Global Existence of Finite Energy Weak Solutions of Quantum Navier-Stokes Equations
aa r X i v : . [ m a t h . A P ] J un GLOBAL EXISTENCE OF FINITE ENERGY WEAK SOLUTIONS OFQUANTUM NAVIER-STOKES EQUATIONS
PAOLO ANTONELLI AND STEFANO SPIRITO
Abstract.
In this paper we consider the Quantum Navier-Stokes system both in two andin three space dimensions and prove global existence of finite energy weak solutions for largeinitial data. In particular, the notion of weak solutions is the standard one. This meansthat the vacuum region are included in the weak formulations. In particular, no extra termlike damping or cold pressure are added to the system in order to define the velocity fieldin the vacuum region. The main contribution of this paper is the construction of a regularapproximating system consistent with the effective velocity transformation needed to getnecessary a priori estimates. Introduction
In this paper we study the Quantum-Navier-Stokes (QNS) system on (0 , T ) × Ω,(1.1) ∂ t ρ + div( ρu ) = 0 ∂ t ( ρu ) + div( ρu ⊗ u ) + ∇ ρ γ − ν div( ρDu ) − k ρ ∇ (cid:18) ∆ √ ρ √ ρ (cid:19) = 0 , with initial data(1.2) ρ (0 , x ) = ρ ( x ) , ( ρu )(0 , x ) = ρ ( x ) u ( x ) . The domain Ω we consider is the d -dimensional torus with d = 2 ,
3. The unknowns ρ, u represent the mass density and the velocity field of the fluid, respectively, ν and κ are positiveconstants and they are called the viscosity and the dispersive coefficients.The above system belongs to a wider class of fluid dynamical evolution equations, calledNavier-Stokes-Korteweg systems, which read(1.3) ∂ t ρ + div( ρu ) = 0 ∂ t ( ρu ) + div( ρu ⊗ u ) + ∇ p = div S + div K , where S = S ( ∇ u ) is the viscosity stress tensor S = h ( ρ ) D u + g ( ρ ) div u I , and K = K ( ρ, ∇ ρ ) the capillarity (dispersive) term, defined throughdiv K = ∇ (cid:18) ρ div( k ( ρ ) ∇ ρ ) −
12 ( ρk ′ ( ρ ) − k ( ρ )) |∇ ρ | (cid:19) − div( k ( ρ ) ∇ ρ ⊗ ∇ ρ ) . Furthermore, similar systems arise also in the description of quantum fluids. For examplethe inviscid system, i.e. (1.1) with ν = 0, is the well known Quantum Hydrodynamics(QHD) model for superfluids [26]. Global existence of finite energy weak solutions for theQHD system has been studied in [1, 2]. Inviscid systems with a general capillarity tensor arealso studied extensively, for example in [6] the local well-posedness in high regularity spacesof the Euler-Korteweg system is treated. Recently in [4] the global well-posedness of the samesystem for small irrotational data was proved. The viscous correction term in (1.1) has been Mathematics Subject Classification.
Primary: 35Q35, Secondary: 35D05, 76N10.
Key words and phrases.
Compressible Fluids, Quantum Navier-Stokes, Vacuum, Existence. also derived in [13], by closing the moments for a Wigner equation with a BGK term. Formore details about the derivation of the QNS system we refer the reader to [22].The main result we are going to prove in our paper is the existence of global in time finiteenergy weak solutions for the Cauchy problem (1.1), (1.2). This is the first result of globalexistence for finite energy weak solutions to a Navier-Stokes-Korteweg system in several spacedimensions. For the one dimensional case, in [23] the global existence of weak solutions for theQNS system (1.1) is proved. Furthermore, in [18] the authors consider a large class of NSKsystems in one dimension, for which they prove the existence of global in time finite energyweak solutions. We also mention [15] where the authors show the existence of global classicalsolutions around constant states in one space dimension. Concerning the multidimensionalsetting, in [20] the existence of global strong solutions to (1.1) is shown, by choosing a linearpressure and κ = ν .A global existence result for (1.1), (1.2) with finite energy initial data was already obtainedby J¨ungel in [21] in the case κ > ν and γ >
3. However, the notion of weak solutions in[21] requires test functions of the type ρφ , with φ smooth and compactly supported. Thisparticular choice of such test functions does not consider the nodal region { ρ = 0 } in theweak formulation, where there are the main difficulties in dealing with the convective termand it was introduced in [9] to prove a global existence result for a Navier-Stokes-Kortewegsystem (1.3) with a specific choice of viscosity and capillarity coefficients.Furthermore, some global existence results by using the classical notion of weak solutionshave been shown by augmenting the system (1.1) with some additional terms: for example,[19] considers a cold pressure term, whereas in [31] damping terms are added. Those aug-mented systems ensure that the velocity field is well defined also in the vacuum region andit lies in some suitable Lebesgue or Sobolev spaces. From such a priori estimates it is thenpossible to infer the sufficient compactness properties for the weak solutions, in particular todeal with the convective term in the vacuum region.When κ = 0 in (1.1), global existence results for finite energy weak solutions have beenrecently obtained by [30] and [27]. One of the main tools to treat the convective term is theMellet-Vasseur inequality [29]. There the authors prove the compactness of finite energy weaksolutions for the Navier-Stokes equations with degenerate viscosity by obtaining a logarithmicimprovement to the usual energy estimates, namely they show the quantity ρ | u | log (cid:0) | u | (cid:1) is uniformly bounded in L ∞ t L x .The presence of the dispersive term in (1.1), however, prevents to directly prove a Mellet-Vasseur type inequality. This was indeed already remarked in [30], where the authors canonly prove an approximate estimate by exploiting the extra damping terms and a truncationtechnique for the mass density.In [3] we overcome this difficulty by using an alternative formulation for (1.1) in terms ofan effective velocity w = u + c ∇ log ρ . In this way it is possible to tune the viscosity andcapillarity coefficients such that the dispersive term vanishes in the new formulation. TheMellet-Vasseur inequality is proved then for the auxiliary system and, by using the a prioribounds obtained from a BD [7, 8] type estimate, we prove the compactness of solutions to(1.1), (1.2). We refer to [3] for a more detailed discussion on the stability properties of (1.1),(1.2). We mention [11, 10], where a similar effective velocity was used to study fluid dynamicalsystems with a two-velocity formulation. We also refer to [23] for a further introduction onmodels where similar effective velocities are considered.In this paper we continue our analysis of system (1.1), (1.2) by showing the global exis-tence of finite energy weak solutions. The main difficulty here is to construct a sequence ofapproximating solutions which satisfy the a priori bounds in [3]. More precisely, we need toconsider an approximating system with the following properties: first of all, it must retainall the a priori estimates, such as the energy and the BD entropy estimates. This further UANTUM NAVIER-STOKES 3 implies that the approximating system must be consistent with the transformation performedin [3] in terms of the effective velocity. Moreover, we need that the auxiliary system satisfiesa Mellet-Vasseur type estimate. Finally, the approximating solutions must be regular. Wenotice that standard approximation procedures based on Faedo-Galerkin method can not beused here since the a priori estimates in [3] heavily depend on the structure of the system.The approximating system we are going to study is the following one ∂ t ρ ε + div( ρ ε u ε ) = 0 ,∂ t ( ρ ε u ε ) + div( ρ ε u ε ⊗ u ε ) + ∇ ( ρ γε + p ε ( ρ ε )) + ˜ p ε ( ρ ε ) u ε = div S ε + div K ε , where p ε ( ρ ε ) is a cold pressure term, ˜ p ε ( ρ ε ) u ε is a damping term, S ε and K ε are the approxi-mating viscosity and capillarity tensors, respectively. As we will see below, the cold pressureterm will give us the higher integrability a priori bounds crucial to prove the global regularityof the approximating solutions. However, this introduces some difficulties in the analysis, firstof all that prevents to obtain a Mellet-Vasseur type estimate. To overcome this problem wethen add the damping term, with a suitable choice of the coefficient ˜ p ε ( ρ ε ) such that in theauxiliary system written in terms of the effective velocity the cold pressure cancels. In orderto show the convergence to zero of the cold pressure and damping terms we need additionala priori estimates. We manage to get further integrability properties for those singular termsby considering a regularized viscous stress tensor, similarly to [27]. On the other hand, thisrequires that also the capillarity tensor has to be regularized accordingly; this is necessaryso that the approximating system is consistent with the transformation through the effectivevelocity, as already remarked above. We will thus consider a regularization for the capillaritytensor such that it can be transformed as a part of the effective viscous tensor. Moreover, thisis the good approximation for the capillarity tensor since this yields the necessary a prioribounds on the mass density.We conclude this introduction by a comparison with the result in [3]. The compactnessholds for any ν, κ > κ < ν . In the two dimensional case we prove theexistence result for the same range κ < ν , while in the three dimensional case we consider ν and µ at the same scale, namely κ < ν < ακ for some α >
1. However, it is worth to pointout that no smallness assumption on ν and κ are assumed.Our paper is structured as follows: in Section 2 we introduce the notations and definitions,in Section 3 we study the approximating system and we show some useful identities. Then,in Section 4 we prove the a priori estimate we need. Finally, in Section 5 we prove the The-orem 2.2 and 2.3 and in Section 6 we prove the global existence of smooth solutions for theapproximating system.2. Notations, Definitions and Main Result
In this section we are going to fix the notations used in the paper, to give the precisedefinition of weak solution for the system (1.1) and to state our main results.
Notations
Given Ω ⊂ R , the space of compactly supported smooth functions will be D ((0 , T ) × Ω). Wewill denote with L p (Ω) the standard Lebesgue spaces and with k · k p their norm. The Sobolevspace of L p functions with k distributional derivatives in L p is W k,p (Ω) and in the case p = 2we will write H k (Ω). The spaces W − k,p (Ω) and H − k (Ω) denote the dual spaces of W k,p ′ (Ω)and H k (Ω) where p ′ is the H¨older conjugate of p . Given a Banach space X we use the theclassical Bochner space for time dependent functions with value in X , namely L p (0 , T ; X ), W k,p (0 , T ; X ) and W − k,p (0 , T ; X ). Finally, Du = ( ∇ u + ( ∇ u ) T ) / Au = ( ∇ u − ( ∇ u ) T ) / C will be P. ANTONELLI AND S. SPIRITO any constant depending on the data of the problem but independent on ε . Moreover, ε willbe always less than a small ε f depending only on γ , ν and κ , which will be chosen in the sequel. Weak Solutions
We first recall two alternative ways to write the third order tensor term, which will bevery useful in the sequel:(2.1) 2 ρ ∇ (cid:18) ∆ √ ρ √ ρ (cid:19) = div( ρ ∇ log ρ ) = ∇ ∆ ρ − ∇√ ρ ⊗ ∇√ ρ ) . Then, by using (2.1), we can consider the following definition of weak solutions.
Definition 2.1.
A pair ( ρ, u ) with ρ ≥ is said to be a weak solution of the Cauchy problem (1.1) - (1.2) if(1) Integrability conditions: ρ ∈ L ∞ (0 , T ; L ∩ L γ ( T d )) , √ ρu ∈ L ∞ (0 , T ; L ( T d )) , √ ρ ∈ L ∞ (0 , T ; H ( T d )) . (2) Continuity equation: Z ρ φ (0) + Z Z ρφ t + √ ρ √ ρu ∇ φ = 0 , for any φ ∈ C ∞ c ([0 , T ); C ∞ ( T d )) .(3) Momentum equation: Z ρ u ψ (0) + Z Z √ ρ ( √ ρu ) ψ t + √ ρu ⊗ √ ρu ∇ ψ + ρ γ div ψ − ν Z Z ( √ ρu ⊗ ∇√ ρ ) ∇ ψ − ν Z Z ( ∇√ ρ ⊗ √ ρu ) ∇ ψ + ν Z Z √ ρ √ ρu ∆ ψ + ν Z Z √ ρ √ ρu ∇ div ψ − κ Z Z ( ∇√ ρ ⊗ ∇√ ρ ) ∇ ψ + 2 κ Z Z √ ρ ∇√ ρ ∇ div φ = 0 , for any ψ ∈ C ∞ c ([0 , T ); C ∞ ( T d )) .(4) Energy Inequality: if E ( t ) = Z ρ | u | + ρ γ γ − κ |∇√ ρ | , then the following energy inequality is satisfied for a.e. t ∈ [0 , T ] E ( t ) ≤ E (0) . UANTUM NAVIER-STOKES 5
Main Result
Let us start by specifying the assumptions on the initial data. Let ν > κ and let η be asmall fixed positive number. We consider an initial density ρ such that(2.2) ρ ≥ T d ,ρ ∈ L ∩ L γ ( T d ) , ∇√ ρ ∈ L ∩ L η ( T d ) . Concerning the initial velocity u we assume that(2.3) u = 0 on { ρ = 0 } , p ρ u ∈ L ∩ L η ( T d ) . The hypothesis of higher integrability on ∇ p ρ and p ρ u imply that(2.4) ρ (cid:18) | v | (cid:19) log (cid:18) | v | (cid:19) is uniformly bounded in L ( T d ) , with v = u + c ∇ log ρ and c >
0. In order to simplify the presentation we assume also that ρ is bounded from above and below, namely there exists ¯ ρ > < ρ ≤ ρ ≤ ¯ ρ . Then, we state our main result in the two dimensional case.
Theorem 2.2.
Let d = 2 . Let ν, κ and γ positive such that κ < ν and γ > . Then for any < T < ∞ there exists a finite energy weak solutions of the system (1.1) on (0 , T ) × T , withinitial data (1.2) satisfying (2.2) , (2.3) and (2.5) . In the three dimensional case we need the a restriction on ν, κ and γ . Theorem 2.3.
Let d = 3 . Let ν, κ and γ positive such that κ < ν < κ and < γ < .Then for any < T < ∞ there exists a finite energy weak solutions of the system (1.1) on (0 , T ) × T , with initial data (1.2) satisfying (2.2) , (2.3) and (2.5) . Let us briefly comment on the extra assumption we have in Theorem 2.3. This assumptionit is not required in the passage to the limit from the approximating solutions ( ρ ε , u ε ) tosolutions of (1.1) but only in the proof of global existence of smooth solutions of the approx-imating system, see Theorem 6.2. As it will be clear from our proof (see Proposition 6.3),we need the viscosity and capillarity constants to be comparable in order to prove regularityof solutions of the approximating system. The constant 9 / ν.κ .3. The Approximating System
In this Section we first introduce the approximating system we are going to study and wethen show how that can be transformed into an equivalent system in terms of the effectivevelocity, analogously to what was done in [3].
Approximating System
The system in (0 , T ) × T d we consider is(3.1) ∂ t ρ ε + div( ρ ε u ε ) = 0 ,∂ t ( ρ ε u ε ) + div( ρ ε u ε ⊗ u ε ) − ν div S ε + ∇ ( ρ γε + p ε ( ρ ε )) + ˜ p ε ( ρ ε ) u ε = κ div K ε . P. ANTONELLI AND S. SPIRITO
The system (3.1) is coupled with initial data on { t = 0 } × T d :(3.2) ρ ε (0 , x ) = ρ ε ( x ) ,ρ ε u ε (0 , x ) = ρ ε ( x ) u ε ( x ) . Let us describe in what follows the various terms appearing in (3.1).The viscosity coefficient h ε ( ρ ε ) is defined as follows(3.3) h ε ( ρ ε ) = ρ ε + ερ ε + ερ γε and we define g ε ( ρ ε ) to be(3.4) g ε ( ρ ε ) = ρ ε h ′ ε ( ρ ε ) − h ε ( ρ ε ) . Then the stress tensors S ε = S ε ( ∇ u ε ) is:(3.5) S ε ( ∇ u ε ) = h ε ( ρ ε ) Du ε + g ε ( ρ ε ) div u ε I . The following inequalities follow from the definitions of h ε ( ρ ε ) and g ε ( ρ ε )(3.6) h ε ( ρ ε ) ≥ , | g ε ( ρ ε ) | ≤ ( γ − h ε ( ρ ε ) ,h ′ ε ( ρ ε ) ρ ε ≤ γh ε ( ρ ε ) , | h ′′ ε ( ρ ε ) | ρ ε ≤ ( γ − h ′ ε ( ρ ε ) . In particular it follows from (3.3) that(3.7) h ε ( ρ ε ) | Du ε | + g ε ( ρ ε ) | div u ε | > h ε ( ρ ε ) | Du ε | . The approximating dispersive term K ε = K ε ( ρ ε , ∇ ρ ε ) is defined asdiv( K ε ( ρ ε , ∇ ρ ε )) = 2 ρ ε ∇ (cid:18) h ′ ε ( ρ ε ) div( h ′ ε ( ρ ε ) ∇√ ρ ε ) √ ρ ε (cid:19) . We notice that, for ε = 0, we recover the quantum term in (2.1). Next Lemma clarifies howthis approximation is consistent with the approximating viscous tensor in (3.5). Lemma 3.1.
The following formulae hold for the tensor K ε : ρ ε ∇ (cid:18) h ′ ε ( ρ ε ) div( h ′ ε ( ρ ε ) ∇√ ρ ε ) √ ρ ε (cid:19) = div( h ε ( ρ ε ) ∇ ( φ ε ( ρ ε )) + ∇ ( g ε ( ρ ε )∆ φ ε ( ρ ε ))= ∇ (cid:0) h ′ ε ( ρ ε )∆ h ε ( ρ ε ) (cid:1) − h ′ ε ( ρ ε ) ∇√ ρ ε ) ⊗ ( h ′ ε ( ρ ε ) ∇√ ρ ε )) where φ ε ( ρ ε ) is such that ρ ε φ ′ ( ρ ε ) = h ′ ε ( ρ ε ) .Proof. By direct computations we getdiv( K ( ρ ε )) = ∇ ( h ′ ε ( ρ ε ) √ ρ ε div( h ′ ε ( ρ ε ) ∇ ρ ε / √ ρ ε )) − h ′ ε ( ρ ε ) ∇√ ρ ε div( h ′ ε ( ρ ε ) ∇√ ρ ε )= ∇ ( h ′ ε ( ρ ε )∆ h ε ( ρ ε )) − ∇ (cid:0) | h ′ ε ( ρ ε ) ∇√ ρ ε | (cid:1) − h ′ ε ( ρ ε ) ∇√ ρ ε ⊗ h ′ ε ( ρ ε ) ∇√ ρ ε ) + 4 ∇ (cid:0) h ′ ε ( ρ ε ) ∇√ ρ ε (cid:1) · ( h ′ ε ( ρ ε ) ∇√ ρ ε )= ∇ ( h ′ ε ( ρ ε )∆ h ε ( ρ ε )) − h ′ ε ( ρ ε ) ∇√ ρ ε ⊗ h ′ ε ( ρ ε ) ∇√ ρ ε ) . To prove the remaining identity, we use the fact that ρ ε φ ′ ( ρ ε ) = h ′ ε ( ρ ε ) we have ∇ ( h ′ ε ( ρ ε )∆ h ε ( ρ ε )) − h ′ ε ( ρ ε ) ∇√ ρ ε ⊗ h ′ ε ( ρ ε ) ∇√ ρ ε )) = ∇ ( h ′ ε ( ρ ε ) div( ρ ε ∇ φ ε ( ρ ε ))) − div( ∇ h ε ( ρ ε ) ⊗ ∇ φ ε ( ρ ε )) = ∇ ( h ′ ε ( ρ ε ) ρ ε ∆ φ ε ( ρ ε )) + ∇ ( ∇ h ε ( ρ ε ) · ∇ φ ε ( ρ ε )) − ∇ ( h ε ( ρ ε ) ∇ φ ε ( ρ ε )) + div( h ε ( ρ ε ) ∇ φ ε ( ρ ε )) = ∇ ( h ′ ε ( ρ ε ) ρ ε ∆ φ ε ( ρ ε )) − ∇ ( h ε ( ρ ε )∆ φ ε ( ρ ε )) + div( h ε ( ρ ε ) ∇ φ ε ( ρ ε )) = ∇ ( g ε ( ρ ε )∆ φ ε ( ρ ε )) + div( h ε ( ρ ε ) ∇ φ ε ( ρ ε )) . (cid:3) UANTUM NAVIER-STOKES 7
The previous Lemma explains how the regularization of the dispersive tensor is consistentwith (3.3) and the transformation through the effective velocity. Indeed, since the viscoustensor S ε ( ∇ u ε ) = h ε ( ρ ε ) Du ε + g ε ( ρ ε ) div u ε , the effective velocity is given by v ε = u ε + c ∇ φ ε ( ρ ε ), where as above φ ε ( ρ ε ) is defined through h ′ ε ( ρ ε ) = ρ ε φ ′ ε ( ρ ε ). Then, from theidentities in Lemma 3.1, it is straightforward to see that div K ε ( ρ ε , ∇ ρ ε ) = div S ε ( ∇ φ ε ( ρ ε )),so that in the effective system this can be incorporated in the effective viscous tensor.The coefficient ˜ p ε ( ρ ε ) in the damping term is defined by˜ p ε ( ρ ε ) = λ ( ε ) (cid:18) ρ ε ε + ρ − ε ε (cid:19) where λ ( ε ) = e − ε . The cold pressure p ε ( ρ ε ) is defined such that p ′ ε ( ρ ε ) = µ ˜ p ε ( ρ ε ) h ′ ε ( ρ ε ) ρ ε , where(3.8) µ = ν − p ν − κ . In particular, by using the definition of h ε ( ρ ε ) and ˜ p ε ( ρ ε ) by direct computations we get thefollowing expression for p ε ( ρ ε )(3.9) p ε ( ρ ε ) = µε λ ( ε ) ρ ε ε + ε µ λ ( ε )8 − ε ρ ε − ε + ε µλ ( ε ) γ ε ( γ − ρ ε + γ − ε − µλ ( ε ) ε ρ − ε ε − ε µ λ ( ε ) ε + 8 ρ − ε − ε − ε γλ ( ε )1 − ε ( γ − ρ − ε + γ − ε = X i =1 p iε ( ρ ε ) . Let f ε ( ρ ε ) such that p ε ( ρ ε ) = ρ ε f ′ ε ( ρ ε ) − f ε ( ρ ε ) . Then, again by direct calculation we have that(3.10) f ε ( ρ ε ) = µε λ ( ε )1 − ε ρ ε ε + ε µ λ ( ε )(8 − ε )(8 − ε ) ρ ε − ε + ε µλ ( ε ) γ (1 + ε ( γ − ε ( γ − ρ ε + γ − ε + ε µλ ( ε ) ε + 1 ρ − ε ε + ε µ λ ( ε )8(8 + ε )(9 + 8 ε ) ρ − ε − ε + ε µγλ ( ε )(1 − ε ( γ − − ε ( γ − ρ − ε + γ − ε = X i =1 f iε ( ρ ε ) . It is straightforward to check that there exists ε f = ε f ( γ ) > f iε ( ρ ε ) and ( f iε ( ρ ε )) ′′ are positive for any i = 1 , ..., ε < ε f . P. ANTONELLI AND S. SPIRITO
Finally we construct the initial data (3.2). Given ( ρ , u ) satisfying (2.2), (2.3) and (2.5)it is easy to construct a sequence of smooth functions ( ρ ε , u ε ) such that(3.11) 1¯ ρ ≤ ρ ≤ ¯ ρ ,ρ ε → ρ strongly in L ( T d ) , { ρ ε } ε in uniformly in bounded in L ∩ L γ ( T d ) , { h ′ ε ( ρ ε ) ∇ p ρ ε } ε is uniformly bounded in L ∩ L η ( T d ) ,h ′ ε ( ρ ε ) ∇ p ρ ε → ∇ p ρ strongly in L ( T d ) , { p ρ ε u ε } is uniformly bounded in L ∩ L η ( T d ) ,ρ ε u ε → ρ ε u ε in L ( T d ) ,f ε ( ρ ε ) → L ( T d ) . In particular the hypothesis on the boundedness of ρ makes easy to prove that h ′ ( ρ ε ) and f ε ( ρ ε ) are uniformly bounded. Moreover, the higher integrability on h ′ ( ρ ε ) ∇ p ρ ε and p ρ ε u ε implies that(3.12) ρ ε (cid:18) | v ε | (cid:19) log (cid:18) | v ε | (cid:19) is uniformly bounded in L ( T d ) , with v ε = u + c ∇ φ ε ( ρ ε ) and c > The effective velocity formulation
We now consider the effective velocity v ε = u ε + c ∇ φ ε ( ρ ε ). Next Lemma shows that thesystem (3.1) can be equivalently written in terms of ( ρ ε , v ε ). Furthermore, with a suitablechoice of the constant c , either the dispersive and the cold pressure terms will vanish. Lemma 3.2.
Let ( ρ ε , u ε ) be a smooth solution of the system (3.1) . Then, ( ρ ε , v ε ) , with v ε = u ε + c ∇ φ ε ( ρ ε ) and c > satisfies the following system, (3.13) ∂ t ρ ε + div( ρ ε v ε ) = c ∆ h ε ( ρ ε ) ∂ t ( ρ ε v ε ) + div( ρ ε v ε ⊗ v ε ) + ∇ ρ γε + ˜ λ ∇ p ε ( ρ ε ) − c ∆( h ε ( ρ ε ) v ε ) + ˜ p ( ρ ε ) v ε − ν − c ) div( h ε ( ρ ε ) Dv ε ) − (2 ν − c ) ∇ ( g ε ( ρ ε ) div v ε ) − ˜ κ div K ε = 0 , where µ > is defined in (3.8) , ˜ κ = κ − νc + c , ˜ λ = ( µ − c ) /µ .Proof. Let c ∈ R . From the first equation in (3.1) we have that(3.14) c ( ρ ε ∇ φ ε ( ρ ε )) t = − c ∇ (div( h ε ( ρ ε ) u ε )) − c ∇ ( g ε ( ρ ε ) div u ε ) . Moreover, it is straightforward to prove that(3.15) c div( ρ ε u ε ⊗ ∇ φ ε ( ρ ε ) + ρ ε ∇ φ ε ( ρ ε ) ⊗ u ε ) = c ∆( h ε ( ρ ε ) u ε ) − c div( h ε ( ρ ε ) Du ε )+ c ∇ div( h ε ( ρ ε ) u ε )and(3.16) c div( ρ ε ∇ φ ε ( ρ ε ) ⊗ ∇ φ ε ( ρ ε )) = c ∆( h ε ( ρ ε ) ∇ φ ε ( ρ ε )) − c div( h ε ( ρ ε ) ∇ φ ε ( ρ ε )) , see also [23]. Then, by using the definition of v ε we have(3.17) ∂ t ( ρ ε v ε ) + div( ρ ε v ε ⊗ v ε ) + ∇ ρ γε = ∂ t ( ρ ε u ε ) + div( ρ ε u ε ⊗ u ε ) + ∇ ρ γε + c ( ρ ε ∇ φ ε ( ρ ε )) t + c div( ρ ε u ε ⊗ ∇ φ ε ( ρ ε ) + ρ ε ∇ φ ε ( ρ ε ) ⊗ u ε ) c ∆( h ε ( ρ ε ) ∇ φ ε ( ρ ε )) − c div( h ε ( ρ ε ) ∇ φ ε ( ρ ε )) + ∇ ρ γε UANTUM NAVIER-STOKES 9 and by using (3.14)-(3.16) and the fact the ( ρ ε , u ε ) satisfies the momentum equation in (3.1)we get(3.18) ∂ t ( ρ ε v ε ) + div( ρ ε v ε ⊗ v ε ) − c ∆( h ε ( ρ ε ) v ε ) + ∇ ρ γε − ν − c ) div( h ε ( ρ ε ) Dv ε ) − (2 ν − c ) ∇ ( g ε ( ρ ε ) div v ε ) + ˜ p ( ρ ε ) v ε = c ˜ p ε ( ρ ε ) ∇ φ ε ( ρ ε ) − p ′ ε ( ρ ε ) ∇ ρ ε + ( κ − νc + c ) div K ε ( ρ ε ) . By using that v ε = u ε + c ∇ φ ε ( ρ ε ), Lemma 3.1 and the definition on p ε ( ρ ε ) we get(3.19) ∂ t ( ρ ε v ε ) + div( ρ ε v ε ⊗ v ε ) − c ∆( h ε ( ρ ε ) v ε ) + ∇ ρ γε + c div( h ε ( ρ ε ) Dv ε ) − (2 ν − c ) div S ε ( v ε ) + ˜ p ( ρ ε ) v ε = ( c − νc + κ ) div K ε ( ρ ε ) − µ − cµ ∇ p ε ( ρ ε ) . Let us notice that, by taking c = µ , then the right hand side in (3.19) vanishes. (cid:3) A priori Estimates
In this Section we are going to show that the approximating system satisfies, uniformly in ε >
0, the a priori estimates used in [3] to prove the compactness of weak solutions to (1.1).First of all we prove the classical energy estimate for system (3.1).
Proposition 4.1.
Let ( ρ ε , u ε ) be a smooth solution of (3.1) . Then, the following estimateholds. (4.1) ddt (cid:18)Z ρ ε | u ε | ρ γε γ − f ε ( ρ ε ) + 2 κ | h ′ ε ( ρ ε ) ∇√ ρ ε | (cid:19) + 2 ν Z h ε ( ρ ε ) | Du ε | + 2 ν Z g ε ( ρ ε ) | div u ε | + Z ˜ p ε ( ρ ε ) | u ε | = 0 . Proof.
Let us multiply the momentum equation in (3.1) by u ε . After integrating by partsand using the first equation we get(4.2) ddt Z ρ ε | u ε | ν Z h ε ( ρ ε ) | Du ε | + 2 ν Z g ε ( ρ ε ) | div u ε | + Z ˜ p ε ( ρ ε ) | u ε | − κ Z div K ε u ε + Z ∇ ( ρ γε + p ε ( ρ ε )) u ε = 0 . Then, we consider the pressure terms. By multiplying the first equation by γρ γ − ε γ − we get(4.3) ddt Z ρ γε γ − − Z ∇ ρ γε u ε = 0 . By multiplying again the first equation by f ′ ε ( ρ ε )(4.4) ddt Z f ε ( ρ ε ) − Z ∇ p ε ( ρ ε ) u ε = 0 . Finally, we deal with dispersive term. By multiplying the first equation by − κ h ′ ε ( ρ ε ) div( h ′ ε ( ρ ε ) ∇√ ρ ε ) / √ ρ ε we get − κ Z ∂ t ρ ε h ′ ε ( ρ ε ) div( h ′ ε ( ρ ε ) ∇√ ρ ε ) √ ρ ε − κ Z div( ρ ε u ε ) h ′ ε ( ρ ε ) div( h ′ ε ( ρ ε ) ∇√ ρ ε ) √ ρ ε = 0 . Then, by using Lemma 3.1, integrating by parts and using the chain rule we get(4.5) ddt Z κ | h ′ ε ( ρ ε ) ∇√ ρ ε | + κ Z div K ε u ε = 0 . By summing up (4.2), (4.3), (4.4) and (4.5) we get (4.1). (cid:3)
Next Lemma gives the energy estimate for the transformed system (4.11).
Proposition 4.2.
Let ( ρ ε , v ε ) be a smooth solution to (3.1) and let us consider ( ρ ε , v ε ) , where v ε is the effective velocity v ε = u ε + c ∇ φ ε ( ρ ε ) , with c ∈ (0 , µ ) . Then we have (4.6) ddt (cid:18)Z ρ ε | v ε | ρ γε γ − λf ε ( ρ ε ) + 2˜ κ | h ′ ε ( ρ ε ) ∇√ ρ ε | (cid:19) + c Z h ε ( ρ ε ) | Av ε | + (2 ν − c ) Z ( h ε ( ρ ε ) | Dv ε | + g ε ( ρ ε ) | div v ε | )+ Z ˜ p ε ( ρ ε ) | v ε | + cγ Z h ′ ε ( ρ ε ) |∇ ρ ε | ρ γ − ε + c ˜ λ Z h ′ ε ( ρ ε ) |∇ ρ ε | f ′′ ε ( ρ ε )+ c ˜ κ Z h ε ( ρ ε ) |∇ φ ε ( ρ ε ) | + c ˜ κ Z g ε ( ρ ε ) | ∆ φ ε ( ρ ε ) | = 0 , where ˜ λ = ( µ − c ) /µ and ˜ κ = c − νc + κ .Proof. Since ( ρ ε , u ε ) is a smooth solution of (3.1) we can use Lemma 3.2 to deduce that ( ρ ε , v ε )satisfies equations (3.13). Then, by multiplying the momentum equation by v ε , integratingby parts and using the first equation we get(4.7) ddt Z ρ ε | v ε | c Z h ε ( ρ ε ) | Av ε | + (2 ν − c ) Z ( h ε ( ρ ε ) | Dv ε | + g ε ( ρ ε ) | div v ε | )+ Z ∇ ρ γε · v ε + ˜ λ Z ∇ p ε ( ρ ε ) · v ε + Z ˜ p ε ( ρ ε ) | v ε | − ˜ κ Z div K ε v ε = 0 , where we used that |∇ v ε | = | Dv ε | + | Av ε | . Then, by multiplying the first equation by γρ γ − ε γ − and integrating by parts we get(4.8) ddt Z ρ γε γ − cγ Z |∇ ρ ε | h ′ ε ( ρ ε ) ρ γ − ε − Z ∇ ρ γε v ε = 0 . By multiplying again the first equation by ˜ λf ′ ε ( ρ ε ) we get(4.9) ddt Z ˜ λf ε ( ρ ε ) − ˜ λ Z ∇ p ε ( ρ ε ) v ε + c ˜ λ Z h ′ ε ( ρ ε ) |∇ ρ ε | f ′′ ε ( ρ ε ) = 0 . Then, we consider the dispersive term. By multiplying the first equationby − κ h ′ ε ( ρ ε ) div( h ′ ε ( ρ ε ) ∇√ ρ ε ) / √ ρ ε we get − κ Z ∂ t ρ ε h ′ ε ( ρ ε ) div( h ′ ε ( ρ ε ) ∇√ ρ ε ) √ ρ ε − κ Z div( ρ ε v ε ) h ′ ε ( ρ ε ) div( h ′ ε ( ρ ε ) ∇√ ρ ε ) √ ρ ε + c κ Z ∆ h ε ( ρ ε ) h ′ ε ( ρ ε ) div( h ′ ε ( ρ ε ) ∇√ ρ ε ) √ ρ ε = 0 . The first two terms are treated as in Proposition 4.1 and we get − κ Z ∂ t ρ ε h ′ ε ( ρ ε ) div( h ′ ε ( ρ ε ) ∇√ ρ ε ) √ ρ ε − κ Z div( ρ ε v ε ) h ′ ε ( ρ ε ) div( h ′ ε ( ρ ε ) ∇√ ρ ε ) √ ρ ε = ddt Z κ | h ′ ε ( ρ ε ) ∇√ ρ ε | − ˜ κ Z div K ε v ε . UANTUM NAVIER-STOKES 11
We then consider the last term, by integrating by parts we get2 Z ∆ h ε ( ρ ε ) h ′ ε ( ρ ε ) div( h ′ ε ( ρ ε ) ∇√ ρ ε ) √ ρ ε = − Z ∇ φ ε ( ρ ε ) ρ ε ∇ (cid:18) h ′ ε ( ρ ε ) div( h ′ ε ( ρ ε ) ∇√ ρ ε ) √ ρ ε (cid:19) = − Z ∇ φ ε ( ρ ε ) div K ε = − Z ∇ φ ε ( ρ ε ) div( h ε ( ρ ε ) ∇ φ ε ( ρ ε ) − Z ∇ φ ε ( ρ ε ) ∇ ( g ε ( ρ ε )∆ φ ε ( ρ ε ) , where Lemma 3.1 has been used. By integrating by parts we get2 Z ∆ h ε ( ρ ε ) h ′ ε ( ρ ε ) div( h ′ ε ( ρ ε ) ∇√ ρ ε ) √ ρ ε = Z h ε ( ρ ε ) |∇ φ ε ( ρ ε ) | + Z g ε ( ρ ε ) | ∆ φ ε ( ρ ε ) | . Resuming, we have(4.10) ddt Z κ | h ′ ε ( ρ ε ) ∇√ ρ ε | + ˜ κ Z div K ( ρ ε ) v ε + c ˜ κ Z h ε ( ρ ε ) |∇ φ ε ( ρ ε ) | + c ˜ κ g ε ( ρ ε ) | ∆ φ ε ( ρ ε ) | = 0By summing up (4.7), (4.8), (4.9) and (4.10) we get (4.6) (cid:3) Let us now choose the constant in the effective velocity to be µ = ν −√ ν − κ . Throughoutthis paper we will denote by w ε the effective velocity with this particular choice of theconstant, i.e. w ε = u ε + µ ∇ φ ε ( ρ ε ). As we already noticed, in this case both the dispersiveterm and the cold pressure term vanish in (3.13), so that the system reads(4.11) ∂ t ρ ε + div( ρ ε w ε ) = µ ∆ h ε ( ρ ε ) ,∂ t ( ρ ε w ε ) + div( ρ ε w ε ⊗ w ε ) − µ ∆( h ε ( ρ ε ) w ε ) + ∇ ρ γε + ˜ p ( ρ ε ) w ε − ν − µ ) div( h ε ( ρ ε ) Dw ε ) − (2 ν − µ ) ∇ ( g ε ( ρ ε ) div w ε ) = 0 . Analogously to what we did in [3], we now prove a Mellet-Vasseur type estimate for (4.11).We will first prove an auxiliary Lemma which will also be useful later in section 6, see Lemma6.3.
Lemma 4.3.
Let ( ρ ε , u ε ) be a solution of the system (3.1) . Then, for any β ∈ C ( R ) thepair ( ρ ε , w ε ) satisfies the following integral equation (4.12) ddt Z ρ ε β (cid:18) | w ε | (cid:19) + µ Z h ε ( ρ ε ) | Aw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + µ Z h ε ( ρ ε ) | Aw ε | β ′ (cid:18) | w ε | (cid:19) + (2 ν − µ ) Z h ε ( ρ ε ) | Dw ε | β ′ (cid:18) | w ε | (cid:19) + (2 ν − µ ) Z g ε ( ρ ε ) | div w ε | β ′ (cid:18) | w ε | (cid:19) + Z ˜ p ε ( ρ ε ) | w ε | β ′ (cid:18) | w ε | (cid:19) + (2 ν − µ ) Z h ε ( ρ ε ) | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) = − Z ∇ ρ γε w ε β ′ (cid:18) | w ε | (cid:19) − ν Z h ε ( ρ ε )( Dw ε · w ε ) · ( Aw ε · w ε ) β ′′ (cid:18) | w ε | (cid:19) − (2 ν − µ ) Z g ε ( ρ ε ) div w ε w ε · ( Dw ε · w ε ) β ′′ (cid:18) | w ε | (cid:19) . Proof.
Let β ∈ C ( R ). By a simple integration by parts we get − µ Z ∆( h ε ( ρ ε ) w ε ) w ε β ′ (cid:18) | w ε | (cid:19) = − µ Z ∆ h ε ( ρ ε ) | w ε | β ′ (cid:18) | w ε | (cid:19) + µ Z ∆ h ε ( ρ ε ) β (cid:18) | w ε | (cid:19) + µ Z h ε ( ρ ε ) |∇ w ε | β ′ (cid:18) | w ε | (cid:19) + µ Z h ε ( ρ ε ) (cid:12)(cid:12)(cid:12)(cid:12) ∇ | w ε | (cid:12)(cid:12)(cid:12)(cid:12) β ′′ (cid:18) | w ε | (cid:19) . Then, by multiplying the first equation in (4.11) by w ε β ′ (cid:16) | w ε | (cid:17) we get(4.13) ddt Z ρ ε β (cid:18) | w ε | (cid:19) + µ Z h ε ( ρ ε ) (cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:18) | w ε | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) β ′′ (cid:18) | w ε | (cid:19) ν − µ ) Z h ε ( ρ ε ) | Dw ε | β ′ (cid:18) | w ε | (cid:19) + (2 ν − µ ) Z g ε ( ρ ε ) | div w ε | β ′ (cid:18) | w ε | (cid:19) + Z ˜ p ε ( ρ ε ) | w ε | β ′ (cid:18) | w ε | (cid:19) + 2( ν − µ ) Z Dw ε w ε ∇ w ε w ε β ′′ (cid:18) | w ε | (cid:19) = − Z ∇ ρ γε w ε β ′ (cid:18) | w ε | (cid:19) − (2 ν − µ ) Z g ε ( ρ ε ) div w ε w ε ∇ (cid:18) | w ε | (cid:19) β ′′ (cid:18) | w ε | (cid:19) . Let us consider the last term on the left-hand side of the equality, we have(4.14) 2( ν − µ ) Z h ε ( ρ ε ) ∂ j w iε + ∂ i w jε ! w i ∂ j w lε w lε β ′′ (cid:18) | w ε | (cid:19) =2( ν − µ ) Z h ε ( ρ ε ) ∂ j w iε + ∂ i w jε ! w iε ∂ j w lε + ∂ l w jε ! w lε β ′′ (cid:18) | w ε | (cid:19) +2( ν − µ ) Z h ε ( ρ ε ) ∂ j w iε + ∂ i w jε ! w iε ∂ j w lε − ∂ l w jε ! w lε β ′′ (cid:18) | w ε | (cid:19) . Concerning the last term in the right-hand side of the inequality we have(4.15) Z g ε ( ρ ε ) div w ε w iε ∂ i w lε w lε β ′′ (cid:18) | w ε | (cid:19) = Z g ε ( ρ ε ) div w ε w iε (cid:18) ∂ i w lε + ∂ l w iε (cid:19) w lε β ′′ (cid:18) | w ε | (cid:19) + Z g ε ( ρ ε ) div w ε w iε (cid:18) ∂ i w lε − ∂ l w iε (cid:19) w lε β ′′ (cid:18) | w ε | (cid:19) = Z g ε ( ρ ε ) div w ε w iε (cid:18) ∂ i w lε + ∂ l w iε (cid:19) w lε β ′′ (cid:18) | w ε | (cid:19) . UANTUM NAVIER-STOKES 13
Finally, by using that(4.16) Z h ε ( ρ ε ) (cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:18) | w ε | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) β ′′ (cid:18) | w ε | (cid:19) = Z h ε ( ρ ε ) | Aw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + Z h ε ( ρ ε ) | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + 2 Z h ε ( ρ ε )( Aw ε · w ε )( Dw ε · w ε ) β ′′ (cid:18) | w ε | (cid:19) . Then, by using (4.14), (4.15) and (4.16), we get from (4.13) exactly (4.12). (cid:3)
Now, we are in position to prove the Mellet & Vasseur type inequality.
Proposition 4.4.
Let ( ρ ε , u ε ) be a smooth solution of (3.1) . Then, there exists and a genericconstant C > independent on ε such that ( ρ ε , w ε ) sastifies (4.17) sup t ∈ (0 ,T ) Z ρ ε (cid:18) | w ε | (cid:19) log (cid:18) | w ε | (cid:19) ≤ C Z Z h ε ( ρ ε ) |∇ w ε | + Z (cid:18)Z ρ (2 γ − δ/ − / (2 − δ )) ε dx (cid:19) − δ Z ρ ε (cid:18) (cid:18) | w ε | (cid:19)(cid:19) δ dx ! dt + Z ρ ε (cid:18) | w ε | (cid:19) log (cid:18) | w ε | (cid:19) for any δ ∈ (0 , .Proof. By choosing β ( t ) = (1 + t ) log(1 + t ) in Lemma 4.3 and keeping only the terms weneed we get that there exists a generic constant C > ε such thatsup t ∈ (0 ,T ) Z ρ ε (cid:18) | w ε | (cid:19) log (cid:18) | w ε | (cid:19) + Z Z ρ ε |∇ w ε | log (cid:18) | w ε | (cid:19) ≤ C (cid:12)(cid:12)(cid:12)(cid:12)Z ∇ ρ γε w ε β ′ (cid:18) | w ε | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + C Z h ε ( ρ ε ) |∇ w ε | + C Z | g ε ( ρ ε ) || div w ε ||∇ w ε | + Z ρ ε (cid:18) | w ε | (cid:19) log (cid:18) | w ε | (cid:19) . Then, by using (3.6), integrating by parts the first term, using H¨older and Young inequalitywe get sup t ∈ (0 ,T ) Z ρ ε (cid:18) | w ε | (cid:19) log (cid:18) | w ε | (cid:19) + Z Z ρ ε |∇ w ε | log (cid:18) | w ε | (cid:19) ≤ C Z Z ρ γ − ε (cid:18) (1 + log (cid:18) | w ε | (cid:19)(cid:19) + 12 Z Z ρ ε |∇ w ε | log (cid:18) | w ε | (cid:19) + C Z Z h ε ( ρ ε ) |∇ w ε | + Z ρ ε (cid:18) | w ε | (cid:19) log (cid:18) | w ε | (cid:19) . Finally, for δ ∈ (0 ,
2) by using H¨older inequality we get(4.18)
Z Z ρ γ − ε (cid:18) (cid:18) | w ε | (cid:19)(cid:19) ≤ Z (cid:18)Z ρ (2 γ − δ/ − / (2 − δ )) ε dx (cid:19) − δ Z ρ ε (cid:18) (cid:18) | w ε | (cid:19)(cid:19) δ dx ! dt ≤ C Z (cid:18)Z ρ (2 γ − δ/ − / (2 − δ )) ε dx (cid:19) (2 − δ ) / dt. Then, (4.17) is proved. (cid:3) Proof of Theorem 2.2 and Theorem 2.3
In this section prove we give the proofs of Theorem 2.2 and Theorem 2.3. Let us start bycollecting and deriving the main bounds which will be needed.
Uniform Bounds
Let ε < ε f and let { ( ρ ε , u ε ) } ε , with ρ ε >
0, be a sequence of smooth solutions of (3.1)with initial data ( ρ ε , u ε ) satisfying (3.11). The global existence of ( ρ ε , u ε ) will be proved inthe next section. By Proposition 4.1 there exists a generic constant C > ε such that(5.1) sup t Z ρ ε | u ε | ≤ C, sup t Z | h ′ ε ( ρ ε ) ∇√ ρ ε | ≤ C, sup t Z ( ρ ε + ρ γε ) ≤ C, Z Z h ε ( ρ ε ) | Du ε | ≤ C, sup t Z f ε ( ρ ε ) ≤ C, Z Z | ˜ p ε ( ρ ε ) || u ε | ≤ C. where (3.6) has been used. In particular, by using (3.3) we have that(5.2) sup t Z |∇√ ρ ε | ≤ C, Z Z ρ ε | Du ε | ≤ C. Then, by (3.11) and Proposition 4.2 we get that there exists a generic constants
C > ε such that(5.3) Z Z h ε ( ρ ε ) | Au ε | ≤ C, Z Z h ′ ε ( ρ ε ) |∇ ρ ε | ρ γ − ε ≤ C, Z Z h ′ ε ( ρ ε ) |∇ ρ ε | f ′′ ε ( ρ ) ≤ C, Z Z h ε ( ρ ε ) |∇ φ ε ( ρ ε ) | ≤ C, where we have used (3.6) and the fact that Aw ε = Au ε . In particular combining (5.1), (5.2),(5.3) and (3.3) we have(5.4) Z Z h ε ( ρ ε ) |∇ u ε | ≤ C, Z Z ρ ε |∇ u ε | ≤ C. UANTUM NAVIER-STOKES 15
Next we consider the pressure. From (5.1) and (5.3), after using (3.3), we get(5.5)
Z Z |∇ ρ γ ε | ≤ C and then by interpolation with (5.1) we have that(5.6) Z Z ρ γ ε ≤ C for any γ > d = 2 and for any γ ∈ (1 ,
3) if d = 3 . Finally, we have the following uniform bounds(5.7)
Z Z |∇ √ ρ | + |∇ ρ | ≤ C. The bounds (5.7) are crucial to handle the passage to the limit in the dispersive term and arenot a straightforward consequence of the a priori estimates. Indeed in order to obtain themwe need a generalization of the inequality
Z Z |∇ √ ρ | + |∇ ρ | ≤ C Z Z ρ |∇ log ρ | proved in [21], see also [31] for an alternative proof. Lemma 5.1.
Let ρ > and h ( ρ ) be a smooth function such that (5.8) h ( ρ ) ≥ , h ′ ( ρ ) > , h ′ ( ρ ) ρ ≤ Ch ( ρ ) , | h ′′ ( ρ ) | ρ ≤ Ch ′ ( ρ ) Then, the following inequality hold (5.9)
Z Z h ′ ( ρ ) |∇ ( h ′ ( ρ ) ∇√ ρ ) | + Z Z ( h ′ ( ρ )) |∇√ ρ | ρ ≤ C Z Z h ( ρ ) |∇ φ ( ρ ) | where ρφ ′ ( ρ ) = h ′ ( ρ ) . Moreover, if in addition we assume that h ′ ( ρ ) ≥ c > then (5.10) Z Z |∇ √ ρ | + Z Z |∇ ρ | ≤ C Z Z h ( ρ ) |∇ φ ( ρ ) | . Proof.
By using (5.8) we have that(5.11)
Z Z ρh ′ ( ρ ) (cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:18) h ′ ( ρ ) ∇√ ρ √ ρ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = Z Z ρh ′ ( ρ ) |∇ φ ( ρ ) | ≤ C Z Z h ( ρ ) |∇ φ ( ρ ) | , where h ′ ( ρ ) = ρφ ′ ( ρ ) has been used. Then, by using the chain rule we have(5.12) ∇ (cid:18) h ′ ( ρ ) ∇√ ρ √ ρ (cid:19) = ∇ ( h ′ ( ρ ) ∇√ ρ ) √ ρ − h ′ ( ρ ) ∇√ ρ ⊗ ∇√ ρρ . Taking the square we get(5.13) (cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:18) h ′ ( ρ ) ∇√ ρ √ ρ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = |∇ ( h ′ ( ρ ) ∇√ ρ ) | ρ − ∇ ( h ′ ( ρ ) ∇√ ρ ) h ′ ( ρ ) ∇√ ρ ⊗ ∇√ ρ √ ρρ + ( h ′ ( ρ )) |∇√ ρ | ρ . Then, by using (5.11)(5.14)
Z Z h ′ ( ρ ) |∇ ( h ′ ( ρ ) ∇√ ρ ) | + ( h ′ ( ρ )) |∇√ ρ | ρ ≤ C Z Z h ( ρ ) |∇ φ ( ρ ) | + 2 Z Z ρh ′ ( ρ ) ∇ ( h ′ ( ρ ) ∇√ ρ ) h ′ ( ρ ) ∇√ ρ ⊗ ∇√ ρ √ ρρ Now we focus on the last term of (5.14). By integrating by parts we get
Z Z h ′ ( ρ ) ∇ ( h ′ ( ρ ) ∇√ ρ ) h ′ ( ρ ) ∇√ ρ ⊗ ∇√ ρ √ ρ = Z Z ∂ i ( h ′ ( ρ ) ∂ j √ ρ ) h ′ ( ρ ) ∂ i √ ρh ′ ( ρ ) ∂ j √ ρ √ ρ = − Z Z ∂ i ( h ′ ( ρ ) ∂ j √ ρ ) h ′ ( ρ ) ∂ i √ ρh ′ ( ρ ) ∂ j √ ρ √ ρ − Z Z h ′ ( ρ ) ∂ j √ ρ∂ i (cid:18) h ′ ( ρ ) ∂ i √ ρ √ ρ (cid:19) h ′ ( ρ ) ∂ j √ ρ. Then,(5.15) 2
Z Z h ′ ( ρ ) ∇ ( h ′ ( ρ ) ∇√ ρ ) h ′ ( ρ ) ∇√ ρ ⊗ ∇√ ρ √ ρ = − Z Z h ′ ( ρ ) ∇√ ρ div (cid:18) h ′ ( ρ ) ∇√ ρ √ ρ (cid:19) h ′ ( ρ ) ∇√ ρ = − Z Z √ ρ p h ′ ( ρ ) p h ′ ( ρ ) √ ρ ∇√ ρ div (cid:18) h ′ ( ρ ) ∇√ ρ √ ρ (cid:19) h ′ ( ρ ) ∇√ ρ ≤ C Z Z ρh ′ ( ρ ) (cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:18) h ′ ( ρ ) ∇√ ρ √ ρ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + 12 Z Z ( h ′ ( ρ )) |∇√ ρ | ρ . Then, by (5.14) we get(5.16)
Z Z h ′ ( ρ ) |∇ ( h ′ ( ρ ) ∇√ ρ ) | + Z Z ( h ′ ( ρ )) |∇√ ρ | ρ ≤ C Z Z h ( ρ ) |∇ φ ( ρ ) | . Now we prove the second part of the Lemma. Assuming that h ′ ( ρ ) > c it is straightforwardto prove that Z Z |∇ ρ | ≤ C Z Z h ( ρ ) |∇ φ ( ρ ) | . By using the chian rule we have(5.17)
Z Z h ′ ( ρ ) |∇ ( h ′ ( ρ ) ∇√ ρ ) | = Z Z h ′ ( ρ ) | h ′ ( ρ ) ∇ √ ρ + 2 h ′′ ( ρ ) √ ρ ∇√ ρ ⊗ ∇√ ρ | = Z Z h ′ ( ρ ) | h ′ ( ρ ) ∇ √ ρ | + 4 Z Z h ′ ( ρ ) h ′ ( ρ ) ∇ √ ρh ′′ ( ρ ) ρ ∇√ ρ ⊗ ∇√ ρ √ ρ + 4 Z Z h ′ ( ρ ) | h ′′ ( ρ ) √ ρ ∇√ ρ ⊗ ∇√ ρ | . By using the fact that | h ′′ ( ρ ) ρ | ≤ Ch ′ ( ρ ) we get(5.18) Z Z h ′ ( ρ ) | h ′ ( ρ ) ∇ √ ρ | ≤ C Z Z h ′ ( ρ ) |∇ ( h ′ ( ρ ) ∇√ ρ ) | + 4 (cid:12)(cid:12)(cid:12)(cid:12)Z Z Z Z h ′ ( ρ ) h ′ ( ρ ) ∇ √ ρh ′′ ( ρ ) ρ ∇√ ρ ⊗ ∇√ ρ √ ρ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z Z h ′ ( ρ ) |∇ ( h ′ ( ρ ) ∇√ ρ ) | + C Z Z ( h ′ ( ρ )) |∇√ ρ | ρ + 12 Z Z h ′ ( ρ ) | h ′ ( ρ ) ∇ √ ρ | . UANTUM NAVIER-STOKES 17
Then, by using (5.16) we get(5.19)
Z Z h ′ ( ρ ) | h ′ ( ρ ) ∇ √ ρ | ≤ C Z Z h ( ρ ) |∇ φ ( ρ ) | . Then, since h ′ ( ρ ) > c we have that(5.20) Z Z |∇ √ ρ | ≤ C Z Z h ( ρ ) |∇ φ ( ρ ) | . Summing up (5.19) and (5.20) we get (5.10). (cid:3)
Preliminary Lemma
In the following lemma we prove the main convergences needed in the proof of the mainTheorems.
Lemma 5.2.
Let { ( ρ ε , u ε ) } ε be a sequence of solutions of (3.1) . Then up to subsequencesthere exists a function √ ρ such that √ ρ ε → √ ρ strongly in L (0 , T ; H ( T d )) . (5.21) Proof.
Let us consider the first equation in (3.1). Since, by Proposition 6.4 ρ ε > ∂ t √ ρ ε = − √ ρ ε u ε − div( √ ρ ε u ε ) + ∇ u ε √ ρ ε , and, by the uniform bounds we have in (5.2) and (5.4), we have that { ∂ t √ ρ ε } ε is uniformly bounded in L (0 , T ; H − ( T d )) . Then, since {√ ρ ε } ε is uniformly bounded in L (0 , T ; H ( T d )) by using Aubin-Lions Lemmawe get (5.21). (cid:3) Lemma 5.3.
Let { ( ρ ε , u ε ) } ε be a sequence of solutions of (3.1) . Then h ′ ε ( ρ ε ) √ ρ ε → √ ρ strongly in L ((0 , T ) × T d )) , (5.23) h ′ ε ( ρ ε ) ∇√ ρ ε → ∇√ ρ strongly in L ((0 , T ) × T d ) , (5.24) h ′′ ε ( ρ ε ) ρ ε ∇√ ρ ε → strongly in L ((0 , T ) × T d ) . (5.25) Proof.
Let us start by proving (5.23). By using (3.3) we get
Z Z | h ′ ε ( ρ ε ) √ ρ ε − √ ρ ε | ≤ Z Z |√ ρ ε − √ ρ | + ε Z Z ρ ε + ε Z Z ρ γ − ε . The first term goes to zero because of (5.21). The second term, simply by using H¨olderinequality and the uniform bound (5.1). Finally, for the last term we have that when d = 2there for any γ > δ = δ ( γ ) small enough such that 2 γ − < (2 − δ ) γ and them the integral is bounded because ρ γε ∈ L r ((0 , T ) × T d ) for any r <
2. When d = 3since γ ∈ (1 ,
3) it holds that 2 γ − < γ . Then, the third term goes to zero by using H¨olderinequality and (5.6). To prove (5.24) we have
Z Z | h ′ ε ( ρ ε ) ∇√ ρ ε − ∇√ ρ | ≤ Z Z |∇√ ρ ε − ∇√ ρ | + Cε Z Z ρ − ε |∇√ ρ ε | + Cε Z Z ρ γ − ε |∇√ ρ ε | . Then, the first term goes to 0 because of (5.21). Concerning the second term we have ε Z Z ρ − ε |∇√ ρ ε | ≤ ε Z Z ρ ε |∇ ρ ε | . Then, by H¨older inequality ε Z Z ρ − ε |∇√ ρ ε | ≤ Cε (cid:18)Z Z √ ρ ε (cid:19) (cid:18)Z Z |∇ ρ ε | (cid:19) ≤ Cε . Now, we treat the last term. From (5.3) we have that
Z Z |∇ ρ ε | h ′ ε ( ρ ε ) ρ γ − ε ≤ C. Then, by using (3.3) we get ε Z Z |∇ ρ ε | ρ γ − ε = 4 ε Z Z |∇√ ρ ε | ρ γ − ε ≤ C, which implies the convergence of the last term. Finally concerning (5.25), by using again(3.3), we have Z Z | h ′′ ε ( ρ ε ) ρ ε ∇√ ρ ε − ∇√ ρ | ≤ Cε Z Z ρ − ε |∇√ ρ ε | + Cε Z Z ρ γ − ε |∇√ ρ ε | . Then, it goes to zero arguing as above. (cid:3)
Lemma 5.4.
Let { ( ρ ε , u ε ) } ε be a sequence of solutions of (3.1) then h ε ( ρ ε ) − ρ ε → in L ((0 , T ) × T d ) , (5.26) g ε ( ρ ε ) → in in L ((0 , T ) × T d ) . (5.27) Proof.
By (3.3) and (3.4) we have that
Z Z | h ε ( ρ ε ) − ρ ε | ≤ Cε Z Z ρ ε + Cε Z Z ρ γε , Z Z | g ε ( ρ ε ) | ≤ Cε Z Z ρ ε + Cε Z Z ρ γε . Then, we conclude by using H¨older inequality and (5.1).
Lemma 5.5.
Let { ( ρ ε , u ε ) } ε be a sequence of solutions of (3.1) then (5.28) ρ γε → ρ γ in L ((0 , T ) × T d ) ,p ε ( ρ ε ) → in L ((0 , T ) × T d ) , ˜ p ε ( ρ ε ) → in L ((0 , T ) × T d ) . UANTUM NAVIER-STOKES 19
Proof.
The convergence of ρ γε follows from (5.21) and the bound (5.5). By the definition of p ε we have that there exists a generic constant C independent on ε such that(5.29) Z Z | p ε ( ρ ε ) | ≤ C X i =1 Z Z | p iε ( ρ ε ) | . Let us recall from (5.1)(5.30) sup t ε λ ( ε ) (cid:18)Z ρ ε + γ − ε + ρ − ε − ε (cid:19) ≤ C. We start by estimating p ε ( ρ ε ), by (3.9) and H´older inequality we have(5.31) Z Z | p ε ( ρ ε ) | ≤ Cε λ ( ε ) Z Z ρ ε ε ≤ Cε λ ( ε ) (cid:18)Z Z ρ ε + γ − ε (cid:19) ε γ − . Then, by using that λ ( ε ) = e − /ε we get Z p ε ( ρ ε ) ≤ C ε ε / (1+ ε ( γ − e − (cid:16) ( γ − ε ε γ − (cid:17) (cid:18) ε λ ( ε ) sup t Z ρ ε + γ − ε (cid:19) ε γ − and then by using (5.30) we get the convergence to 0 of p ε ( ρ ε ). The term p ε ( ρ ε ) is treated atthe same way. Now, we deal with convergence of the term p ε ( ρ ε ). First of all, we have that Z Z | p ε ( ρ ε ) | ≤ Cε λ ( ε ) Z Z ρ ε + γ − ε . Then, we recall from (5.3) the following uniform bound(5.32)
Z Z |∇ ρ ε | h ′ ε ( ρ ε ) f ′′ ε ( ρ ε ) ≤ C which contains the following uniform bound ε λ ( ε ) Z Z |∇ ρ ε | ρ ε +2 γ − ε ≤ C which means that ε λ ( ε ) Z Z (cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:18) ρ ε + γ − ε (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C. Then, by Sobolev embedding ε λ ( ε ) Z (cid:18)Z ρ ε +6 γ − ε (cid:19) dt ≤ C. Now, by H¨older inequality we get
Z Z | p ε ( ρ ε ) | ≤ Cε λ ( ε ) Z (cid:18)Z ρ ε +6 γ − ε (cid:19) dt ! ε γ − ε γ − ≤ C ε ε ε γ − ε γ − e − ( γ − ε ε γ − ε λ ( ε ) Z (cid:18)Z ρ ε +6 γ − ε (cid:19) dt ! ε γ − ε γ − . Then, we have that p ε ( ρ ε ) vanishes as ε goes to 0. Let us consider the term p ε ( ρ ε ). We have Z Z | p ε ( ρ ε ) | ≤ Cε λ ( ε ) Z Z ρ − ε ε ≤ Cε λ ( ε ) (cid:18)Z Z ρ − ε − ε (cid:19) ε ≤ C ε ε ε e − ε ε (cid:18) ε λ ( ε ) sup t Z ρ − ε − ε (cid:19) ε . Then we get that p ε ( ρ ε ) goes to 0. Now, we consider the term p ε ( ρ ε ). By (3.9) and H¨olderinequality(5.33) Z Z | p ε ( ρ ε ) | ≤ Cε λ ( ε ) Z Z ρ − ε − ε . In the bound (5.32) is contained the following bound ε λ ( ε ) Z Z ρ − − ε |∇ ρ ε | ρ − ε − − ε ≤ C which means that ε λ ( ε ) Z Z (cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:18) ρ − ε − ε (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C which by Sobolev embedding implies(5.34) ε λ ( ε ) Z (cid:18)Z ρ − ε − ε dx (cid:19) dt ≤ C. Then,(5.35)
Z Z | p ε ( ρ ε ) | ≤ C ε ε ε ε e − ε ε ε ε λ ( ε ) Z (cid:18)Z ρ − ε − ε (cid:19) dt ! ε ε . and then p ε ( ρ ε ) vanishes as ε goes to 0. Finally, the term p ε ( ρ ε ) is treated as the term p ε ( ρ ε ).The same proof of the convergence of the term p ε ( ρ ε ) and p ε ( ρ ε ) show the convergence of thedamping coefficient ˜ p ε ( ρ ε ). (cid:3) Lemma 5.6.
Let { ( ρ ε , u ε ) } ε be a sequence of solutions of (3.1) then up to subsequences thereexists a vector m such that (5.36) ρ ε u ε → m strongly in L (0 , T ; L p ( T d )) with p ∈ [1 , / . Proof.
To prove (5.36) we first notice that from the bounds (5.1)(5.37) {∇ ( ρ ε u ε ) } ε is uniformly bounded in L (0 , T ; L ( T d )) . Then, we need to estimate the time derivative of ρ ε u ε . Precisely, we are going to prove that(5.38) Z k ∂ t ( ρ ε u ε ) k W − , ≤ C By using the first equation in (3.1) we get(5.39) ∂ t ( ρ ε u ε ) = − div( ρ ε u ε ⊗ u ε ) − ∇ ρ γε − ∇ p ε ( ρ ε ) − ˜ p ε ( ρ ε ) u ε + 2 ν div( h ε ( ρ ε ) Du ε ) + 2 ν ∇ ( g ε ( ρ ε ) div u ε )+ κ div( h ε ( ρ ε ) ∇ ( φ ε ( ρ ε )) + ∇ ( g ε ( ρ ε )∆ φ ε ( ρ ε ))= X i =1 I εi First of all we notice that(5.40) | g ε ( ρ ε ) | ≤ Ch ε ( ρ ε ) h ε ( ρ ε ) is uniformly bounded in L t,x . Then, we estimates each term. From (5.1) we have that(5.41) { I ε } ε is uniformly bounded in L (0 , T ; W − , ( T d )) . By using Lemma (pressure) we get that for i = 2 , , { I εi } ε is uniformly bounded in L (0 , T ; W − , ( T d )) . UANTUM NAVIER-STOKES 21
Regarding the stress tensor by using (5.40) and (5.2) we have for i = 5(5.43) { I ε } ε is uniformly bounded in L (0 , T ; W − , ( T d )) . Then, by using the (5.3) and (5.40) we have also that for i = 7 , { I εi } ε is uniformly bounded in L (0 , T ; W − , ( T d )) . Then, by using Aubin-Lions the lemma is proved. (cid:3)
Lemma 5.7.
Let ( ρ ε , u ε ) be a sequence of solutions of (1.1) and let w ε = u ε + c ∇ φ ε ( ρ ε ) .Then, up to subsequences we have that (5.45) √ ρ ε u ε → √ ρu strongly in L ((0 , T ) × T d ) , where u is defined m/ρ on { ρ > } and on { ρ = 0 } .Proof. Let us consider the Mallet-Vasseur type estimate (4.17) in Proposition 4.4. By using(5.1)-(5.5) and by taking δ > t ∈ (0 ,T ) Z ρ ε (cid:18) | w ε | (cid:19) log (cid:18) | w ε | (cid:19) ≤ C. By Lemma 5.2 and Lemma 5.3 we can extract a further subsequence such that(5.47) √ ρ ε → √ ρ a.e. in (0 , T ) × T d , ∇√ ρ ε → ∇√ ρ a.e. in (0 , T ) × T d ,h ′ ε ( ρ ε ) ∇√ ρ ε → ∇√ ρ a.e. in (0 , T ) × T d ,m ,ε = ρ ε u ε → m a.e. in (0 , T ) × T d . Then, it follows that(5.48) m ,ε := m ,ε + 2 µ √ ρ ε h ′ ε ( ρ ε ) ∇√ ρ ε → m + 2 µ √ ρ ∇√ ρ =: m , a.e. in (0 , T ) × T d . Arguing as in [29] by using (5.1)-(5.4) and Fatou Lemma we have that(5.49) Z Z lim inf ε m ,ε ρ ε ≤ lim inf ε Z Z m ,ε ρ ε < ∞ . This implies that m = 0 a.e. on { ρ = 0 } . Let us define the following limit velocity u = m ρ on { ρ > } , { ρ = 0 } . In this way we have that m = ρu and m / √ ρ ∈ L ∞ (0 , T ; L ( T d )). Then from (5.48) wehave that m = m + 2 µ √ ρ ∇√ ρ and since ∇√ ρ is finite almost everywhere we also havethat m = 0 on the set { ρ = 0 } . This in turn implies that after defining the following limitvelocity w = m ρ + 2 µ √ ρ ∇√ ρρ on { ρ = 0 } { ρ = 0 } , we have that m = ρw and m √ ρ = √ ρu + 2 µ ∇√ ρ ∈ L ∞ (0 , T ; L ( T d )) . Now we can prove (5.45). First, by using (5.47), (5.46) and Fatou Lemma we get that(5.50) sup t Z ρ | w | log (cid:18) | w | (cid:19) ≤ sup t Z ρ ε | w ε | log (cid:18) | w ε | (cid:19) ≤ C. Then, we note that for any fixed
M > √ ρ ε w ε χ | w ε |≤ M → √ ρwχ | w |≤ M a.e. in (0 , T ) × Ω. Indeed, in { ρ = 0 } it holds(5.52) √ ρ ε w ε = m ,ε √ ρ ε + h ′ ε ( ρ ε ) ∇√ ρ ε → m √ ρ + ∇√ ρ a.e.While, in { ρ = 0 } we have(5.53) |√ ρ ε w ε χ | w ε | Let ( ρ , u ) be initial data for (1.1) satisfying (2.2) and (2.3) and let ( ρ ε , u ε ) be the sequenceof initial data constructed in Section 3 satisfying (3.11). For any ε < ε f by using Theorem6.1 in the two dimensional case and Theorem 6.2 in the three dimensional one, there existsa sequence of global smooth solutions { ( ρ ε , u ε ) } ε , ρ ε > 0, of (3.1)-(3.2) and ( ρ, u ), with u defined zero on the set { ρ = 0 } , such that the convergences stated in Lemma 5.2-5.7 hold.We still denote with ( ρ ε , u ε ) the subsequence chosen in the convergence lemma. Let us provethat ( ρ, u ) is a finite energy weak solution of (1.1)-(1.2). Let us consider the first equationof (3.1). ∂ t ρ ε + div( ρ ε u ε ) = 0 . The convergence to the weak formulation of (1.1) holds because of Lemma 5.2 and Lemma5.6. Next, let us consider the momentum equation ∂ t ( ρ ε u ε ) + div( ρ ε u ε ⊗ u ε ) − ν div( ρ ε Du ε ) + ∇ ρ γε − κ div K ε , = 2 ν div(( h ε ( ρ ε ) − ρ ε ) Du ε ) + 2 ν ∇ ( g ε ( ρ ε ) div u ε ) − ∇ p ε ( ρ ε ) − ˜ p ε ( ρ ε ) u ε . UANTUM NAVIER-STOKES 23 Then, Z Z | h ε ( ρ ε ) − ρ ε || Du ε | ≤ (cid:18)Z Z | h ε ( ρ ε ) − ρ ε | (cid:19) (cid:18)Z Z | h ε ( ρ ε ) || Du ε | + | ρ ε || Du ε | (cid:19) and this term converges to zero because of Lemma 5.4, (5.1) and (5.2). Then, Z Z | g ε ( ρ ε ) || div u ε | ≤ (cid:18)Z Z | g ε ( ρ ε ) | (cid:19) (cid:18)Z Z | g ε ( ρ ε ) || div u ε | (cid:19) (cid:18)Z Z | g ε ( ρ ε ) | (cid:19) (cid:18)Z Z | h ε ( ρ ε ) ||∇ u ε | (cid:19) and this term converges to zero because of Lemma 5.4 and (5.4). Note that (3.6) has beenused. Then, the pressure term p ε ( ρ ε ) goes to 0 because of Lemma 5.5. Concerning thedamping term we have Z Z | ˜ p ε ( ρ ε ) u ε | ≤ (cid:18)Z Z | ˜ p ε ( ρ ε ) | (cid:19) (cid:18)Z Z | ˜ p ε ( ρ ε ) || u ε | (cid:19) ≤ C (cid:18)Z Z | ˜ p ε ( ρ ε ) | (cid:19) which goes to zero thanks to Lemma 5.5. Now, we consider the terms in the left-handside. The only convergence to prove is the convergence in the dispersive term. Indeed,since the strong convergence in L t,x of √ ρ ε u ε holds the convergence of the other terms isstraightforward, see [3] for more details. Let us consider the following term where φ ∈ C ∞ ([0 , T ) × T d ) Z Z div K ε · φ = Z Z ∇ ( h ′ ε ( ρ ε )∆ h ε ( ρ ε )) φ − Z Z div( h ′ ε ( ρ ε ) ∇√ ρ ε ⊗ h ′ ε ( ρ ε ) ∇√ ρ ε ) φ = 4 Z Z h ′′ ε ( ρ ε ) ρ ε h ′ ε ( ρ ε ) |∇√ ρ ε | div φ + 2 Z Z h ′ ε ( ρ ε ) √ ρ ε h ′ ε ( ρ ε ) ∇√ ρ ε ∇ div φ + 4 Z Z h ′ ε ( ρ ε ) ∇√ ρ ε ⊗ h ′ ε ( ρ ε ) ∇√ ρ ε ∇ φ and by using Lemma 5.3 it easy to conclude that4 Z Z h ′′ ε ( ρ ε ) ρ ε h ′ ε ( ρ ε ) |∇√ ρ ε | div φ → , Z Z h ′ ε ( ρ ε ) √ ρ ε h ′ ε ( ρ ε ) ∇√ ρ ε ∇ div φ → Z Z √ ρ ∇√ ρ ∇ div φ, Z Z h ′ ε ( ρ ε ) ∇√ ρ ε ⊗ h ′ ε ( ρ ε ) ∇√ ρ ε ∇ φ → Z Z ∇√ ρ ⊗ ∇√ ρ : ∇ φ. (cid:3) Global Regularity for the approximating system In this section we prove the global in time existence of smooth solutions for the approxi-mating system (3.1). In the two-dimensional case the following theorem holds: Theorem 6.1. Let ν, κ > such that κ < ν and γ > . Then for ε < ε f = ε f ( ν, κ, ν, γ ) there exists a global smooth solution of (3.1) - (3.2) . Concerning the three dimensional case, we have the following result Theorem 6.2. Let ν, κ > such that κ < ν < (9 / κ and γ ∈ (1 , . Then, for ε < ε f = ε f ( ν, κ, ν, γ ) there exists a global smooth solution of (3.1) - (3.2) . In order to prove Theorems 6.1 and 6.2 the main estimate to show is that the density isbounded from above and below, namely (6.7). Indeed, the higher order a priori estimatecan be done with minor change as in [27], Lemma 2.5. Then, Theorem 6.1 and Theorem6.2 follow by a standard continuity argument on local smooth solutions of (3.1)-(3.2) with ρ ε > 0. To prove (6.7) we exploit the fact that the first equation in (4.11) is uniformlyparabolic since h ′ ε ( ρ ε ) ≥ 1. Then, to apply the parabolic regularity estimates we need that ρ ε w ε , w ε /ρ ε ∈ L ∞ t ( L px ) with p > d . On the other hand we already know by the energyestimate that ρ ε and 1 /ρ ε are L ∞ t ( L qx ) for some very large q depending on ε . Then, it sufficesto infer that ρ ε | w ε | d + δ is in L ∞ t ( L x ). This is proved in the next Lemma and it is exactly herethat we need the restriction on κ and ν in three dimensions. This is due to the fact that inthe second equation of (4.11) there is the symmetric part of the gradient of w ε . Lemma 6.3. Let ( ρ ε , u ε ) be a smooth solution of 3.1. Then, ( ρ ε , w ε ) satisfies the followingestimates:In the two dimensional case, for any γ > and ν, κ > there exists a small ¯ δ = ¯ δ ( γ, ν, κ ) and a constant C , possibly depending on ε , such that for any δ < ¯ δ (6.1) sup t Z ρ ε | w ε | δ ≤ C. In the three dimensional case, for any γ ∈ (1 , and ν, κ > such that κ < ν < κ thereexists a constant C , possibly depending on ε , such that (6.2) sup t Z ρ ε | w ε | δ ≤ C. Proof. For convenience of the reader we write again the integral equality of Lemma 4.3 ddt Z ρ ε β (cid:18) | w ε | (cid:19) + µ Z h ε ( ρ ε ) | Aw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + µ Z h ε ( ρ ε ) | Aw ε | β ′ (cid:18) | w ε | (cid:19) + (2 ν − µ ) Z h ε ( ρ ε ) | Dw ε | β ′ (cid:18) | w ε | (cid:19) + (2 ν − µ ) Z g ε ( ρ ε ) | div w ε | β ′ (cid:18) | w ε | (cid:19) + Z ˜ p ε ( ρ ε ) | w ε | β ′ (cid:18) | w ε | (cid:19) + (2 ν − µ ) Z h ε ( ρ ε ) | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) = − Z ∇ ρ γε w ε β ′ (cid:18) | w ε | (cid:19) − ν Z h ε ( ρ ε )( Dw ε · w ε ) · ( Aw ε · w ε ) β ′′ (cid:18) | w ε | (cid:19) − (2 ν − µ ) Z g ε ( ρ ε ) div w ε w ε · ( Dw ε · w ε ) β ′′ (cid:18) | w ε | (cid:19) . UANTUM NAVIER-STOKES 25 Let β ( t ) = t δ and with δ > 0. Then we have that β ′ ( t ) = β ′′ ( t ) tδ . By integrating by partsthe pressure term and using Young inequality we get ddt Z ρβ (cid:18) | w ε | (cid:19) + µ Z h ε ( ρ ε ) | Aw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + (2 ν − µ ) Z h ε ( ρ ε ) | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + (2 ν − µ ) Z h ε ( ρ ε ) | Dw ε | β ′ (cid:18) | w ε | (cid:19) + (2 ν − µ ) Z g ε ( ρ ε ) | div w ε | β ′ (cid:18) | w ε | (cid:19) + µ Z h ε ( ρ ε ) | Aw ε | β ′ (cid:18) | w ε | (cid:19) + Z ˜ p ε ( ρ ε ) | w ε | β ′ (cid:18) | w ε | (cid:19) ≤ δ Z ρ γε div w ε | w ε | β ′′ (cid:18) | w ε | (cid:19) + Z ρ γε w ε ( Dw ε · w ε ) β ′′ (cid:18) | w ε | (cid:19) + ν Z h ε ( ρ ε ) | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + ν Z h ε ( ρ ε ) | Aw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + (2 ν − µ )( γ − ε Z ρ γε | div w ε | | w ε | β ′′ (cid:18) | w ε | (cid:19) + (2 ν − µ )( γ − ε Z ρ γε | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + 3(2 ν − µ )16 ε Z ρ ε | Dw ε | | w ε | β ′′ (cid:18) | w ε | (cid:19) + 2 ν − µ ε Z ρ ε | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) . Then, by writing everything in term of β ′′ (cid:16) | w ε | (cid:17) , using (3.6) and the fact that | Aw ε · w ε | ≤| Aw ε | | w ε | we have ddt Z ρ ε β (cid:18) | w ε | (cid:19) + (cid:16) µ + µ δ (cid:17) Z h ε ( ρ ε ) | Aw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + (2 ν − µ ) Z h ε ( ρ ε ) | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + 2 ν − µ δ ε Z ρ ε | Dw ε | | w ε | β ′′ (cid:18) | w ε | (cid:19) + (cid:18) ν − µ δ (cid:19) ε Z ρ γε | Dw ε | | w ε | β ′′ (cid:18) | w ε | (cid:19) + 5(2 ν − µ )16 δ ε Z ρ ε | Dw ε | | w ε | β ′′ (cid:18) | w ε | (cid:19) + (2 ν − µ )( γ − δ ε Z ρ γε | div w ε | | w ε | β ′′ (cid:18) | w ε | (cid:19) ≤ C ( τ )2 δ Z ρ γ − ε | w ε | β ′′ (cid:18) | w ε | (cid:19) + 3 τ δ Z ρ ε | Dw ε | | w ε | β ′′ (cid:18) | w ε | (cid:19) + C ( α ) Z ρ γ − ε | w ε | β ′′ (cid:18) | w ε | (cid:19) + α Z ρ ε | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + ν Z h ε ( ρ ε ) | Aw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + (cid:18) ν + (2 ν − µ )16 (cid:19) ε Z ρ ε | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + (cid:18) ν + (2 ν − µ )( γ − (cid:19) ε Z ρ γε | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + ν Z ρ ε | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + (2 ν − µ )( γ − ε Z ρ γε | div w ε | | w ε | β ′′ (cid:18) | w ε | (cid:19) + 3(2 ν − µ )16 ε Z ρ ε | Dw ε | | w ε | β ′′ (cid:18) | w ε | (cid:19) . Finally, by absorbing terms from the right hand-side to the left hand-side we get(6.3) ddt Z ρ ε β (cid:18) | w ε | (cid:19) + (cid:16) µ − ν + µ δ (cid:17) Z h ε ( ρ ε ) | Aw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + ( ν − µ ) Z ρ ε | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + (cid:18) ν − µ − (2 ν − µ )( γ − (cid:19) ε Z ρ γε | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + (cid:18) ν − µ (cid:19) ε Z ρ ε | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + 2 ν − µ δ Z ρ ε | Dw ε | | w ε | β ′′ (cid:18) | w ε | (cid:19) + (cid:18) ν − µ δ (cid:19) ε Z ρ γε | Dw ε | | w ε | β ′′ (cid:18) | w ε | (cid:19) + (cid:18) ν − µ )16 δ − ν − µ )16 (cid:19) ε Z ρ ε | Dw ε | | w ε | β ′′ (cid:18) | w ε | (cid:19) + (2 ν − µ )( γ − (cid:18) δ − (cid:19) ε Z ρ γε | div w ε | | w ε | β ′′ (cid:18) | w ε | (cid:19) ≤ C ( τ )2 δ Z ρ γ − ε | w ε | β ′′ (cid:18) | w ε | (cid:19) + 3 τ δ Z ρ ε | Dw ε | | w ε | β ′′ (cid:18) | w ε | (cid:19) + C ( α ) Z ρ γ − ε | w ε | β ′′ (cid:18) | w ε | (cid:19) + α Z ρ ε | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) By considering δ ∈ (0 , 1] and choosing τ = (2 ν − µ ) / α = ν − µ , after using Younginequality we get(6.4) ddt Z ρ ε | w ε | δ + (cid:16) µ − ν + µ δ (cid:17) Z h ε ( ρ ε ) | Aw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + (cid:18) ν − µ + 2 ν − µ (cid:18) δ − ( γ − (cid:19)(cid:19) ε Z ρ γε | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) + (cid:18) ν − µ (cid:19) ε Z ρ ε | Dw ε · w ε | β ′′ (cid:18) | w ε | (cid:19) ≤ C ( ν, κ, δ ) (cid:18)Z ρ ˜ γε + Z ρ ε | w ε | δ (cid:19) with ˜ γ := (cid:18) γ − − δ δ (cid:19) (1 + δ ) > δ ∈ (0 , . Now we consider the two dimensional case. Given ν, κ > ν > κ and γ > δ small enough µ − ν + µ δ > δ − ( γ − > . By Proposition 4.1 we have that sup t Z ρ ε ε ≤ C ( ε ) . UANTUM NAVIER-STOKES 27 Then, by choosing if needed ε f small enough such that for any ε < ε f it holds ˜ γ < /ε wearrive at ddt Z ρ ε | w ε | δ ≤ C ( ε, κ, µ, ν ) + Z ρ ε | w ε | δ and we get (6.1) by Gronwall Lemma. Then, we consider the three dimensional case. Sinceit seems not possible to avoid a restriction on ν, κ we do not aim to optimality, which can beobtained by optimizing the the Young inequalities and minimizing in δ . Going back to (6.4)we argue as follows. First we note that for any δ ∈ (0 , 1) it holds(6.5) µ − ν + µ δ > µ − ν and the left-hand side of (6.5) is positive if the following restriction on ν and κ holds:(6.6) κ < ν < κ Then, by using that γ ∈ (1 , 3) we can choose δ = min { / ( γ − , } . Notice that δ ∈ (1 / , ν − µ + 2 ν − µ (cid:18) δ − ( γ − (cid:19) > δ = + δ ′ , with an abuse of notation avoiding the prime, we have ddt Z ρ ε | w ε | δ ≤ C ( ν, κ, δ ) (cid:18)Z ρ ˜ γε + Z ρ ε | w ε | δ (cid:19) Then, by choosing if needed ε f small enough such that for any ε < ε f it holds ˜ γ < /ε weget (6.2) by using Gronwall Lemma. We stress that ε f depends only on γ, ν and κ . (cid:3) Now, we are in position to give the proof of Proposition 6.4. Proposition 6.4. Let ( ρ ε , u ε ) be a smooth solution of the system (3.1) . then, there exists aconstant C > possibly dependent on ε such that (6.7) 1 C ≤ ρ ε ≤ C Proof. First we want to prove that(6.8) ρ ε w ε and w ε ρ ε ∈ L ∞ (0 , T ; L p ( T d ) with p > d When d = 2, by H¨older and Young inequality we get Z | ρ ε w ε | δ ≤ Z ρ ˜ δε + Z ρ ε | w ε | δ with ˜ δ = ˜ δ ( γ, ν, κ ) > 0. Then by choosing if needed ε small enough such that ˜ δ < /ε weget the desired estimate. Then, (cid:12)(cid:12)(cid:12)(cid:12) | w ε | ρ ε (cid:12)(cid:12)(cid:12)(cid:12) δ = (cid:12)(cid:12)(cid:12)(cid:12) ρ ε (cid:12)(cid:12)(cid:12)(cid:12) δ δ | ρ ε | δ δ | w ε | δ . Then, by H¨older and Young inequality Z (cid:12)(cid:12)(cid:12)(cid:12) | w ε | ρ ε (cid:12)(cid:12)(cid:12)(cid:12) δ ≤ Z (cid:12)(cid:12)(cid:12)(cid:12) ρ ε (cid:12)(cid:12)(cid:12)(cid:12) ˜ δ + Z ρ ε | w ε | δ , with ˜ δ = (cid:18) δ δ (cid:19) (cid:18) δ δ (cid:19) ∗ . Then, again by choosing if necessary ε small enough such that ˜ δ ≤ /ε we get the desiredestimate. Now we are in position to prove (6.7). The proof is standard and it is based on De Giorgi type estimate. We use the same approach as in [27], Lemma 2.4. Let us start byproving that ρ ε is bounded. Let m ε = ρ ε w ε . Then the first equation in (4.11) is the following(6.9) ∂ t ρ ε − div( h ′ ε ( ρ ε ) ∇ ρ ε ) = div m ε . Let k > k ρ ε k ∞ and A k ( t ) = { ρ ε > k } . Then by using H´older inequality and the fact that h ′ ε ( ρ ε ) > ddt Z | ( ρ ε − k ) + | + 12 Z h ′ ε ( ρ ε ) |∇ ( ρ ε − k ) + | ≤ Z A k ( t ) | m ε | !(cid:18)Z | m ε | p (cid:19) p | A k ( t ) | − p . By using (6.8) and denoting r k = sup t ∈ (0 ,T ) | A k ( t ) | we get(6.11) ddt Z | ( ρ ε − k ) + | + 12 Z h ′ ε ( ρ ε ) |∇ ( ρ ε − k ) + | ≤ C r − p k . Let σ ∈ (0 , T ) such that Z Z | ( ρ ε − k ) + ( σ ) | := sup t ∈ (0 ,T ) Z | ( ρ ε − k ) + ( t ) | . Then, Z | ( ρ ε − k ) + ( σ ) | + Z |∇ ( ρ ε − k ) + ( σ ) | ≤ r − p k , where the fact that h ′ ε ( ρ ε ) > l > k > k ρ ε k ∞ and q ≥ | A l ( t ) | ( l − k ) ≤ k ( ρ ε − k ) + ( t ) k ≤ k ( ρ ε − k ) + ( σ ) k ≤ k ( ρ ε − k ) + ( σ ) k q | A k ( σ ) | − q ≤ k∇ ( ρ ε − k ) + ( σ ) k | A k ( σ ) | − q ≤ Cr − p − q k . If we show that(6.13) r l ≤ ( l − k ) r αk for some α > ρ ε is bounded. Then, in the threedimensional case by Sobolev embedding we are forced to take q = 6 in (6.12). Then since p > α = 23 − p > . Note that since p is depending only on ν, κ and γ then α has the dependence as well. Inthe two dimensional case by Sobolev embedding we can take any q < ∞ . Then, given p = p ( ν, κ, γ ) > 2, it always possible to find q big enough such that α = 1 − p − q > . Now we prove that ρ ε is bounded away from 0. By using (6.9) it follows that the equationfor q ε := 1 /ρ ε is the following.(6.14) ∂ t q ε − div( h ′ ε ( ρ ε ) ∇ q ε ) + 2 h ′ ε ( ρ ε ) |∇ ρ ε | ρ ε = − div w ε q ε + w ε · ∇ q ε . UANTUM NAVIER-STOKES 29 Let k > k /ρ ε k , m ε := q ε w ε and A k ( t ) := { q ε > k } . Then we get(6.15) ddt Z | ( q ε − k ) + | + Z h ′ ε ( ρ ε ) |∇ ( q ε − k ) + | + Z h ′ ε ( ρ ε ) |∇ ρ ε | ρ ε q ε = − Z q ε div w ε ( q ε − k ) + + Z w ε · ∇ q ε ( q ε − k ) + = 2 Z A k ( t ) w ε ∇ q ε ( q ε − k ) + + Z A k ( t ) q ε w ε · ∇ ( q ε − k ) + ≤ Z |∇ ( q ε − k ) + || w ε || q ε | , where it has been used that | ( q ε − k ) + | ≤ | q ε | . By using H¨older inequality, Young inequalityand the fact that h ′ ε ( ρ ε ) ≥ ddt Z | ( q ε − k ) + | + 12 Z h ′ ε ( ρ ε ) |∇ ( q ε − k ) + | ≤ Z A k ( t ) | m ε | !(cid:18)Z | m ε | p (cid:19) p | A k ( t ) | − p , with p > d and the last term in the right-hand side of (6.15) has been dropped because it ispositive. Then, by using (6.8) and defining as before r k = sup t ∈ (0 ,T ) | A k ( t ) | we get ddt Z | ( q ε − k ) + | + 12 Z h ′ ε ( ρ ε ) |∇ ( q ε − k ) + | ≤ C r − p k . Arguing as in the proof of boundedness of ρ ε we can conclude that q ε is bounded and then ρ ε is bounded away from 0. (cid:3) Acknowledgement We would like to thank Prof. Jing Li for some useful comments. References [1] P. Antonelli and P. Marcati. On the finite energy weak solutions to a system in quantum fluid dynamics. Commun. Math. Phys., 287, (2009), 657-686.[2] P. Antonelli and P. Marcati, The Quantum Hydrodynamics system in two space dimensions , Arch. Rat.Mech. Anal., 203, (2012), 499–527.[3] P. 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Spirito) GSSI - Gran Sasso Science Institute, Viale Francesco Crispi 7, 67100, L’Aquila, Italy E-mail address ::