[Paper Review] On the Motion of a Self-Gravitating Incompressible Fluid with Free Boundary and Constant Vorticity: An Appendix

Lydia Bieri, Shuang Miao+2 more|arXiv (Cornell University)|Nov 23, 2015

Advanced Mathematical Physics Problems1 references3 citations

TL;DR

This paper extends a prior analysis of self-gravitating incompressible fluids with free boundaries by incorporating constant vorticity, demonstrating that the lifespan of solutions remains bounded below by $ T \gtrsim \epsilon^{-2} $ for $ \epsilon $-size perturbations of equilibrium states. The authors adapt a nonlinear coordinate transformation and energy method from the irrotational case, showing that constant vorticity introduces only a bounded linear term that can be removed via a further linear transformation, preserving the absence of quadratic nonlinearities in the transformed system.

ABSTRACT

In a recent work [1] the authors studied the dynamics of the interface separating a vacuum from an inviscid incompressible fluid, subject to the self-gravitational force and neglecting surface tension, in two space dimensions. The fluid is additionally assumed to be irrotational, and we proved that for data which are size $ε$ perturbations of an equilibrium state, the lifespan $T$ of solutions satisfies $T \gtrsim ε^{-2}$. The key to the proof is to find a nonlinear transformation of the unknown function and a coordinate change, such that the equation for the new unknown in the new coordinate system has no quadratic nonlinear terms. For the related irrotational gravity water wave equation with constant gravity the analogous transformation was carried out by the last author in [3]. While our approach is inspired by the last author's work [3], the self-gravity in the present problem is a new nonlinearity which needs separate investigation. Upon completing [1] we learned of the work of Ifrim and Tataru [2] where the gravity water wave equation with constant gravity and constant vorticity is studied and a similar estimate on the lifespan of the solution is obtained. In this short note we demonstrate that our transformations in [1] can be easily modified to allow for nonzero constant vorticity, and a similar energy method as in [1] gives an estimate $T\gtrsimε^{-2}$ for the lifespan $T$ of solutions with data which are size $ε$ perturbations of the equilibrium. In particular, the effect of the constant vorticity is an extra linear term with constant coefficient in the transformed equation, which can be further transformed away by a bounded linear transformation. This note serves as an appendix to the aforementioned work of the authors.

Motivation & Objective

• To extend the lifespan estimate $ T \gtrsim \epsilon^{-2} $ for self-gravitating incompressible fluids with free boundaries to the case of non-zero constant vorticity.
• To demonstrate that the nonlinear transformation and energy method developed in prior work [1] can be adapted to handle constant vorticity by showing the vorticity term introduces only a bounded linear perturbation.
• To prove that the transformed system retains no quadratic nonlinearities, ensuring the same energy estimates and lifespan lower bound apply.
• To establish that the Taylor sign condition remains valid under the assumption $ \omega_0^2 < \pi $, which is necessary for local well-posedness.

Proposed method

• Adapt the nonlinear transformation and coordinate change from the irrotational case to include a constant vorticity term $ 2\omega_0 $ in the velocity field.
• Decompose the velocity as $ \mathbf{v} = \mathbf{v}_0 + \mathfrak{v} $, where $ \mathbf{v}_0 = \omega_0(y, -x) $, to decouple the vorticity contribution.
• Apply a complex-variable formulation using the Hilbert transform $ H $, showing that $ (I - \overline{H})(z_t + i\omega_0 z) = 0 $ on the boundary.
• Introduce a new unknown $ \tilde{\delta} $ via a time-dependent transformation $ \tilde{\delta} = e^{-i\omega_0 t} \delta $, which removes the oscillatory phase from the equation.
• Derive a transformed system of the form $ (\partial_t^2 + ia\partial_\alpha - (\pi - \omega_0^2))\tilde{\delta} = \widetilde{\mathcal{M}}_1 $, with no quadratic nonlinearities.
• Verify that all nonlinear terms in the energy estimates remain cubic or higher, ensuring the same $ \epsilon^{-2} $ lifespan bound holds.

Experimental results

Research questions

• RQ1 Can the nonlinear transformation and energy method used in the irrotational case be extended to include constant vorticity in self-gravitating incompressible fluids with free boundaries?
• RQ2 Does the presence of constant vorticity introduce quadratic nonlinearities that would invalidate the $ \epsilon^{-2} $ lifespan estimate?
• RQ3 How does the vorticity term affect the structure of the boundary equations and the applicability of the Hilbert transform framework?
• RQ4 Is the Taylor sign condition still satisfied when $ \omega_0^2 < \pi $, ensuring local well-posedness?
• RQ5 Can the linear vorticity term be removed via a bounded linear transformation without introducing new nonlinearities?

Key findings

• The lifespan of solutions to the self-gravitating incompressible fluid with free boundary and constant vorticity satisfies $ T \gtrsim \epsilon^{-2} $ for $ \epsilon $-size perturbations of equilibrium states.
• The constant vorticity term contributes only a bounded linear term in the transformed equation, which can be removed via a bounded linear transformation.
• The nonlinear structure of the system remains free of quadratic nonlinearities after the transformation, preserving the energy method framework from the irrotational case.
• The energy estimates for the transformed system are identical in form to those in the irrotational case, ensuring the same quantitative lifespan bound.
• All nonlinear contributions in the energy estimates are cubic or higher, confirming that the $ \epsilon^{-2} $ bound is robust under the inclusion of constant vorticity.
• The assumption $ \omega_0^2 < \pi $ is sufficient to ensure the validity of the Taylor sign condition, which is necessary for local well-posedness.