[Paper Review] A priori Estimates for the Compressible Euler Equations for a Liquid with Free Surface Boundary and the Incompressible Limit

Hans Lindblad, Chenyun Luo|arXiv (Cornell University)|Nov 16, 2016

Navier-Stokes equation solutions12 references3 citations

TL;DR

This paper establishes uniform a priori energy estimates for the compressible Euler equations with a free surface boundary, generalizing Christodoulou and Lindblad's work to the compressible case without requiring irrotationality. The key contribution is proving that solutions converge to incompressible solutions as sound speed → ∞, with energy bounds uniform in the sound speed, enabling long-time existence for slightly compressible fluids.

ABSTRACT

In this paper, we prove a new type of energy estimates for the compressible Euler's equation with free boundary, with a boundary part and an interior part. These can be thought of as a generalization of the energies in Christodoulou and Lindblad [CL] to the compressible case and do not require the fluid to be irrotational. In addition, we show that our estimates are in fact uniform in the sound speed. As a consequence, we obtain convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of Ebin [Eb] to when you have a free boundary. In the incompressible case our energies reduces to those in [CL] and our proof in particular gives a simplified proof of the estimates in [CL] with improved error estimates. Since for an incompressible irrotational liquid with free surface there are small data global existence results our result leaves open the possibility of long time existence also for slightly compressible liquids with a free surface.

Motivation & Objective

• To establish a priori energy estimates for the compressible Euler equations with a free surface boundary, valid without assuming irrotationality.
• To generalize Christodoulou and Lindblad's energy framework to the compressible case by introducing a new energy structure based on enthalpy and density functions.
• To prove that these energy estimates are uniform in the sound speed κ, enabling the analysis of the incompressible limit.
• To show that solutions of the compressible free-boundary problem converge to solutions of the incompressible Euler equations as κ → ∞.
• To recover and simplify the incompressible energy estimates of Christodoulou and Lindblad as a special case of the new framework.

Proposed method

• Introduce the enthalpy h(ρ) = ∫₁^ρ p′(λ)λ⁻¹ dλ and define e(h) = log ρ(h), transforming the compressible equations into a form resembling incompressible Euler equations.
• Define a new energy functional E_r* that combines interior and boundary contributions, including weighted terms involving √e′(h)D_t^r h for improved control.
• Use the material derivative D_t = ∂_t + v^k ∂_k and derive energy estimates via integration by parts, treating boundary terms carefully using the second fundamental form θ.
• Establish control of boundary terms via the condition −∇_N h ≥ ε > 0, ensuring the physical stability of the free surface.
• Apply elliptic estimates and Sobolev embedding to bound L^∞ norms in terms of energy norms, enabling closure of a priori estimates.
• Prove uniformity in the sound speed κ by showing that the energy estimates and bounds (1.30)–(1.34) remain valid as κ → ∞, under the condition that e_κ(h) → 0 and |e_κ^{(k)}(h)| ≤ c₀√e_κ′(h) for k ≤ 6.

Experimental results

Research questions

• RQ1 Can a priori energy estimates be established for the compressible Euler equations with a free surface boundary without assuming irrotationality?
• RQ2 Can the energy estimates be made uniform in the sound speed κ, enabling the rigorous derivation of the incompressible limit?
• RQ3 Does the new energy structure recover and simplify the incompressible energy estimates of Christodoulou and Lindblad as a limiting case?
• RQ4 Under what conditions does the solution of the compressible free-boundary problem converge to the incompressible solution as κ → ∞?
• RQ5 Can the a priori bounds be closed uniformly in time, suggesting the possibility of long-time existence for slightly compressible fluids?

Key findings

• The paper establishes a priori energy estimates for the compressible Euler equations with free surface boundary, valid for 0 ≤ r ≤ 4, under the assumptions (1.8), (1.30)–(1.34), with the bound |dE_r/dt| ≤ C_r(K, 1/ε, M, c₀, vol D_t, E_{r-1}*) E_r*.
• The energy estimates are uniform in the sound speed κ, as shown in Theorem LABEL:uniform_energybounds and Proposition LABEL:uniform_kk, which allows passage to the incompressible limit.
• As κ → ∞, the compressible solutions (v_κ, h_κ) converge to solutions of the incompressible Euler equations, generalizing Ebin’s result to the free-boundary case.
• In the incompressible limit (e_κ(h) → 0), the energy functional reduces to that of Christodoulou and Lindblad, and the proof provides a simplified and improved version of their estimates.
• The condition −∇_N h ≥ ε > 0 is essential for well-posedness; without it, the problem is ill-posed, as shown by a counterexample in [CL].
• For n ≤ 3, the a priori L^∞ bounds (1.32)–(1.34) can be closed for small time t ≤ T, depending only on initial energy and L^∞ norms, under slightly stronger initial conditions (Proposition LABEL:a_priori_estimate_strong).