Digital Object Identifier (DOI) 10.1007/s00220-014-1961-9
Mathematical
Physics
An Asymptotic Limit of a Navier–Stokes System
with Capillary Effects
Ansgar Jüngel1, Chi-Kun Lin2, Kung-Chien Wu3
1 Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria. E-mail: juengel@tuwien.ac.at
2 Department of Applied Mathematics and Center of Mathematical Modeling and Scientific Computing, National Chiao Tung University, Hsinchu 30010, Taiwan. E-mail: cklin@math.nctu.edu.tw
3 Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 82444, Taiwan. E-mail: kungchienwu@gmail.com
Received: 6 February 2013 / Accepted: 6 August 2013
Published online: 9 March 2014 – © Springer-Verlag Berlin Heidelberg 2014
Abstract: A combined incompressible and vanishing capillarity limit in the barotropic compressible Navier–Stokes equations for smooth solutions is proved. The equations are considered on the two-dimensional torus with well prepared initial data. The momentum equation contains a rotational term originating from a Coriolis force, a general Korteweg-type tensor modeling capillary effects, and a density-dependent viscosity. The limiting model is the viscous quasi-geostrophic equation for the “rotated” velocity potential. The proof of the singular limit is based on the modulated energy method with a careful choice of the correction terms.
1. Introduction
The aim of this paper is to prove a combined incompressible and vanishing capillar-ity limit for a two-dimensional Navier–Stokes–Korteweg system, leading to the viscous quasi-geostrophic equation. We consider the (dimensionless) mass and momentum equa-tions for the particle densityρ(x, t) and the mean velocity u(x, t) = (u1(x, t), u2(x, t))
of a fluid in the two-dimensional torusT2:
∂tρ + div(ρu) = 0 in T2, t > 0, (1)
∂t(ρu) + div(ρu ⊗ u) + ρu⊥+∇ p(ρ) = div(K + S), (2)
with initial conditions
ρ(·, 0) = ρ0, u(·, 0) = u0 inT2.
Here,ρu⊥describes the Coriolis force, u⊥= (−u2, u1), the function p(ρ) = ργ/γ with
γ > 1 denotes the pressure of an ideal gas obeying Boyle’s law, K is the Korteweg-type tension tensor and S is the viscous stress tensor.
More precisely, the free surface tension tensor is given by div K = κ0ρ∇(σ(ρ)σ(ρ)),
whereκ0> 0, which can be written in conservative form as div K = κ0div S(ρ) − 1 2S (ρ)|∇ρ|2 I − ∇σ (ρ) ⊗ ∇σ (ρ) , (3)
where S(ρ) = ρσ(ρ)2,σ (ρ) is a (nonlinear) function, and I denotes the unit matrix inR2×2. For a general introduction and the physical background of Navier–Stokes– Korteweg systems, we refer to [8,13,23]. In standard Korteweg models,κ(ρ) = σ(ρ)2 defines the capillarity coefficient [13, Formula (1.29)]. In the shallow-water equation, of-tenσ(ρ) = ρ is used such that div K = ρ∇ρ (see, e.g., [5,31]). Bresch and Desjardins [6] employed general functionsσ (ρ) and suitable viscosities allowing for additional en-ergy estimates (also see [24]). Ifσ(ρ) = √ρ, the third-order term can be interpreted as a quantum correction, and system (1) and (2) (without the rotational term) corresponds to the so-called quantum Navier–Stokes model, derived in [9] and analyzed in [23].
The viscous stress tensor is defined by
div S= 2 div(μ(ρ)D(u)), where D(u) = 1
2(∇u + ∇u) and μ(ρ) denotes the density-dependent viscosity. Often,
the viscosity in the Navier–Stokes model is assumed to be constant for the mathematical analysis [15]. Density-dependent viscosities of the formμ(ρ) = ρ were chosen in [5] and were derived, in the context of the quantum Navier–Stokes model, in [9]. The choice μ(ρ) = σ(ρ) allows one to exploit a certain entropy structure of the system [6].
In the special caseσ(ρ) = √ρ and without rotational term, the existence of global (in time) weak solutions to (4) and (5) was shown in [23]. We discuss the existence of local smooth solutions in the Appendix.
Without capillary effects, system (1) and (2) reduces to the viscous shallow-water or viscous Saint-Venant equations, whose inviscid version was introduced in [33]. The viscous model was formally derived from the three-dimensional Navier–Stokes equa-tions with a free moving boundary condition [18]. This derivation was generalized later to varying river topologies [31]. The existence of global weak or strong solutions to the Korteweg-type shallow-water equations was proved in [6,8,19,20,22] under various assumptions on the nonlinear functions. In [8], the authors obtained several existence results of weak solutions under various assumptions concerning the density dependency of the coefficients. The notion of weak solution involves test functions depending on the density; this allows one to circumvent the vacuum problem. Duan et al. [12] showed the existence of local classical solutions to the shallow-water model without capillary effects. For more details and references on the shallow-water system, we refer to the review [4].
The combined incompressible and vanishing capillarity limit studied in this work is based on the scaling t → εt, u → εu, μ(ρ) → εμ(ρ) and on the choice κ0 = ε2α
(0< α, ε < 1), which gives ∂tρε+ div(ρεuε) = 0 in T2, t > 0, (4) ∂t(ρεuε) + div(ρεuε⊗ uε) + 1 ερεu⊥ε + 1 ε2γ∇(ρ γ ε) − 2ε2(α−1)ρε∇(σ(ρε)σ(ρε)) = 2 div(μ(ρε)D(uε)), (5)
with the initial conditions
The conditionα < 1 is needed to control the capillary energy; see the energy identity in Lemma1below.
When lettingε → 0, it holds ρε → 1 and ρεuε → ∇⊥φ = (−∂φ/∂x2, ∂φ/∂x1)
in appropriate function spaces, whereφ solves the viscous quasi-geostrophic equation [32, Chap. 6] (see Sect.2for details)
∂t(φ − φ) + (∇⊥φ · ∇)(φ) = μ(1)2φ in T2, t > 0, (7)
φ(·, 0) = φ0 in T2. (8)
The objective of this paper is to make this limit rigorous. Our proof requires the (local) existence of a smooth solution to (7) and (8), which is shown in the Appendix. For a proof of global weak solutions in the whole spaceR2, we refer to [16, Theorem 1.1].
Several derivations of inviscid quasi-geostrophic equations have been published; see, e.g., [10,14,34]. The reader is also referred to the monograph [30] for a more complete discussion of this model. The viscous equation was derived rigorously for weak solutions from the shallow-water system in [5]. The proof is essentially based on the presence of the additional viscous part div(ρ∇u) and a friction term in the momentum equation. The novelty of the present paper is that these expressions are not needed and that more general expressions can be considered. In particular, we allow for viscous terms of the type div(μ(ρ)D(u)), and no friction is prescribed.
In the literature, singular limits in PDEs arising in fluid mechanics have been studied extensively. The first works on the incompressible limit were obtained by Klainerman and Majda [25] and Ukai [35]. The low Mach number limit of viscous compressible flows was proved by Desjardins and Grenier [11] and by Levermore et al. [26], allowing for dispersive corrections to the stress tensor (third-order terms in the velocity and temperature). Only few works are concerned with compressible rotating fluids. Bresch et al. [7] proved the combined low Mach and low Rossby limit in the compressible Navier–Stokes equations for well-prepared initial data. The same limit for ill-prepared data was shown by Feireisl et al. [16]. Finally, let us mention the work [17] in which the Mach and Rossby numbers are proportional to certain powers of a small parameter and, depending on the powers, its limit leads to the two-dimensional incompressible Navier–Stokes system or to a linear fourth-order equation for the limiting functionφ.
In the following, we describe our main result. In order to simplify the presentation, we assume that the nonlinearities are given by power-law functions:
σ (ρ) = ρs, μ(ρ) = ρm
forρ ≥ 0,
where s > 0 and m > 0. The exponents s and m cannot be chosen freely; we need to suppose that
0< s ≤ 1, m = s +1 2 ≤
γ + 1
2 . (9)
This assumption includes the quantum Navier–Stokes model s = 1/2, m = 1 and the shallow-water model with s = 1, m = 3/2. Furthermore, we assume that the initial data are sufficiently regular (ensuring the local-in-time existence of smooth solutions)
ρ0
ε ∈ Hk(T2), u0ε∈ Hk−1(T2), φ0∈ Hk+1(T2), where k > 2,
and that they are well prepared:
Gε(φε0) → φ0, ε−1(ρε0− 1) → φ0, ρ0 εu0ε → ∇⊥φ0, εα−1∇ ρ0 ε → 0 (10)
in L2(T2) as ε → 0, where ρε0= 1 + εφε0(this definesφε0), Gε(φε) = √ 2 ε sign(φε) h(1 + εφε), ρε= 1 + εφε, (11) and the internal energy h(ρ) is defined by h(ρ) = p(ρ)/ρ = ργ −2and h(1) = h(1) =
0 (see (13) for an explicit expression). Note that the convergenceε−1(ρε0− 1) → φ0in
L2(T2) implies that Gε(φ0ε) → φ0in L1(T2) if ρε0is bounded in L∞(T2) (see (17)). Theorem 1. Let 0< α < 1 and γ > 1. We suppose that (9) holds and that the initial
data satisfy (10). Furthermore, let(ρε, uε) be the classical solution to (4)–(6) and letφ
be the classical solution to (7) and (8), both on the time interval(0, T ). Then, as ε → 0, ρε→ 1 in L∞(0, T ; Lγ(T2)),
ρεuε→ ∇⊥φ in L∞(0, T ; L2γ /(γ +1)(T2)).
Furthermore, if s< 12andγ ≥ 2(1 − s) or if s = 1 and γ ≥ 2,
ρε→ 1 in L∞(0, T ; Lp(T2)),
ρεuε→ ∇⊥φ in L∞(0, T ; Lq(T2)), for all 1≤ p < ∞ and 1 ≤ q < 2.
The proof is based on the modulated energy method, first introduced by Brenier in a kinetic context [2] and later extended to various models, e.g. [1,3,28]. The idea of the method is to estimate, through its time derivative, a suitable modification of the energy by introducing in the energy the solution of the limit equation. We suggest the following form of the modulated energy:
Hε(t) = T2 ρε 2|uε− ∇ ⊥φ|2 + 1 2|Gε(φε) − φ| 2+ 2ε2(α−1)|∇σ (ρ ε)|2 d x + 2 t 0 T2μ(ρε)|D(uε) − D(∇ ⊥φ)|2 d x, (12)
These terms express the differences of the kinetic, internal, and Korteweg energies as well as the viscosity. Differentiating the modulated energy with respect to time and employing the evolution equations, elaborated computations lead to the inequality
Hε(t) ≤ C
t 0
Hε(s)ds + o(1), t > 0,
where o(1) denotes terms vanishing in the limit ε → 0, uniformly in time. The Gronwall lemma then implies the result.
The paper is organized as follows. In Sect.2, we derive the energy identities for the shallow-water system and the quasi-geostrophic equation and give a formal derivation of the latter model from the former one. Theorem1is proved in Sect.3. In the Appendix, we discuss the existence of local smooth solutions to (4) and (5) and give an existence proof for local smooth solutions to (7) and (8).
2. Auxiliary Results
In this section, we derive the energy estimates for (4) and (5) and derive formally the quasi-geostrophic equation (7). Based on the definition h(ρ) = p(ρ)/ρ, h(1) =
h(1) = 0, we can give an explicit formula for this function:
h(ρ) = 1
γ (γ − 1)
ργ− 1 − γ (ρ − 1), ρ ≥ 0. (13)
The energy identity for (4) and (5) is given as follows.
Lemma 1. Let(ρε, uε) be a smooth solution to (4) and (6) on(0, T ). Then the energy
identity
d Eε
dt + Dε= 0, t ∈ (0, T ),
holds, where the energy Eεand energy dissipation Dεare defined by, respectively,
Eε = T2 1 ε2h(ρε) + 1 2ρε|uε| 2+ 2ε2(α−1)|∇σ (ρ ε)|2 d x, Dε = 2 T2μ(ρε)|D(uε)| 2 d x.
Proof. Multiply (4) byε−2h(ρε) −12|uε|2− 2ε2(α−1)σ(ρε)σ(ρε), integrate over T2,
and then integrate by parts: 0= 2 T 1 ε2∂th(ρε) − 1 ε2h (ρ ε)∇ρε· (ρεuε) −1 2|uε| 2∂ tρε+ρεuε· ∇uε· uε + 4ε2(α−1)∇σ (ρε) · ∇∂tσ(ρε) − 2ε2(α−1)div(ρεuε)σ(ρε)σ(ρε) d x.
Multiplying (5) by uεand integrating overT2gives, since u⊥ε · uε= 0, 0= T2 ∂t(ρuε) · uε− ρε(uε⊗ uε) : ∇uε+ 1 ε2ρ γ −1 ε ∇ρε· uε
+ 2ε2(α−1)σ(ρε)σ(ρε) div(ρεuε) − 2μ(ρε)D(uε) : ∇uε
d x,
where “:” means summation over both matrix indices. Observing that h satisfies h(ρε) = ργ −2ε and using the identity D(uε) : ∇uε= |D(uε)|2, the sum of the above two equations becomes d dt T2 1 ε2h(ρε) + 1 2ρε|uε| 2+ 2ε2(α−1)|∇σ (ρ ε)|2 d x + 2 T2μ(ρε)|D(uε)| 2 d x= 0,
which proves the lemma.
Lemma 2. Let(ρε, uε) be a smooth solution to (4) and (6) on(0, T ). Then there exists
C > 0 such that for all 0 < ε < 1,
ρε− 1L∞(0,T ;Lγ(T2)) ≤ Cεmin{1,2/γ } ifγ > 1, (14)
ρε− 1L∞(0,T ;L2(T2)) ≤ Cε if γ ≥ 2. (15)
Proof. Ifγ = 2, h(ρ) = 12(ρ − 1)2, and the result follows immediately from Lemma1. Letγ > 2. We claim that h(ρ) ≥ |ρ −1|γ/(γ (γ −1)) for ρ ≥ 0. Then the result follows again from the energy identity. Indeed, the function f(ρ) = ργ−1−γ (ρ −1)−|ρ −1|γ is convex in(21, ∞) and concave in (0,12). Since the values f (0) = γ − 2 and f (12) = γ /2 − 1 are positive, f ≥ 0 on [0,1
2]. Furthermore, f (1) = f(1) = 0 which implies,
together with the convexity, that f ≥ 0 in [12, ∞), proving the claim. Finally, let γ < 2. By [29, p. 591], h(ρ) ≥ cR|ρ − 1|2forρ ≤ R and h(ρ) ≥ cR|ρ − 1|γ forρ > R, for
some cR > 0 and R > 0. Hence, using Hölder’s inequality and γ < 2,
ρε− 1γLγ(T2)≤ C {ρε≤R} |ρε− 1|2d x γ /2 + {ρε>R} |ρε− 1|γd x ≤ C {ρε≤R} h(ρε)dx γ /2 + C {ρε>R} h(ρε)dx ≤ C(εγ +ε2) ≤ Cεγ,
where here and in the following C > 0 denotes a generic constant not depending on ε. Estimate (15) forγ ≥ 2 follows from
ρε− 12L2(T2)= T2(ρε− 1) 2 d x≤ C T2h(ρε)dx ≤ Cε 2,
which finishes the proof.
We perform the formal limitε → 0 in (4) and (5). For this, we observe that (4) can be written in terms ofφε = (ρε− 1)/ε as follows:
∂tφε+ div(φεuε) +
1
εdiv uε= 0.
We apply the operator div⊥(defined by div⊥(v1, v2) = −∂v1/∂x2+∂v2/∂x1) to (5) and
observe that div⊥(ρεu⊥ε)/ε = div uε/ε + div(φεuε) = −∂tφε, by the above equation.
Then we find that
∂tdiv⊥(ρεuε) + div⊥div(ρεuε⊗ uε) − ∂tφε
= 2ε2(α−1)
div⊥ρε∇(σ(ρε)σ(ρε))+ 2 div⊥div(μ(ρε)D(uε)). (16) By the energy estimate,ρε → 1 (in L∞(0, T ; Lγ(T2))). Assuming that φε → φ and
uε→ ∇⊥φ in suitable function spaces and employing the relations
the formal limit in (16) yields the limit equation (7). The initial condition reads as φ(·, 0) = φ0, whereφ0= lim
ε→0φε(·, 0) in T2. The energy and the energy dissipation of (7) equal E0= 1 2 T2(|∇φ| 2 +φ2)dx, D0= 2μ(1) T2|D(∇ ⊥φ)|2 d x.
Multiplying the limiting equation byφ and using the properties
T2(∇ ⊥φ · ∇)(φ)φdx = 0, T2(φ) 2d x= 2 T2|D(∇ ⊥φ)|2d x,
we find the energy identity of the viscous quasi-geostrophic equation:
d E0
dt + D0= 0, t > 0.
3. Proof of Theorem1
First, we prove the following lemma.
Lemma 3. Let T > 0, γ > 1, and 0 < α < 1. Then lim
ε→0Hε(t) = 0 uniformly in (0, T ),
where Hεis defined in (12).
Proof. Using the definitions of the energy and energy dissipation as well as the relation 1 2Gε(φε) 2= ε−2h(ρ ε), we write Hε(t) = (Eε+ E)(t) + t 0 (Dε+ D)(s)ds +1 2 T2(ρε− 1)|∇ ⊥φ|2d x − T2(Gε(φε) − φε)φdx − T2ρεuε· ∇ ⊥φdx − T2φεφdx + 2 t 0 T2(μ(ρε) − μ(1))|D(∇ ⊥φ)|2 d xds − 4 t 0 T2μ(ρε)D(uε) : D(∇ ⊥φ)dxds = I1+· · · + I8.
The aim is to estimate d Hε/dt. To this end, we treat the integrals Ij or their
deriva-tives term by term. By the energy estimates, dtd(I1+ I2) = 0. The integral I3cancels
with a contribution originating from I5; see below. The estimate of I4, . . . , I8(or their
derivatives) is performed in several steps.
The key point is the estimate of the modulated potential energy I4. We show by
elementary estimations that I4= o(1) as ε → 0. The estimate of the modulated kinetic
energy I5is new although parts of the estimates resemble those in [28]. In the estimations
of I6, I7, and I8, some terms cancel with those coming from I5. These estimates are also
Step 1: estimate of I4. L’Hôpital’s rule shows that forγ > 1, lim z→0 h(1 + z) z2 = 1 2, limz→0 1 z h(1 + z) z2 − 1 2 = γ − 2 6 .
Therefore, there exists a nonnegative function f , defined on[0, ∞), such that h(1+z) =
1 2z
2f(z) for z ≥ 0, and a function g, defined on [0, ∞), such that f (z) − 1 = zg(z) for z≥ 0. Furthermore, the inequalities f (z) ≥ f (0) = 1 and |g(z)| ≤ C(1+z(γ −3)+) hold,
where z+= max{0, z}. Finally, we claim that f (z) = 2h(1+z)/z2≥ 2(1+z)γ −2/(γ (γ − 1)) for z ≥ 0 and γ ≥ 4. Indeed, the function w(z) = h(1+z)−z2(1+z)γ −2/(γ (γ −1))
is convex in[0, ∞) and w(0) = w(0) = 0, which implies that w(z) ≥ 0 in [0, ∞), proving the claim. With these preparations, we can estimate the difference Gε(φε) − φε
appearing in I4: |Gε(φε) − φε| = sign(φε) √ 2 ε h(1 + εφε) − |φε| = |φε| f(εφε) − 1 = |φ√ε| | f (εφε) − 1| f(εφε) + 1 = |φε√| |εφε| |g(εφε)| f(εφε) + 1 .
In view of the bounds for f and g as well as the relationεφε= ρε− 1, we infer that
|Gε(φε) − φε| ≤ Cε|ρε− 1|2 1 +ρ (γ −3)+
ε
√
f(εφε) + 1. (17)
This bound allows us to estimate I4. Indeed, if 1< γ < 4, by (14),
I4(t) ≤ C
εφL∞(0,T ;L∞(T2))ρε− 12
L∞(0,T ;L1(T2))≤ Cε2 min{1,2/γ }−1= o(1) uniformly in(0, T ). Here and in the following, the constant C > 0 depends on φ and its derivatives but not onε. If γ ≥ 4, we have, using the upper bound of f (z) for γ ≥ 4, (17), and 1 +εφε = ρε, |Gε(φε) − φε| ≤Cε|ρε− 1|2 1 +ρ γ −3 ε Cρε(γ −2)/2+ 1 ≤ C ε|ρε− 1| 2 1 +ρε(γ −3)−(γ −2)/2.
We employ estimates (14) and (15) and Hölder’s inequality to conclude that
I4(t) ≤ CφL∞(0,T ;L∞(T2))ε−1ρε− 1L∞(0,T ;L2(T2))ρε− 1L∞(0,T ;Lγ(T2))
×1 +ρε(γ −4)/2L∞(0,T ;Lγ(T2))
≤ Cε2/γφ
Step 2: estimate of d I5/dt. Inserting the momentum Eq. (5) and integrating by parts, it follows that d I5 dt = − T2∂t(ρεuε) · ∇ ⊥φdx − T2ρεuε· ∇ ⊥∂ tφdx = − T2ρε(uε⊗ uε) : ∇∇ ⊥φdx + 1 ε T2ρεu ⊥ ε · ∇⊥φdx + 1 ε2γ T2∇ρ γ ε · ∇⊥φdxd − 2ε2(α−1) T2ρε∇ σ(ρε)σ(ρε)· ∇⊥φdx + 2 T2μ(ρε)D(uε) : ∇∇ ⊥φdx − T2ρεuε· ∇ ⊥∂ tφdx = J1+· · · + J6.
We treat the integrals J1, . . . , J6term by term. The integral J2can be written as J2=1
ε
T2ρεuε· ∇φdx. The third integral vanishes since div∇⊥= 0:
J3= − 1 ε2γ T2ρ γ ε div(∇⊥φ)dx = 0.
Using the identity (3) and div∇⊥= 0, we compute
J4= ε2(α−1) T2 S(ρε) −12S(ρε)|∇ρε|2 div(∇⊥φ) − (∇σ(ρε) ⊗ ∇σ(ρε)) : ∇∇⊥φ d x ≤ C Hε.
Integration by parts and using div∇⊥= 0 again yields
J5= − T2μ(ρε)uε· (∇ ⊥φ + ∇ div(∇⊥φ))dx − T2μ (ρ ε)(∇ρε⊗ uε+ uε⊗ ∇ρε) : ∇∇⊥φdx = − T2μ(ρε)uε· ∇ ⊥φdx − 2 T2 μ(ρε) √ρεσ(ρε) ∇σ(ρε) ⊗ (√ρεuε) + (√ρεuε) ⊗ ∇σ (ρε): ∇∇⊥φdx.
The assumptions on μ and σ (see9) yieldμ(ρε)/(√ρεσ(ρε)) = ρmε−s−1/2. Hence, applying the Cauchy–Schwarz inequality, the last integral is bounded from above by
We conclude that
J5≤ −
T2μ(ρε)uε· ∇
⊥φdx + o(1).
The integral J6remains unchanged. Finally, we estimate J1. To this end, we add and
substract the expression∇⊥φ such that J1= K1+· · · + K4, where K1= − T2ρε(uε− ∇ ⊥φ) ⊗ (u ε− ∇⊥φ) : ∇∇⊥φdx, K2= − T2ρε∇ ⊥φ ⊗ u ε: ∇∇⊥φdx, K3= − T2ρε uε⊗ ∇⊥φ : ∇∇⊥φdx, K4= T2ρε∇ ⊥φ ⊗ ∇⊥φ : ∇∇⊥φdx.
The first integral can be bounded by the modulated energy:
K1≤ C T2ρε|uε− ∇ ⊥φ|2d x ≤ C H ε. A reformulation yields K2= − T2ρεuε· (∇⊥φ · ∇)∇⊥φd x.
We employ the continuity Eq. (4) to find
K3= −1 2 T2ρεuε· ∇|∇ ⊥φ|2 d xd= 1 2 T2div(ρεuε)|∇ ⊥φ|2 d x = −1 2 T2∂t(ρε− 1)|∇ ⊥φ|2 d x = −1 2 d dt T2(ρε− 1)|∇ ⊥φ|2 d x +1 2 T2(ρε− 1)∂t|∇ ⊥φ|2 d x = −d I3 dt + o(1).
Finally, using again div∇⊥= 0,
K4= − T2ρε (∇⊥φ · ∇)∇⊥φ· ∇⊥φdx = − T2(ρε− 1) (∇⊥φ · ∇)∇⊥φ· ∇⊥φdx − T2 (∇⊥φ · ∇)∇⊥φ· ∇⊥φdx = − T2(ρε− 1) (∇⊥φ · ∇)∇⊥φ· ∇⊥φdx −1 2 T2∇ ⊥φ · ∇(|∇⊥φ|2)dx = − T2(ρε− 1) (∇⊥φ · ∇)∇⊥φ· ∇⊥φdx +1 2 T2div(∇ ⊥φ)|∇⊥φ|2 d x = o(1).
In the last step, we have employed estimate (14) forρε− 1. Summarizing the estimates for K1, . . . , K4, we have shown that
J1≤ C Hε−d I3 dt − T2 (∇⊥φ · ∇)∇⊥φ· (ρεuε)dx + o(1).
Then, summarizing the estimates for J1, . . . , J6, we obtain d I5 dt ≤ C Hε− d I3 dt + 1 ε T2ρεuε· ∇φdx − T2 (∂t +∇⊥φ · ∇)∇⊥φ + μ(1)∇⊥φ · (ρεuε)dx − T2 μ(ρε) − μ(1)ρεuε· ∇⊥φdx + o(1).
The last integral can be estimated by employing the assumptions onμ and Hölder’s inequality: T2 μ(ρε) − μ(1)ρε √ρε √ρ εuε· ∇⊥φdx ≤ Cρεm−1/2− ρ1ε/2L2(T2)√ρεuεL2(T2). We claim that the first factor on the right-hand side is of order o(1). To prove this statement, we consider first12 < m < 1:
ρm−1/2 ε − ρ1ε/22L2(T2)≤ T2ρ 2m−1 ε |ρε− 1|2(1−m)d x ≤ ρε2mLγ−1(T2)ρε− 1 2(1−m) Lp(T2),
where p = 2γ (1 − m)/(γ − 2m + 1). The inequality p ≤ γ is equivalent to γ ≥ 1. Note that the Hölder inequality can be applied since we supposed that 2m− 1 ≤ γ ; see (9). Second, let 1< m ≤ 2 (the case m = 1 being trivial). We compute
ρm−1/2 ε − ρε1/22L2(T2)≤ T2ρε|ρε− 1| 2(m−1) d x≤ ρεLγ(T2)ρε− 12(m−1) Lq(T2), where q = 2γ (m − 1)/(γ − 1), and q ≤ γ if and only if m ≤ (γ + 1)/2. Finally, if
2≤ m ≤ (γ + 1)/2, we find that ρm−1/2 ε − ρε1/22L2(T2)≤ C T2ρε(1 + ρ m−2 ε )2|ρε− 1|2d x ≤ C(1 + ρε2mLγ−3(T2))ρε− 1 2 Lr(T2),
with r = 2γ /(γ − 2m + 3) satisfying r ≤ γ if and only if m ≤ (γ + 1)/2. We conclude that T2 μ(ρε) − μ(1)ρε √ρε √ρ εuε· ∇⊥φdx ≤ Cρε− 1βLγ(T2)
for someβ > 0, and together with (14), this shows that the integral is of order o(1). Therefore, d I5 dt ≤ C Hε− d I3 dt + 1 ε T2ρεuε· ∇φdx − T2 (∂t+∇⊥φ · ∇)∇⊥φ + μ(1)∇⊥φ · (ρεuε)dx + o(1). (18)
Step 3: estimate of d I6/dt. Employing (4) and (7), we can write
d I6 dt = − T2∂tφεφdx − T2φε∂tφdx = 1ε T2 div(ρεuε)φdx − T2 (∂t+∇⊥φ · ∇)(φ) − μ(1)2φ φεd x = −1 ε T2ρε uε· ∇φdx + T2 (∂t+∇⊥φ · ∇)(∇⊥φ) − μ(1)∇⊥φ · ∇⊥φεd x. (19)
We observe that the first integral on the right-hand side cancels with the corresponding integral in (18). To deal with the second integral, we employ again the momentum Eq. (5). We write
1
γ∇ργε = (γ − 1)∇h(ρε) + ∇(ρε− 1) = (γ − 1)∇h(ρε) + ε∇φε. Then, because of(u⊥ε)⊥= −uε, (5) is equivalent to
∇⊥φε= ρεuε− εFε⊥, where Fε = ∂t(ρεuε) + div(ρεuε⊗ uε) +γ − 1 ε2 ∇h(ρε) − 2 div(μ(ρε)D(uε)) − ε2(α−1) ∇S(ρ ε) −1 2∇(S (ρε)|ρε|2) − div∇σ (ρ ε) ⊗ ∇σ (ρε).
Replacing∇⊥φεin the second integral in (19) by the above expression gives
T2 (∂t +∇⊥φ · ∇)(∇⊥φ) − μ(1)∇⊥φ · ∇⊥φεd x = T2 (∂t+∇⊥φ · ∇)(∇⊥φ) − μ(1)∇⊥φ · (ρεuε− εFε⊥)dx.
We claim that the integral containing Fε⊥is bounded in an appropriate space. Indeed, letψ be a smooth (vector-valued) test function. The first term of Fεis written in weak form as follows: T 0 T2∂t(ρεuε) · ψdxds = − T 0 T2ρεuε· ∂tψdxds + T2(ρεuε)(t) · ψ(t)dx − T2ρ 0 εu0ε· ψ(0)dx.
These integrals are bounded ifρεuε is bounded in L∞(0, T ; L1(T2)). This is the case, since mass conservation and the energy estimate show that
T2|ρεuε|dx ≤ 1 2 T2ρεd x + 1 2 T2ρε|uε| 2 d x
is uniformly bounded in(0, T ). An integration by parts gives
T 0 T2div(ρε uε⊗ uε) · ψdxds = − T 0 T2ρε uε⊗ uε: ∇ψdxds,
and this integral is uniformly bounded, by the energy estimate. Furthermore, again integrating by parts, T 0 T2 γ − 1 ε2 ∇h(ρε) − 2 div(μ(ρε)D(uε)) · ψdxds = − T 0 T2 γ − 1 ε2 h(ρε)I − 2μ(ρε)D(uε) : ∇ψdxds,
which is uniformly bounded since we can estimate
T 0 T2|μ(ρε)D(uε)|dxds ≤ 1 2 T 0 T2μ(ρε)dxds + 1 2 T 0 T2μ(ρε)|D(uε)| 2 d xds
andμ(ρε) ≤ C(1 + ργε). Also the remaining terms are bounded since ε2(α−1) T 0 T2 ∇(S(ρε) − S(1)) −1 2∇(S (ρε)|∇ρε|2) − div(∇σ(ρε) ⊗ ∇σ(ρε)) · ψdxds = −ε2(α−1) T 0 T2 (S(ρε) − S(1)) div ψ +12S(ρε)|∇ρε|2divψ − (∇σ (ρε) ⊗ ∇σ(ρε)) : ∇ψd xds.
Using the Hölder continuity of S(z) = (s/2)z2s, z ≥ 0, the first summand can be
estimated by C|ρε−1|min{1,2s}. We infer that the corresponding integral is of order o(1). We formulate the second summand as
1 2ε 2(α−1)(2s − 1) t 0 T2|∇σ(ρε)| 2divψdxds.
In view of the energy estimate, this integral as well as the third summand are uniformly bounded. This shows that
T2 (∂t+∇⊥φ · ∇)(∇⊥φ) − μ(1)∇⊥φ · ∇⊥φεd x = T2 (∂t+∇⊥φ · ∇)(∇⊥φ) − μ(1)∇⊥φ · (ρεuε)dx + o(1),
and consequently, (19) becomes d I6 dt = − 1 ε T2ρε uε· ∇φdx + T2 (∂t+∇⊥φ · ∇)(∇⊥φ) − μ(1)∇⊥φ · (ρεuε)dx + o(1). Step 4: estimate of d I7/dt. The function μ satisfies |μ(z)−μ(1)| = |zm−1| ≤ |z −1|m
if m ≤ 1 and |μ(z) − μ(1)| ≤ C(1 + zm−1)|z − 1| if m > 1, for z ≥ 0. Therefore, if
m≤ 1, taking into account (14),
d I7 dt ≤ 2ρε− 1 m L∞(0,T ;Lγ(T2))D(∇⊥φ) 2 L∞(0,T ;L2γ /(γ −m)(T2)) ≤ Cεm min{1,2/γ }.
Moreover, if 1< m ≤ (γ + 1)/2, using Hölder’s inequality,
d I7 dt ≤ C 1 +ρεL∞(0,T ;L(m−1)γ /(γ −1)(T2)) ρε− 1L∞(0,T ;Lγ(T2)) ≤ Cεmin{1,2/γ }.
The norm ofρε is uniformly bounded since (m − 1)γ /(γ − 1) ≤ γ is equivalent to
m≤ γ .
Step 5: estimate of d I8/dt. Integration by parts yields d I8 dt = T2μ (ρε)∇ρε⊗ uε : ∇∇⊥φdx + 2 T2μ(ρε)uε· ∇ ⊥φdx = T2µ μ(ρε) √ρεσ(ρε)∇σ(ρε) ⊗ ( √ρ εuε) : ∇∇⊥φdx + 2 T2(μ(ρε) − μ(1)ρε)uε· ∇ ⊥φdx + 2μ(1) T2ρε uε· ∇⊥φdx.
By definition ofμ and σ (see9), it follows that
d I8 dt ≤ C∇σ (ρε)L2(T2) √ρ εuεL2(T2)+ Cρεm−1/2− ρ1ε/2L2(T2)√ρεuεL2(T2) + 2μ(1) T2ρε uε· ∇⊥φdx.
Because of the energy estimate, the first summand is of order o(1). The second summand has been estimated in Step 2, and it has been found that it is also of order o(1). This shows that d I8 dt ≤ 2μ(1) T2ρεuε· ∇ ⊥ψdx + o(1).
Step 6: conclusion. Adding the estimates for d I4/dt, . . . , d I8/dt, most of the integrals
cancel, and we end up with
d Hε
dt ≤ C Hε+
d I4 dt + o(1).
Integrating over(0, t) gives
Hε(t) ≤ Hε(0) + C
t 0
Hε(s)ds + I4(t) − I4(0) + o(1).
By Step 1, I4(t) = o(1). Furthermore, I4(0) = o(1) by assumption. It holds that Hε(0) = o(1) since ρ0 ε(u0ε− ∇⊥φ0)L2(T2)≤ ρ0 εu0ε− ∇⊥φ0L2(T2)+(1 − ρ0 ε)∇⊥φ0L2(T2) ≤ ρ0 εu0ε− ∇⊥φ0L2(T2) +1 − ρε0L2(T2)∇⊥φ0L∞(T2) = o(1)
and since the initial data are well prepared. Then the Gronwall lemma implies that
Hε(t) = o(1) finishing the proof.
We are now in the position to prove Theorem1which is a consequence of Lemma3. We observe that by (14),ρε→ 1 in L∞(0, T ; Lγ(T2)) and, using the Hölder inequality and 2γ /(γ + 1) < γ , ρεuε− ∇⊥φL∞(0,T ;L2γ /(γ +1)(T2)) ≤ √ρεL∞(0,T ;L2γ(T2))√ρε(uε− ∇⊥φ)L∞(0,T ;L2(T2)) +ρε− 1L∞(0,T ;L2γ /(γ +1)(T2))∇⊥φL∞(0,T ;L∞(T2)) ≤ C√ρε(uε− ∇⊥φ)L∞(0,T ;L2(T2)) +Cρε− 1L∞(0,T ;Lγ(T2)). (20) We conclude thatρεuε → ∇⊥φ in L∞(0, T ; L2γ /(γ +1)(T2)).
Next, letγ ≥ 2(1 − s) and 0 < s < 1/2. Because of assumption (9), i.e.γ ≥ 2s, we have 2γ /(γ + 2(1 − s)) ≤ γ , and hence,
ρε→ 1 in L∞(0, T ; L2γ /(γ +2(1−s))(T2))
asε → 0. Furthermore, since α < 1, ∇σ(ρε) → 0 in L∞(0, T ; L2(T2)) as ε → 0 and thus, by Hölder’s inequality,
∇(ρε− 1)L∞(0,T ;L2γ /(γ +2(1−s))(T2))
= σ(ρε)−1∇σ(ρε)L∞(0,T ;L2γ /(γ +2(1−s))(T2))
≤ ρε1L−s∞(0,T ;Lγ(T2))∇σ (ρε)L∞(0,T ;L2(T2)) → 0. (21) We infer that ρε → 1 in L∞(0, T ; W1,2γ /(γ +2(1−s))(T2)). Because of the continu-ous embedding W1,2γ /(γ +2(1−s))(T2) → Lγ /(1−s)(T2), this implies that ρε → 1 in
L∞(0, T ; Lγ /(1−s)(T2)). Since 2γ /(γ + 2(1 − s)2) ≤ γ /(1 − s), this gives ρε → 1
in L∞(0, T ; L2γ /(γ +2(1−s)2)(T2)). Applying the same procedure as in (21) again, we obtain
∇(ρε− 1)L∞(0,T ;L2γ /(γ +2(1−s)2)(T2))
Hence,ρε→ 1 strongly in L∞(0, T ; W1,2γ /(γ +2(1−s)2)(T2)) and in L∞(0, T ; Lγ /(1−s)2 (T2)). Repeating this argument, we conclude that ρε → 1 in L∞(0, T ; Lp(T2)) for all
p< ∞.
For the momentum, we obtain for p≥ 1
ρεuε− ∇⊥φL∞(0,T ;L2 p/(p+1)(T2))
≤ √ρεL∞(0,T ;L2 p(T2))√ρε(uε− ∇⊥φ)L∞(0,T ;L2(T2)) +ρε− 1L∞(0,T ;L2 p/(p+1)(T2))∇⊥φL∞(0,T ;L∞(T2))
≤ C√ρε(uε− ∇⊥φ)L∞(0,T ;L2(T2)) +Cρε− 1L∞(0,T ;Lp(T2)).
This shows thatρεuε→ ∇⊥φ in L∞(0, T ; Lq(T2)) for all q < 2.
Finally, letγ ≥ 2 and s = 1. Then ρε→ 1 in L∞(0, T ; H1(T2)) and, by the contin-uous embedding H1(T2) → Lp(T2) for all p < ∞, also ρε → 1 in L∞(0, T ; Lp(T2)) for all p< ∞. The theorem is proved.
Acknowledgements. A. Jüngel acknowledges partial support from the Austrian Science Fund (FWF), grants P22108, P24304, and I395 and from the National Science Council of Taiwan (NSC) through the Tsungming Tu Award. C.-K. Lin was supported by the NSC, grant NSC101-2115-M-009-008-MY2. K.-C. Wu acknowledges partial support from the Tsz-Tza Foundation (Taiwan), National Science Council under grant 102-2115-M-017-004-MY2 (Taiwan) and ERC grant MATKIT (European Union). He thanks Clément Mouhot for his kind invitation to visit the University of Cambridge during the academic years 2011–2013. Part of this work was written during the stay of the A. Jüngel at the Center of Mathematical Modeling and Scientific Computing, National Chiao Tung University (Hsinchu, Taiwan), and at the Institute of Mathematics, Academia Sinica (Taipei, Taiwan); he thanks Chi-Kun Lin and Tai-Ping Liu for their kind hospitality.
A. Local Existence of Smooth Solutions
The local existence of smooth solutions to the Navier–Stokes–Korteweg system (4) and (5) can be shown similarly as in [27]. We only sketch the proof since it is highly technical and does not involve new ideas. First, we rewrite (4) and (5), settingρ = ρε, u = uε, andε = 1. Taking the divergence of (5) and replacing div∂t(ρu) by (4), which has been
differentiated with respect to time, we obtain ∂2
t tρ −
1
γργ + 2ρσ(ρ)22ρ = − div div(ρu ⊗ u) − div(ρu⊥) + 2 div div(μ(ρ)D(u)) + F[ρ],
where F[ρ] = 2 div(ρ∇(σ(ρ)σ(ρ)))−2ρσ(ρ)22ρ involves only three derivatives. This formulation allows one to treat the momentum equation as a nonlinear fourth-order wave equation for which existence and regularity results can be applied. In order to derive some regularity for the velocity, Li and Marcati [27] assumed that curl u = 0. Then u is reconstructed from the problem
divv = −1
ρ(∂tρ + ∇ρ · u), curl v = 0,
T2v(t)dx = ¯u(t).
Theorem 2.1 in [27] gives the existence of a unique solution u ∈ Hs+1(T2) to this problem, provided that the right-hand side satisfies −(∂tρ + ∇ρ · u)/ρ ∈ Hs(T2).
Actually, Li and Marcati replace the right-hand side by−(∂tρ + ∇ρ · u)/ψ, where ψ
solves the mass equation
∂tψ + ψ div v + u · ∇ρ = 0, t > 0, ψ(0) = ρ0.
The reason is that this equation can be solved explicitly, yielding strictly positive solu-tionsψ. The existence proof is based on an iteration scheme: Given (ρp, ψp, up, vp),
solve divvp+1= fp(t), curl vp+1= 0, T2vp+1(t)dx = ¯u(t), ∂tψp+1+ψp+1divvp+ up· ∇ρp= 0, t > 0, ψ(0) = ρ0, ∂2 t tρp+1−γ1ργp+1+ψpσ(ψp)22ρp+1= gp(t), t > 0, ρp+1(0) = ρ0, ∂tρp+1(0) = −ρ0div u0− ∇ρ0· u0, ∂tup+1+ u⊥p+1= hp(t),
where fp(t), gp(t), and hp(t) contain the remaining terms (see [27, Sect. 3] for
de-tails). The existence of solutions to these linear problems follows from ODE theory and the theory of wave equations. The main effort is now to derive uniform estimates in Sobolev spaces Hk(T2). This is done by multiplying the above equations by suitable test functions and assuming that T > 0 is sufficiently small. By compactness, there exists a subsequence of(ρp, ψp, up, vp) which converges in a suitable Sobolev space
to(ρ, ψ, u, v) as p → ∞. This limit allows us also to show that ρ = ψ ≥ 0 and u = v. This shows the existence of local smooth solutions under the assumption of irrotational flow curl u = 0.
Next, we prove the existence of local smooth solutions to the quasi-geostrophic Eq. (7). We setμ := μ(1) > 0.
Theorem 2 (Local existence for the quasi-geostrophic equation). Let φ0 ∈ C∞(T2). Then there exists T > 0 and a smooth solution φ to (7) and (8) for 0≤ t ≤ T .
Proof. The idea of the proof is to apply the theory of linear semigroups. Let p > 2
and let Ap : W2,p(T2) → R, Ap(u) = −μu + u. Then Apis a sectorial operator
satisfying (λ) = 1 for all λ ∈ σ(Ap), where σ (Ap) denotes the spectrum of Ap.
Consequently, Appossesses the fractional powers Aβpforβ ≥ 0, defined on the domain Xβ,p= D(Aβp). This space, endowed with its graph norm, satisfies Xβ,p → Wk,q(T2)
if k− 2/q < 2β − 2/p, q ≥ p [21, Theorem 1.6.1]. Let max{1 − 1/p, 1/2 + 1/(2p)} < β < 1 and set X := Xβ,p. The operator A
pgenerates an analytical semigroup e−t Ap
(t ≥ 0) [21, Theorem 1.3.4], and the following estimates hold for all t> 0 [21, Theorem 1.4.3]: Ape−t Apu Lp(T2)≤ Ct−βe−δtuLp(T2), (e−t Ap− I )v Lp(T2)≤ CtβApvLp(T2)≤ CtβvX for 0< δ < 1, u ∈ Lp(T2), and v ∈ X.
Next, we reformulate (7). Set u= φ − φ. Then (7) can be written as a system of two second-order equations:
−φ + φ = u in T2, t > 0, (22)
We employ a fixed-point argument. Let T > 0 and R > 0. We introduce the spaces
Y = C0([0, T ]; X) and BR = {u ∈ Y : u − u0Y ≤ R}, where u0 = −φ0+φ0∈ C∞(T2). Given u ∈ Y ⊂ C0([0, T ]; Lp(T2)), let φ ∈ L∞(0, T ; W2,p(T2)) be the
unique solution to (22) satisfying the elliptic estimateφW2,p(T2) ≤ CuLp(T2). Then define
J(u) = e−t Apu0+ t
0
e(t−s)ApF(φ(s), u(s))ds, where
F(φ, u) = (∇⊥φ · ∇)(φ − u) + μ(u − φ) + u.
Using the continuous embedding W2,p(T2) → W1,2p(T2) and the elliptic estimate for
φ, we infer the estimate
F(φ, u)L∞(0,T ;Lp(T2)) ≤ CuL∞(0,T ;W1,2p(T2)) 1 +uL∞(0,T ;W1,2p(T2)) ≤ CuL∞(0,T ;X) 1 +uL∞(0,T ;X) = CuY(1 + uY).
The last inequality follows from the embedding X → W1,2p(T2) which holds for β > 1/2 + 1/(2p).
We show that J maps BR into BR and that J : BR → BR is a contraction for
sufficiently small T > 0. Let T > 0 be such that (e−t Ap−I )u
0Lp(T2) ≤ CTβu0X ≤ R/2. Then, for u ∈ BR, J(u) − u0 Y ≤ sup 0<t<T (e−t Ap− I )u0 Lp(T2) + sup 0<t<T t 0 A pe−t ApF(φ(s), u(s)) Xds ≤ R 2 + sup0<t<T t 0 (t − s)−βe−δ(t−s)F(φ(s), u(s))Xds ≤ R 2 + C T1−β 1− β uY(1 + uY) ≤ R,
if T > 0 is sufficiently small, using that u ∈ BR. Thus J(u) ∈ BR. In a similar way, we
show that, for given u,v ∈ BR,
J(u) − J(v)Y ≤
C T1−β
1− β (uY +vY)u − vY.
Again, choosing T > 0 small enough, J becomes a contraction, and the fixed-point theorem of Banach provides the existence and uniqueness of a mild solution on[0, T ]. It remains to prove that the mild solution is smooth. Sinceβ > 1 − 1/p, we have
X → W2,p/2(T2) and hence u ∈ L∞(0, T ; W2,p/2(T2)) ⊂ L∞(0, T ; W1,p(Td)).
Furthermore, ∇φ ∈ L∞(0, T ; W1,p(T2)) ⊂ L∞(0, T ; L∞(T2)) (here, we use p > 2). This shows that ∂tu + Ap(u) ∈ L∞(0, T ; Lp(T2)). Parabolic theory implies that u ∈ Lq(0, T ; W2,p(T2)) for all q < ∞. This improves the regularity of φ to φ ∈ Lq(0, T ; W4,p(T2)). Hence, ∂tu + Ap(u) ∈ Lq(0, T ; L∞(T2)), and a bootstrap
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