The Rayleigh-Taylor condition for the evolution of irrotational fluid interfaces

Abstract

For the free boundary dynamics of the two-phase Hele-Shaw and Muskat problems, and also for the irrotational incompressible Euler equation, we prove existence locally in time when the Rayleigh–Taylor condition is initially satisfied for a 2D interface. The result for water waves was first obtained by Wu in a slightly different scenario (vanishing at infinity), but our approach is different because it emphasizes the active scalar character of the system and does not require the presence of gravity.

There are several interesting problems in fluid mechanics regarding the evolution of the interface between two fluids [the Hele-Shaw cell (1, 2) and the Muskat problem (3)] or between a fluid and vacuum or another fluid with zero density, as in models of water waves. In all of them the first important question to be asked is to guarantee local existence, usually within the chain of Sobolev spaces. However, such a result turns out to be false for general initial data. First, Rayleigh (4), Taylor (5), and Saffman and Taylor (2), and later Beale et al. (6), Wu (7, 8), Christodoulou and Lindblad (9), Ambrose (10), Lindblad (11), Ambrose and Masmoudi (12), Coutand and Shkoller (13), Córdoba and Gancedo (14), Shatah and Zeng (15) and Zhang and Zhang (16) gave a condition that must be satisfied to have a solution locally in time; namely, the normal component of the pressure gradient jump at the interface has to have a distinguished sign. This is known as the Rayleigh–Taylor condition.

In refs. 17 and 18 we have obtained local existence in the 2D case: for the Hele-Shaw and Muskat problems, our result addresses the more difficult case when the two fluids have different densities and viscosities; for water waves, we give a different proof of the important theorem of Wu (7) where gravity plays a crucial role in the sign of the Rayleigh–Taylor condition. In our proof, however, we consider the two cases, with or without gravity, and with initial data always satisfying the Rayleigh–Taylor condition.

When that condition is not imposed initially there are several cases where ill-posedness has been proved. Let us point out the works of Ebin (19, 20), Caflisch and Orellana (21), Siegel et al. (22), and Córdoba and Gancedo (14).

We regard these models as transport equations for the density, considered as an active scalar, with a divergence-free velocity field given by Darcy's law (Hele-Shaw and Muskat) or Bernoulli's law (irrotational incompressible Euler equation). It follows that the vorticity is then a delta distribution at the interface multiplied by an amplitude. The dynamics of that interface is governed by the Birkhoff–Rott integral of the amplitude from which we may subtract any component in the tangential direction without modifying its evolution (see ref. 23). We treat the case without surface tension which leads to equality of the pressure on the free boundary, and in both problems it is assumed that the initial interface does not self-intersect. We quantify that property by imposing that the arc-chord quotient be initially strictly positive. It is part of the evolution problem to check carefully that such a positivity prevails for a short time (see ref. 24), as does the Rayleigh–Taylor condition, although depending, in both cases, on the initial data.

1. Equations

The free boundary is given by the discontinuity of the densities and the viscosities (in the case of the free boundary for the irrotational incompressible Euler equation, the viscosity is zero) of the fluids

and μ¹, μ², ρ¹, ρ² (μ¹ ≠ μ² and ρ¹ ≠ ρ²) are constants.

Let the free boundary be parameterized by

where

with the initial data z(α,0) = z ₀(α). We will study also the case of a closed curve:

We consider that the velocity v = v(x ₁,x ₂,t) is irrotational, i.e., ω = ∇ × v = 0, in the interior of each domain Ω^j (j = 1,2). Therefore, the vorticity ω has its support on the curve z(α,t) and it has the form

i.e., ω is a measure defined by

with η(x) a test function.

Then z(α,t) evolves with a velocity field coming from the Biot–Savart law, which can be explicitly computed; it is given by the Birkhoff–Rott integral of the amplitude ϖ along the interface curve:

where PV denotes principal value (see ref. 25).

We have

where v ^j(z(α,t),t) denotes the limit velocity field obtained approaching the boundary in the normal direction inside Ω^j and BR(z,ϖ)(α,t) is given by Eq. 3. It provides us with the velocity field at the interface from which we can subtract any term in the tangential direction without modifying the geometric evolution of the curve

A wise choice of c(α,t) namely:

allows us to accomplish the fact that the length of the tangent vector to z(α,t) only depends on the variable t:

Next, to close the system we apply Darcy's law or, respectively, Bernoulli's law for the case of Euler equations, which leads to an equation relating the parametrization z(α,t) with the amplitude ϖ(α,t).

1.1. Darcy's Law.

Darcy's law is the following momentum equation for the velocity v

where p is the pressure, μ is the dynamic viscosity, κ is the permeability of the medium, ρ is the liquid density, and g is the acceleration due to gravity. Together with the incompressibility condition ∇ · v = 0, Eq. 7 implies the identity

(see also ref. 17). Let us introduce the following notation:

Then, taking the limit in Darcy's law we obtain

which gives us

so that

where $A_{\mu }=\frac{{\mu }_2-{\mu }_1}{{\mu }_2+{\mu }_1}$.

1.2. Bernoulli's Law.

In this section we deduce the evolution equation for the amplitude of vorticity ϖ(α,t) from Bernoulli's law. Let us consider an irrotational flow satisfying the Euler equations

and the incompressibility condition ∇ · v = 0, and let ϕ be such that v(x,t) = ∇ϕ(x,t) for x ≠ z(α,t). Then, we have the expression

From the Biot–Savart law, for x ≠ z(α,t), we get

Let us define

where again ϕ^j(z(α,t),t) denotes the limit obtained approaching the boundary in the normal direction inside Ω^j. It is clear that

therefore,

Now, we observe that

where

Then, using Bernoulli's law inside each domain and taking limits approaching the common boundary, one finds

and since

(see also ref. 18), we get

where we have introduced the following notation:

Then, it is clear that ϕ_t ^j(z(α,t),t) = ∂_t(ϕ^j(z(α,t),t)) − z _t(α,t) · ∇ϕ^j(z(α,t),t), and using Eq. 9 we find

Eqs. 4 and 5 in Eq. 10 give us

where $A_{\rho }=\frac{{\rho }_2-{\rho }_1}{{\rho }_2+{\rho }_1}$. Easily we find the identity:

Then taking a derivative in Eq. 11 and using the identity above we get the desired formula for ϖ, which in the case A _ρ = 1, i.e., ρ₁ = 0, reads as follows

That is, we have obtained the standard water waves model where g is the acceleration due to gravity.

2. Rayleigh–Taylor Condition

Our next step is to find the formula for the difference of the gradients of the pressure in the normal direction:

2.1. Darcy's Law.

Applying Darcy's law and approaching the boundary, we get

2.2. Bernoulli's Law.

We will consider the case A _ρ = 1, which yields −∇p(x,t) = 0 inside Ω¹(t) and therefore ∇p ¹(z(α,t),t) = 0. Next, we define the Lagrangian coordinates for the free boundary with the velocity v ²

We have the same curve with different parameterizations Z(γ,t) = z(α(γ,t),t) and two equations for the velocity of the curve, namely,

and another one given by

Next, we introduce the function (see refs. 6 and 12)

and we observe that the dot product of Eqs. 15 and 14 with the tangential vector gives

Taking a time derivative in Eq. 15 yields

and therefore

3. Local Existence

Our main results consist on the existence of a positive time τ (depending on the initial conditions) for which we have a solution of the periodic Muskat problem (Eqs. 5, 6, and 8) and of the free boundary of the irrotational incompressible Euler equations in vacuum (Eqs. 5, 6, and 12) during the time interval [0,τ] so long as the initial data belong to H ^k(𝕋) for k sufficiently large, F(z ₀)(α,β) < ∞, and

where p ^j denote the pressure in Ω^j and the function F(z), which measures the arc-chord condition (see ref. 24), is defined by

with

Theorem 3.1. Let z ₀(α) ∈ H ^k (ℍ) for k ≥ 3, F(z ₀)(α,β) < ∞, and

Then there exists a time τ > 0 so that there is a solution to Eqs.5, 6, and 8 in C ¹([0,τ]; H ^k(𝕋)) with z(α,0) = z ₀(α).

Theorem 3.2. Let z ₀(α) ∈ H ^k(𝕋), ϖ₀(α) ∈ H ^k−1, and $\phi (\alpha ,0)={\phi }_0(\alpha )\in H^{k-\frac{1}{2}}$ defined in Eq. 16 for k ≥ 4, F(z ₀)(α,β) < ∞, g ≥ 0, and

Then there exists a time τ > 0 so that there is a solution to Eqs.5, 6, and 12 with z(α, t) ∈ C ¹([0,τ]; H ^k(𝕋)), ϖ(α,t) ∈ C ¹([0,τ]; H ^k−1(𝕋)) for z(α,0) = z ₀(α) and ϖ(α,0) = ϖ₀(α).

Remark 3.1. Notice that the parametrization is defined by properties Eqs. 1 or 2. But in the Hele-Shaw and Muskat problems we can show easily that

for a closed curve, making impossible the task of prescribing a sign as in the Rayleigh–Taylor condition.

4. Sketch of the Proof

First, we consider the operator T(u)(α) = 2BR(z,u)(α) · ∂αz(α) associated to a smooth H ³ curve z satisfying the arc-chord condition. T is a smoothing compact operator in Sobolev space whose adjoint T*, acting on u, is described in terms of the Cauchy integral of u along the curve z, and whose real eigenvalues have an absolute value strictly less than one (see ref. 26).

In our proof it is crucial to get control of the norm of the inverse operators (I − ξT)⁻¹, |ξ| ≤ 1. The arguments rely on the boundedness properties of the Hilbert transforms associated to C ^1,α curves, for which we need precise estimates obtained with arguments involving conformal mappings, Hopf maximum principle, and Harnack inequalities (see refs. 27 and 28).

We then provide upper bounds for the amplitude of the vorticity, the Birkhoff–Rott integral, the parametrization of the curve, the arc-chord condition, and the Rayleigh–Taylor condition, namely:

4.1. A Priori Estimates for Theorem 3.1.

and

where the operator Λ is defined by the Fourier transform $\widehat{\Lambda f}(\xi )=\left|\xi \right|\widehat{f}(\xi )$ and σ(α,t) is the difference of the gradients of the pressure in the normal direction. The first inequality in our list above is where we use the precise control of the norm of the inverse operator. The third estimate depends crucially on the positive sign of the Rayleigh–Taylor condition and the chosen well-adapted parametrization. The second and the fourth follow from the standard Sobolev embedding.

We then define the quantity E(t) for the Muskat problem given by

to get

4.2. A Priori Estimates for Theorem 3.2.

In the case of Bernoulli's law our previous comments can be applied in the following setting: For some universal constant q

We define the quantity E(t) in this case by

If we take

and using that

we obtain

with p ∈ ℕ.

4.3. The Evolution of the Rayleigh–Taylor Condition.

It is clear that in the evolution of the quantities E(t) in both contour dynamics problems in Eqs. 19 and 20, everything depends on the sign of σ(α,t). Therefore, to integrate the systems, it is crucial to study the evolution of

Let us then introduce the Rayleigh–Taylor condition in a definition of energy as follows:

Similar arguments (see ref. 29) allow us to accomplish the fact that m′(t) = σ_t(α_t,t) for almost all t. Since we can use Eqs. 13 and 17 and the above estimates, it is possible to control ∥σ_t(α_t,t)∥_L^∞ by means of E _RT(t). It yields finally

for C a universal constant.

4.4. Existence.

To conclude the existence proof we introduce regularized evolution equations (allowing us to take limits) satisfying uniformly the above a priori estimates and for which the local existence follows by standard arguments. Furthermore, in the case of the Hele-Shaw and Muskat problem, to take advantage of the positivity of σ, we have to use a pointwise inequality satisfied by the nonlocal operator Λ (see ref. 29).