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A priori energy estimate with decay in weighted norms for the water-waves problem with contact points

Mei Ming

Abstract

We prove a weighted a priori energy estimate for the two dimensional water-waves problem with contact points in the absence of gravity and surface tension. When the surface graph function and its time derivative have some decay near the contact points, we show that there is corresponding decay for the velocity, the pressure and other quantities in a short time interval. As a result, we have fixed contact points and contact angles. To prove the energy estimate, a conformal mapping is used to transform the equation for the mean curvature into an equivalent equation in a flat strip with some weights. Moreover, the weighted limits at contact points for the velocity, the pressure etc. are tracked and discussed. Our formulation can be adapted to deal with more general cases.

A priori energy estimate with decay in weighted norms for the water-waves problem with contact points

Abstract

We prove a weighted a priori energy estimate for the two dimensional water-waves problem with contact points in the absence of gravity and surface tension. When the surface graph function and its time derivative have some decay near the contact points, we show that there is corresponding decay for the velocity, the pressure and other quantities in a short time interval. As a result, we have fixed contact points and contact angles. To prove the energy estimate, a conformal mapping is used to transform the equation for the mean curvature into an equivalent equation in a flat strip with some weights. Moreover, the weighted limits at contact points for the velocity, the pressure etc. are tracked and discussed. Our formulation can be adapted to deal with more general cases.
Paper Structure (16 sections, 29 theorems, 463 equations)

This paper contains 16 sections, 29 theorems, 463 equations.

Table of Contents

  1. Introduction
  2. Main result of the paper
  3. Organization of the paper
  4. Notations
  5. Preliminaries
  6. Weighted elliptic estimates related to D-N operator
  7. Weighted elliptic estimates on the strip domain
  8. Weighted estimates for D-N operator
  9. Reformulation of the problem and preparations
  10. Geometric formulation in the strip domain
  11. The proper weights
  12. Preparations for the energy estimate
  13. Weighted estimates for the conformal mapping ${\mathcal{T}}$
  14. Weighted estimates for the velocity and other quantities
  15. Modification of the equation
  16. ...and 1 more sections

Key Result

Theorem 1

(Informal version of Theorem energy estimate) Assume that the contact angles ${\omega}, {\omega}_r\in (0, \pi/2)$, the integer $k$ and real ${\gamma}$ satisfy Let a solution to the water-waves system Euler-P condi be given by the free surface graph function $\zeta$ and its time derivative $\partial_t\zeta$ with ${\alpha}^{-1}\partial_{\widetilde{x}}\widetilde{\zeta}\in H^{k+3/2}({\mathbb R})$ and

Theorems & Definitions (66)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 3
  • ...and 56 more