[Paper Review] Local well-posedness for the great lake equation with transport noise

Dan Crisan, Oana Lang|arXiv (Cornell University)|Mar 6, 2020

Stochastic processes and financial applications23 references4 citations

TL;DR

This paper establishes local well-posedness for the stochastic great lake equation with transport noise on the 2D torus, under a weighted incompressibility condition inspired by shallow water dynamics. By introducing a generalized vorticity-to-stream function operator and using a viscous approximation with compactness arguments, the authors prove pathwise unique, probabilistically strong solutions that preserve initial smoothness in weighted Sobolev spaces.

ABSTRACT

This work is a continuation of the authors' work for the stochastic 2D Euler equation driven by transport type noise. Here we lift the incompressibility constraint. Instead we assume a weighted incompressibility condition. This condition is inspired by a physical model for a fluid in a basin with a free upper surface and a spatially varying bottom topography. Moreover, we assume a different form of the vorticity to stream function operator that generalizes the standard Biot-Savart operator which appears in the Euler equation. These two properties are exhibited in the physical model called the great lake equation. For this reason we refer to the model analysed in this paper as the stochastic great lake equation. The new vorticity to stream function operator generalizes the curl operator and it is shown to have good regularity properties. We also show that the initial smoothness of the solution is preserved. The arguments are based on constructing a family of viscous solutions which is proved to be relatively compact and to converge to a truncated version of the original equation. Finally, we show that the truncation can be removed up to a positive stopping time.

Motivation & Objective

• To extend the stochastic 2D Euler framework to include variable bottom topography and non-constant incompressibility via a weighted constraint.
• To model fluid dynamics in a basin with free surface and spatially varying depth using a stochastic vorticity equation with transport noise.
• To generalize the Biot-Savart operator to a vorticity-to-stream function map that respects the geometry induced by the bottom topography function $b$.
• To establish pathwise unique, probabilistically strong solutions in weighted Sobolev spaces $\mathcal{W}_{b}^{k,2}(\mathbb{T}^2)$, preserving initial regularity.
• To remove truncation in the equation up to a positive stopping time, ensuring the solution remains well-defined locally in time.

Proposed method

• Formulate the stochastic great lake equation using Stratonovich integrals to preserve Kelvin's circulation theorem and energy conservation.
• Introduce a generalized vorticity-to-stream function operator that replaces the standard $\boldsymbol{curl}$ operator and incorporates the depth function $b$.
• Use a viscous approximation scheme to construct a family of solutions $\omega^{\nu_n, R, n}_t$ with controlled regularity and compactness.
• Apply Itô's formula to the $L^2_b$-norm of high-order derivatives to derive a priori estimates, including control of the Itô correction term.
• Employ the Burkholder-Davis-Gundy inequality and Gronwall's lemma to bound the second moment of the $\mathcal{W}_{b}^{k,2}$-norm of the solution.
• Leverage relative compactness in $D([0,T], L^2_b(\mathbb{T}^2))$ and Lemma 20 to pass to the limit and recover the solution in the weighted Sobolev space.

Experimental results

Research questions

• RQ1 Can the stochastic great lake equation with transport noise be shown to have a unique local solution in weighted Sobolev spaces?
• RQ2 How does the generalized vorticity-to-stream function operator behave under stochastic evolution, and does it preserve regularity?
• RQ3 Can the viscous approximation method be adapted to handle the weighted incompressibility condition $\nabla \cdot (b u) = 0$?
• RQ4 What role does the non-degenerate bottom topography $b$ play in preserving the topological structure of the fluid domain?
• RQ5 Is the truncation in the equation removable up to a positive stopping time, ensuring the solution remains valid locally in time?

Key findings

• The stochastic great lake equation admits a pathwise unique and probabilistically strong solution in the weighted Sobolev space $\mathcal{W}_{b}^{k,2}(\mathbb{T}^2)$ for initial data in the same space.
• The initial smoothness of the vorticity is preserved almost surely along the solution trajectory, as shown by uniform a priori bounds on the $\mathcal{W}_{b}^{k,2}$-norm.
• A priori estimates for the fourth moment of the $\mathcal{W}_{b}^{k,2}$-norm are established, with $\mathbb{E}[\sup_{s\in[0,T]}\|\omega_s^{\nu_n,R,n}\|_{b,k,2}^4] \leq \mathcal{C}(R,T)$, uniformly in $n$.
• The generalized vorticity-to-stream function operator is shown to have good regularity properties, ensuring the velocity field remains controlled by the vorticity.
• The truncation in the equation can be removed up to a positive stopping time, confirming the solution exists locally in time without artificial cutoffs.
• The non-degeneracy of $b$ ensures that the geometry of the domain is affected, but the topology remains unchanged, which is crucial for the well-posedness result.