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The moving contact line problem for the $2D$ nonlinear shallow water equations

Tatsuo Iguchi, David Lannes

TL;DR

This work analyzes the moving contact line problem for the two-dimensional nonlinear shallow water equations with a fixed partially immersed solid. The authors formulate the problem as a free-boundary exterior-domain system, then fix the domain via a Hanzawa-type diffeomorphism and develop a two-tier framework of Alinhac good unknowns (including a second-order variant) to overcome derivative losses at the free boundary. They establish a priori energy estimates at the quasilinear regularity threshold under irrotational and subcritical initial flow and a transversal contact-line condition, leveraging weak dissipativity and Rellich-type identities to obtain crucial boundary regularity. The results yield well-posedness insights for wave–structure interactions in shallow water, with a rigorous path from linearized theory to the nonlinear a priori bounds and a clear mechanism for controlling the moving contact line. The techniques have potential impact for coastal engineering and marine energy applications by providing a robust mathematical foundation for simulations involving sliding contact lines and free boundaries in shallow-water regimes.

Abstract

We consider the initial value problem for a nonlinear shallow water model in horizontal dimension d = 2 and in the presence of a fixed partially immersed solid body on the water surface. We assume that the bottom of the solid body is the graph of a smooth function and part of it is in contact with the water. As a result, we have a contact line where the solid body, the water, and the air meet. In our setting of the problem, the projection of the contact line on the horizontal plane moves freely due to the motion of the water surface even if the solid body is fixed. This wave-structure interaction problem reduces to an initial boundary value problem for the nonlinear shallow water equations in an exterior domain with a free boundary, which is the projection of the contact line. The objective of this paper is to derive a priori energy estimates locally in time for solutions at the quasilinear regularity threshold under assumptions that the initial flow is irrotational and subcritical and that the initial water surface is transversal to the bottom of the solid body at the contact line. The key ingredients of the proof are the weak dissipativity of the system, the introduction of second order Alinhac good unknowns associated with a regularizing diffeomorphism, and a new type of hidden boundary regularity for the nonlinear shallow water equations. This last point is crucial to control the regularity of the contact line; it is obtained by combining the use of the characteristic fields related to the eigenvalues of the boundary matrix together with Rellich type identities.

The moving contact line problem for the $2D$ nonlinear shallow water equations

TL;DR

This work analyzes the moving contact line problem for the two-dimensional nonlinear shallow water equations with a fixed partially immersed solid. The authors formulate the problem as a free-boundary exterior-domain system, then fix the domain via a Hanzawa-type diffeomorphism and develop a two-tier framework of Alinhac good unknowns (including a second-order variant) to overcome derivative losses at the free boundary. They establish a priori energy estimates at the quasilinear regularity threshold under irrotational and subcritical initial flow and a transversal contact-line condition, leveraging weak dissipativity and Rellich-type identities to obtain crucial boundary regularity. The results yield well-posedness insights for wave–structure interactions in shallow water, with a rigorous path from linearized theory to the nonlinear a priori bounds and a clear mechanism for controlling the moving contact line. The techniques have potential impact for coastal engineering and marine energy applications by providing a robust mathematical foundation for simulations involving sliding contact lines and free boundaries in shallow-water regimes.

Abstract

We consider the initial value problem for a nonlinear shallow water model in horizontal dimension d = 2 and in the presence of a fixed partially immersed solid body on the water surface. We assume that the bottom of the solid body is the graph of a smooth function and part of it is in contact with the water. As a result, we have a contact line where the solid body, the water, and the air meet. In our setting of the problem, the projection of the contact line on the horizontal plane moves freely due to the motion of the water surface even if the solid body is fixed. This wave-structure interaction problem reduces to an initial boundary value problem for the nonlinear shallow water equations in an exterior domain with a free boundary, which is the projection of the contact line. The objective of this paper is to derive a priori energy estimates locally in time for solutions at the quasilinear regularity threshold under assumptions that the initial flow is irrotational and subcritical and that the initial water surface is transversal to the bottom of the solid body at the contact line. The key ingredients of the proof are the weak dissipativity of the system, the introduction of second order Alinhac good unknowns associated with a regularizing diffeomorphism, and a new type of hidden boundary regularity for the nonlinear shallow water equations. This last point is crucial to control the regularity of the contact line; it is obtained by combining the use of the characteristic fields related to the eigenvalues of the boundary matrix together with Rellich type identities.

Paper Structure

This paper contains 56 sections, 19 theorems, 293 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General setting
  3. Problem setting
  4. Coordinate transformation
  5. Main result
  6. Organization of this article
  7. Basic properties of the shallow water model
  8. Conservation of energy
  9. Analysis of the vorticity
  10. An equivalent formulation I
  11. An equivalent formulation II
  12. Fixing the boundary
  13. In the case of the formulation I
  14. In the case of the formulation II
  15. Energy estimate of a linearized problem
  16. ...and 41 more sections

Key Result

Theorem 1

Let $m\geq3$ be an integer and $c_0, M_0, \eta_0, \eta_0^{\rm in}$ positive constants satisfying $\eta_0^{\rm in}<\eta_0<1$. Then, there exist a sufficiently small positive time $T$ and a large constant $C$ such that under Assumption ass:BSB for any regular solution $u=(\zeta,v^{\rm T})^{\rm T},v_{\ we have the following a priori bounds where $w$ is defined by defw.

Figures (1)

  • Figure 1: Waves interacting with a fixed solid body

Theorems & Definitions (49)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Proposition 1
  • proof
  • Remark 3
  • Remark 4
  • Proposition 2
  • Remark 5
  • proof : Proof of Proposition \ref{['prop:BEE']}
  • ...and 39 more