular solution, for an arbitrary smooth function f . Physically, the relationship (1.8) represents the dynamical coalescence of the CH momentum density into particle-like coherent structures (Young measures) which undergo elastic collisions as a result of their nonlinear interactions. Mathematically, the singular solutions of CH are captured by recognizing that the singular solution ansatz This momentum map property explains, for example, why the singular solutions (1.8) form an invariant manifold for any value of M and why their dynamics form a canonical Hamiltonian system, [4].
The complete integrability of the CH equation as a Hamiltonian system follows from its isospectral problem.
Theorem 3 (Isospectral problem for CH [2]). The CH equation in (1.1) follows from the compatibility conditions for the following CH isospectral eigenvalue problem and evolution equation for the real eigenfunction ψ(x, t), with real isospectral parameter, λ.
Remark 4. The complete integrability of the CH equation as a Hamiltonian system and its soliton paradigm explain the emergence of peakons in the CH dynamics. Namely, their emergence reveals the initial condition's soliton (peakon) content.

Steepening Lemma: the mechanism for peakon formation
In the following we will continue working on the entire real line R, although similar results are also available for a periodic domain with only minimal effort. We use the notation u 2 , u 1,2 and u ∞ to denote, respectively, Remark 5 (Local well-posedness of CH). As reviewed in [4], the deterministic CH equation (1.1) is locally well posed on R, for initial conditions in H s with s > 3/2. In particular, with such initial data, CH solutions are C ∞ in time and the Hamiltonian h(m) in (1.4) is bounded for all time, In fact, CH solutions preserve the Hamiltonian in (1.4) given by the u(·, t) 1,2 norm By a standard Sobolev embedding theorem, (1.13) also implies the useful relation that (1.14) The mechanism for the emergent formation of the peakons seen in Figure 1.1 may also be understood as a variant of classical formations of weak solutions in fluid dynamics by showing that initial conditions exist for which the solution of the CH equation (1.1) can develop a vertical slope in its velocity u(t, x), in finite time. The mechanism turns out to be associated with inflection points of negative slope, such as occur on the leading edge of a rightward-propagating, spatially-confined velocity profile. In particular, Lemma 6 (Steepening Lemma [2]). Suppose the initial profile of velocity u(x, 0) has an inflection point at x = x to the right of its maximum, and otherwise it decays to zero in each direction and that u(·, 0) 1,2 < ∞. Moreover we assume that u , where M is the constant defined in (1.14). Then, the negative slope at the inflection point will become vertical in finite time.
Proof. Consider the evolution of the slope at the inflection point t → x(t) that starts at time 0 from an inflection point x = x of u(x, 0) to the right of its maximum so that Define s t := u x (x(t), t), t ≥ 0. From the spatial derivative of the advective form of the CH equation (1.6) one obtains . This, in turn, yields an equation for the evolution of t → s t . Namely, by using u xx (x(t), t) = 0 and (1.14) one finds Lets be the solution of the equation therefore, s t ≤s t for all t > 0 (as long as both are well defined). However, equation (1.16) admits the explicit solutions Remark 7. Suppose the initial condition is anti-symmetric, so the inflection point at u = 0 is fixed and dx/dt = 0, due to the symmetry (u, x) → (−u, −x) admitted by equation (1.1). In this case, the total momentum vanishes, i.e. M = 0, and no matter how small |s(0)| (with s(0) < 0), the verticality s → −∞ develops at x in finite time.
Remark 8. The Steepening Lemma of [2] proves that in one dimension any initial velocity distribution whose spatial profile has an inflection point with negative slope (for example, any antisymmetric smooth initial distribution of velocity on the real line) will develop a vertical slope in finite time. Note that the peakon solution (1.7) has no inflection points, so it is not subject to the steepening lemma.
The Steepening Lemma underlies the mechanism for forming these singular solutions, which are continuous but have discontinuous spatial derivatives. Indeed, the numerical simulations in Figure 1.1 show that the presence of an inflection point of negative slope in any confined initial velocity distribution triggers the steepening lemma as the mechanism for the formation of the peakons. Namely, according to Figure 1.1, the initial (positive) velocity profile "leans" to the right and steepens, then produces a peakon that is taller than the initial profile, so it propagates away to the right, since the peakon moves at a speed equal to its height. This process leaves a profile behind with an inflection point of negative slope; so it repeats, thereby producing a wave train of peakons with the tallest and fastest ones moving rightward in order of height. In fact, Figure 1.1 shows that this recurrent process produces only peakon solutions, as in (1.7). This is a result of the isospectral property of CH as a completely integrable Hamiltonian system, [2]. Namely, the eigenvalues of the initial profile u(x, 0) for the associated CH isospectral problem are equal to the asymptotic speeds of the peakons in the wave train (1.7).
The peakon solutions lie in H 1 and have finite energy. We conclude that solutions with initial conditions in H s with s > 3/2 go to infinity in the H s norm in finite time, but they remain in H 1 and presumably continue to exist in a weak sense for all time in H 1 .

Advective form of the Stochastic Camassa-Holm (SCH) equation
Following [3], we derive the SCH equation by introducing the stochastic Hamiltonian function, The second term generates spatially correlated random displacements, by pairing the momentum density with the Stratonovich noise in (2.1) via a set of time-independent prescribed functions ξ i (x), i = 1, 2, . . . , N , representing the spatial correlations. Thus, the resulting SCH equation is given by where m := u − u xx and the stochastic vector field v, defined by represents random spatially correlated shifts in the velocity, [3,5,6,7]. Thus, the noise introduced in (2.2) and (2.3) represents an additional stochastic perturbation in the momentum transport velocity.

Peakon solutions and isospectrality for the SCH equation
The canonical Hamiltonian equations for the stochastic peakon trajectories and momenta are thus given by Substituting these stochastic canonical Hamiltonian equations for q a (t) and p a (t) into the singular momentum solution for CH in (1.9) recovers the SCH equation (2.2) and the stochastic vector field v in (2.3).
Thus, the SCH equation (2.2) admits peakon wave train solutions whose peaks in velocity follow the stochastic trajectories given by the stochastic vector field v in (2.3) and satisfy stochastic canonical Hamiltonian equations. The corresponding canonical Hamiltonian equations in the absence of noise describe the trajectories and momenta of CH wave trains. For numerical studies of the interactions of stochastic peakon solutions, see [6,7].
Remarkably, a certain amount of the isospectral structure for the deterministic CH equation is preserved by the addition of the stochastic transport perturbation we have introduced in (2.2) and (2.3).
Theorem 10 (Isospectral problem for SCH). The SCH equation in (2.2) follows from the compatibility condition for the deterministic CH isospectral eigenvalue problem (1.11), and a stochastic evolution equation for the real eigenfunction ψ,

6)
and real isospectral parameter, λ, provided dλ = 0 and ξ i (x) = C i + A i e x + B i e −x , for constants A i , B i and C i .
Remark 11. Theorem 10 means that the SCH equation (2.2) with stochastic vector field v (2.3) with ξ i x (x)−ξ i xxx (x) = 0 has the same countably infinite set of conservation laws as for the deterministic CH equation (1.1). However, the SCH Hamiltonian h(m) in (2.1) is not conserved by the SCH equation (2.2), because it depends explicitly on time. Consequently, for those choices of ξ i (x), the SCH equation is equivalent to consistency of the linear equations (2.4) and (2.5). Therefore, SCH is solvable by the isospectral method for each realisation of the stochastic process in (2.6). However, SCH is almost certainly not completely integrable as a Hamiltonian system, even for constant ξ i . See [1] for an example of an integrable stochastic deformation of the CH equation.
The issue now and for the remainder of the paper is to find out whether the wave breaking property which is the mechanism for the creation of peakon wave trains in the deterministic case also survives the introduction of stochasticity.

Wave breaking estimates for SCH
In the following we will assume the conditions under which the stochastic integrals appearing in equation (2.2) for u as well as the equation for u x are well defined and summable. In particular, we assume that the vector fields ξ i are smooth and bounded and that ( ∂ x A i is obtained by formally differentiating A i ). In the following we will assume that there is a local solution of equation (2.2) such that the operators A i , ∂ x A i , i ∈ Z + are well defined. Let us deduce first the equation for the velocity slope, u x . We have the following Lemma: Lemma 12 (Evolution of the velocity slope). Under the above conditions, we have Proof. Expanding out the SCH equation in terms of u and v gives Therefore, applying the smoothing operator K * := (1 − ∂ 2 x ) −1 , given by the convolution with the Green's function K(x, y) in (1.5) for the Helmholtz operator (1 − ∂ 2 x ), to both sides of the previous equation yields in which the derivative ∂ x is understood to act on everything standing to its right. Consequently, we have Then, since we deduce (2.9).
Remark 13. Observe that Hence, the last term in the expression of (2.8) can be controlled by the supremum norm of u.
Just as in the deterministic case, we define next the process t → ν t as the inflection point of u to the right of its maximum so that u xx (ν t , t) = 0, and s t = u x (ν t , t) < 0.
In what follows, we will assume, without proof, that the process t → ν t is a semi-martingale. The argument to show that validity of this property is based on the implicit function theorem. Indeed one can show that ν satisfies the equation provided the equation is well defined. That is, assume we have an inflection point, not an inflection interval; which means we have assumed u xxx (ν t , t) = 0. Using the semimartingale property of ν and the Itô-Ventzell formula (see, e.g., [8]), we deduce that Hence, by (2.9) and (2.12), we find that where the operators B i are given by We will henceforth consider the particular case when the vector fields ξ i are spatially homogeneous, so that B i (u) = 0. (This is also the isospectral case, which we discussed in the previous section.) In this case, just as in the deterministic case, we have u(·, t) 1,2 = u(·, 0) 1,2 , for all x ∈ R. (2.14) and, again, (2.14) implies that M := sup t∈[0,∞) u(·, t) ∞ < ∞. (2.15) This bound arises because the stochastic term vanishes when computing d u(·, t) 2 1,2 . More precisely, the stochastic term is given by the expression whose spatial integral over the real line vanishes for constant ξ i , for the class of solutions u(·, t) which vanish at infinity and whose gradient also vanishes at infinity. Note that the constant M in (2.15) is independent of the realization of the Brownian motions W i , i ∈ Z + . By a standard Sobolev embedding theorem, (1.13) also implies the useful relation that M := sup t∈[0,∞) u(·, t) ∞ < ∞. (2.16) Proposition 14. As in the deterministic case, suppose the initial profile of velocity u(x, 0) has an inflection point at x = x to the right of its maximum, and it decays to zero in each direction; so that u(·, 0) 1,2 < ∞. Consider the expectation of the slope at the inflection point,s t = E [s t ].
Proof. By changing from Stratonovitch to Itô integration, we obtain from (2.13) that