Conservation of energy for the Euler–Korteweg equations

In this article we study the principle of energy conservation for the Euler–Korteweg system. We formulate an Onsager-type sufficient regularity condition for weak solutions of the Euler–Korteweg system to conserve the total energy. The result applies to the system of Quantum Hydrodynamics.

The positive direction of Onsager's conjecture has been settled already in the 1990's by Constantin et al. [9] (after a partial result of Eyink [17]). The method of mollification and estimation of commutator errors was employed to prove that, if a weak solution u of the incompressible Euler system belongs to L 3 ([0, T ], B α,∞ 3 (T 3 )) ∩ C ([0, T ], L 2 (T 3 )) with α > 1 3 , then the energy u L 2 (T 3 ) is conserved in time. The method of proof as well as the observation that Besov spaces provide a suitable environment for this kind of problem were later used by several authors in the context of other systems of fluid dynamics: like inhomogeneous incompressible Euler and compressible Euler [18], incompressible and compressible Navier-Stokes (resp. [14,15,26,31]), incompressible magnetohydrodynamics [8,25], and general systems of first order conservation laws [22]. Onsager's conjecture was recently studied for incompressible Euler equations in bounded domains, cf. [3]. An overview of these results can be found in [12].
In the present paper we adapt the strategy of Constantin et al. [9] and Feireisl et al. [18] to obtain an Onsager-type sufficient condition on the regularity of weak solutions to the Euler-Korteweg equations so that they conserve the total energy. We consider the isothermal Euler-Korteweg system in the from Here ρ ≥ 0 is the scalar density of a fluid, u is its velocity, h = h(ρ) is the energy density and κ = κ(ρ) > 0 is the coefficient of capillarity. We place the assumption on the functions h and κ: where, depending on the actual form of h and κ, the set T can be chosen to be [0, ∞) or (0, ∞). For instance when κ(ρ) = 1 ρ , as for the QHD system below, then T = (0, ∞) and we have to be away from vacuum.
While the analysis of the above system dates back to the 19th century, when the mathematical theory of phase interfaces and capillary effects was introduced, it still attracts much attention. A modern derivation of the system can be found in [16]. Concerning smooth solutions: in [6,7] local-in-time well-posedness and stability of special solutions are analysed, respectively. A relative energy identity is developed in [21], exploiting the variational structure of the system, and is used to show that solutions of (1.1) converge to smooth solutions of the compressible Euler system (before shock formation) in the vanishing capillarity limit κ → 0, see [20].
The situation with weak solutions is much less understood. Most results concern the Quantum Hydrodynamics system, obtained from (1.1) when κ(ρ) = ε 2 0 4ρ , with ε 0 denoting the Planck constant. This takes the form The interesting connection between QHD and the Schrödinger equation is used in [19] to provide conservative weak solutions for the special case of zero pressure, p(ρ) = 0. Existence of weak solutions for a (relatively limited) class of pressure functions is provided in [1,2]. The existence of wild solutions is possible for (1.1), as pointed out in the recent work of Donatelli et al. [13], where the method of "convex integration" is adapted to show non-uniqueness in the class of dissipative global weak solutions. The possibility of both conservative and dissipative solutions raises the issue of studying the Onsager conjecture for the Euler-Korteweg system (1.1). We use Besov spaces B α,∞ p ( ), with 1 ≤ p < ∞, 0 < α < 1 (see Sect. 2.1 for the definition) and prove the following theorem: Then the energy is locally conserved, i.e. (1) Notice that if α ≥ β, then 3α ≥ α + 2β > 1 and so we must have α > 1 3 . (2) If in addition we assume the following conditions on u and ρ lim |ξ |,τ →0 then, as pointed out by Shvydkoy [30], see also Duchon and Robert [15], one can allow for the case α = β = 1 3 . For details see e.g. Proposition 3 in [15]. The short proof of the main theorem is presented in the following section: it is preceded by an outline of Besov spaces and their basic relevant properties, some preliminary material on the structure of the Euler-Korteweg system, followed by he main part of the proof in Sect. 2.3.

Besov spaces
is finite, cf. [4]. In fact, we can replace the semi-norm in (2.1) with the following one Indeed, if ξ * and r * realize the suprema in (2.1) with |ξ * | < r * , then taking |ξ * | < r < r * would contradict the supremality of r * . Therefore neccesarily |ξ * | = r * , thus producing (2.2). We choose to think of the Besov norm in terms of (2.2), as it is more convienient for our purposes.
) be a standard mollification kernel and we denote Note that the function w ε is well-defined on ε = {x ∈ : dist(x, ∂ ) > ε}. The following inequalities will be extensively used in the proof of the main theorem.
where ∇ stands here for the space-time gradient.
Proof Inequality (2.3) follows directly from the definition of the norm in the space Therefore, by virtue of Fubini and (2.3) For the last of the claimed inequalities we consider the convolution ∇u ε = ∇η ε * u as a bounded linear operator T : On the other hand, writing ∇u ε = η ε * ∇u, we can think of T as mapping W 1, p ( ) into L p ( ). It then has unit norm. Therefore, as the Besov space B α,∞ p is an interpolation space of exponent α for L p and W 1, p (cf. [4,Corollary 4.13]), T is bounded as an operator where the last inequality follows from Lemma 2.1.

Preliminaries
System (1.1) can be written in conservative form where S is the Korteweg stress tensor with I denoting the d-dimensional identity matrix and the local pressure defined as It is routine to show that a strong solution (ρ, u) of the above system will satisfy the following local balance of total (kinetic and internal) energy (2.7) Theorem 1.1 gives sufficient conditions for regularity of weak solutions so that they obey the above energy equality in the sense of distributions. To prove the theorem we employ the strategy of [9], which was used in many works in the subject, including [18,22], where variants of the following lemma are an important ingredient. (2.8)

Proof
We observe that by Taylor's theorem we have where the constant C does not depend on the choice of x and t. Similarly (2.10)

Mollification of the last inequality with respect to (s, y) yields, by virtue of Jensen's inequality
Combining (2.9) and (2.11) and using the triangle inequality we deduce the estimate Finally, we observe that

(2.15)
We observe the following identities Thus the left-hand side of Eq. (2.15) can be written as Hence, upon multiplying with u ε , Eq. (2.15) becomes · u ε , and we have used the mollified continuity equation (2.17) Using (2.17) we can write the first two terms of (2.16) as (2.18) Combining Eqs. (2.16) and (2.18) we obtain

(2.19)
We now rewrite the mollified continuity Eq. (2.17) in the form

After multiplying this equation with
and rearranging, we obtain where

Combining Eqs. (2.19) and (2.20) we obtain
(2.21) It follows that to prove the theorem it is sufficient to show that each commutator error term converges to zero in the distributional sense on (0, T ) × T d as ε → 0.

Commutator estimates
Let ϕ ∈ C 1 c ((0, T ) × T d ) and take ε > 0 small enough so that supp ϕ ⊂ (ε, T − ε) × T d . We will show that for each 1 ≤ i ≤ 7 we have The terms R ε 1 and R ε 2 are dealt with in the same way as in [18]. We recall these estimates for the reader's convenience. For R ε 1 we observe that (2.22) The first part of R ε 1 therefore can be estimated by virtue of an integration by parts, Hölder inequality and estimates (2.4) and (2.5) as Note that since we assume Besov regularity of u also in time, in the above we can estimate the L 3 -norm of ∂ t u ε according to (2.5).
For the second part of R ε 1 according to (2.22), we estimate (using integration by parts, Fubini (2.3) and (2.5)) A similar estimation can be carried out for R ε 2 . We write To use the above decomposition to estimate R ε 2 we require that the product ρu belongs to the space B β,∞ 3 To provide this regularity we needed to assume not only that ρ and u are in appropriate Besov spaces, but also belong to L ∞ . Indeed, observe that since Thus the first part of R ε 2 can be estimated as Likewise, for the second part of R ε 2 we get These estimates show that R ε 1 and R ε 2 vanish as ε → 0. To estimate terms R ε 3 and R ε 4 we integrate by parts and apply Lemma 2.3 to get the following and similarly ).
For brevity the above calculations include only the first term coming from (2.22), with the second term easily seen to produce estimates of the same order. Thus the proof of the theorem is complete.