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Traveling wave solutions to a general incompressible Navier-Stokes-Fourier system with free boundary

Jae Ho Choi, Ian Tice

Abstract

We study traveling wave solutions to the free boundary problem associated to a generalized Navier-Stokes Fourier system, which models a viscous, incompressible, heat-conducting fluid. The fluid is assumed to occupy a horizontally infinite strip-like domain with flat rigid bottom and moving upper surface. The fluid is acted upon by gravity as well as external sources of bulk force and boundary stress and an external heat source. Additionally, we allow for temperature-dependent viscosity and capillary coefficients, the latter of which gives rise to Marangoni stresses on the free surface. We develop a small data well-posedness theory in Sobolev spaces that shows that if the sources of force, stress, and heat are small, then there exists a unique solution depending continuously on these data.

Traveling wave solutions to a general incompressible Navier-Stokes-Fourier system with free boundary

Abstract

We study traveling wave solutions to the free boundary problem associated to a generalized Navier-Stokes Fourier system, which models a viscous, incompressible, heat-conducting fluid. The fluid is assumed to occupy a horizontally infinite strip-like domain with flat rigid bottom and moving upper surface. The fluid is acted upon by gravity as well as external sources of bulk force and boundary stress and an external heat source. Additionally, we allow for temperature-dependent viscosity and capillary coefficients, the latter of which gives rise to Marangoni stresses on the free surface. We develop a small data well-posedness theory in Sobolev spaces that shows that if the sources of force, stress, and heat are small, then there exists a unique solution depending continuously on these data.
Paper Structure (20 sections, 21 theorems, 193 equations)

This paper contains 20 sections, 21 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Traveling waves
  4. Reformulation in a fixed coordinate system
  5. Main theorems
  6. Proof techniques and challenges
  7. Solvability of the $\widetilde{\gamma}$-Stokes System \ref{['mothereqn']}
  8. The bilinear form
  9. Introduction of the pressure term
  10. Promotion of regularity of the weak solution
  11. Characterization of solvability of \ref{['odp']}
  12. Fourier Symbols: The Hilbert Space Setting
  13. Instantiation of Fourier Symbols
  14. Estimates of Fourier Symbols via ODE Theory
  15. Solvability of the Overdetermined Problem with Free Boundary Terms
  16. ...and 5 more sections

Key Result

Theorem 1.1

Suppose that $n \geq 2$, $\mathbb{N} \ni s \geq 1+\lfloor n/2 \rfloor$, and $\mathcal{X}^{s}$ is given by Xs. If $(\mu,\kappa,\sigma'(0)) \in \mathbb{R}^{+} \times \mathbb{R}^{+} \times \mathbb{R}$ satisfy where $Q_{1}$ and $Q_{2}$ are bilinear forms defined in Q1 and Q2, then there exist open sets and $\mathcal{O}^{s}\subseteq \mathcal{X}^{s}$ such that the following hold:

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • ...and 27 more