[Paper Review] On the lifespan of three-dimensional gravity water waves with vorticity

Daniel Ginsberg|arXiv (Cornell University)|Dec 4, 2018

Advanced Mathematical Physics Problems27 references4 citations

TL;DR

This paper establishes long-term regularity for three-dimensional gravity water waves with small initial data and non-zero vorticity, under the condition that vorticity vanishes at the free surface. By deriving a Zakharov/Craig-Sulem-type formulation adapted to vortical flows, the author proves a lifespan of order $ T hicksim \ u^{-1} $, where $ \nu $ is the initial vorticity size, recovering the $ \varepsilon^{-N} $ lifespan in the irrotational limit.

ABSTRACT

We prove a long-term regularity result for three-dimensional gravity water waves with small initial data but nonzero initial vorticity. We consider solutions whose vorticity vanishes on the free boundary and use this to derive a system for the evolution of the free boundary which reduces to the Zakharov/Craig-Sulem formulation in the irrotational case. We are able to continue the solution until a time determined by the size of the initial vorticity in such a way that if the vorticity is zero, one recovers a lifespan $T\sim ε^{-N}$ where $N$ can be taken arbitrarily large if the initial data is taken to be arbitrarily smooth.

Motivation & Objective

• To establish long-term regularity for three-dimensional gravity water waves with non-zero initial vorticity.
• To extend the irrotational water wave theory (Zakharov/Craig-Sulem formulation) to the case with non-zero vorticity.
• To derive a lifespan estimate that depends on the size of the initial vorticity, recovering the $ \varepsilon^{-N} $ lifespan in the irrotational limit.
• To ensure the Taylor sign condition holds throughout the evolution, preventing Rayleigh-Taylor instability.

Proposed method

• Derive a system of equations for the free boundary evolution that generalizes the Zakharov/Craig-Sulem formulation to include vorticity.
• Use the assumption that vorticity vanishes on the free boundary to close energy estimates and control nonlinear terms.
• Apply a Nash-Moser-type iteration scheme to prove local well-posedness in Sobolev spaces.
• Employ a weighted energy method with a modified bilinear form $ B[u,\varphi] $ to control the vorticity and boundary dynamics.
• Construct a Green’s function for the elliptic system arising in the boundary layer analysis, proving pointwise decay estimates for the fundamental solution.
• Use the smallness of the boundary curvature and vorticity in $ L^3 $ and $ L^{3/2} $ norms to ensure coercivity and boundedness of the bilinear form.

Experimental results

Research questions

• RQ1 Can the lifespan of three-dimensional gravity water waves be extended beyond the short-time regime when initial vorticity is non-zero but small?
• RQ2 Does the presence of vorticity with vanishing trace on the free surface still allow for a long-time existence result similar to the irrotational case?
• RQ3 Can the Zakharov/Craig-Sulem formulation be generalized to include non-zero vorticity while preserving the structure needed for energy estimates?
• RQ4 What is the dependence of the lifespan on the size of the initial vorticity in the three-dimensional setting?
• RQ5 Is the Taylor sign condition preserved under the evolution of vortical water waves, ensuring stability?

Key findings

• The lifespan of the solution is bounded below by $ T \sim \nu^{-1} $, where $ \nu $ is the size of the initial vorticity, under smallness assumptions on initial data.
• When the initial vorticity is zero, the lifespan recovers the $ \varepsilon^{-N} $ behavior for arbitrarily large $ N $, matching the irrotational case.
• The system reduces to the classical Zakharov/Craig-Sulem formulation in the irrotational limit, ensuring consistency with prior results.
• The bilinear form $ B[u,\varphi] $ is both bounded and coercive under smallness conditions on $ \|h\|_{W^{4,\infty}} $, enabling the use of the Lax-Milgram theorem.
• Pointwise decay estimates for the Green’s function of the elliptic system are established, with $ |G(z,z')| \lesssim |z-z'|^{-1} $ and $ |D_z G(z,z')| \lesssim |z-z'|^{-2} $, crucial for controlling nonlinearities.
• The smallness of $ \|\nabla H\|_{L^{3/2}} $ and $ \|H\|_{L^3} $ on the boundary is shown to be sufficient for coercivity and is preserved up to time $ T \sim \varepsilon_0^{-N} $, aligning with the lifespan estimate.