Well-posedness for a multi-dimensional viscous liquid-gas two-phase flow model
aa r X i v : . [ m a t h . A P ] F e b WELL-POSEDNESS FOR A MULTI-DIMENSIONAL VISCOUSLIQUID-GAS TWO-PHASE FLOW MODEL
CHENGCHUN HAO AND HAI-LIANG LI
Abstract.
The Cauchy problem of a multi-dimensional ( d >
2) compressible vis-cous liquid-gas two-phase flow model is concerned in this paper. We investigate theglobal existence and uniqueness of the strong solution for the initial data close toa stable equilibrium and the local in time existence and uniqueness of the solutionwith general initial data in the framework of Besov spaces. A continuation criterionis also obtained for the local solution. Introduction
The models of two-phase or multi-phase flows have a very broad applications ofhydrodynamics in industry, engineering, biomedicine and so on, where the fluids un-der investigation contain more than one component. Indeed, it has been estimatedthat over half of anything which is produced in a modern industrial society depends,to some extent, on a multi-phase flow process for their optimum design and safe op-erations. In nature, there is a variety of different multi-phase flow phenomena, suchas sediment transport, geysers, volcanic eruptions, clouds and rain. In addition, themodels of multi-phase flows also naturally appear in many contexts within biology,ranging from tumor biology and anticancer therapies, development biology and plantphysiology, etc. The principles of single-phase flow fluid dynamics and heat transferare relatively well understood, however, the thermo-fluid dynamics of two-phase flowsis an order of magnitude more complicated subject than that of the single-phase flowdue to the existence of moving and deformable interface and its interactions with twophases [3, 14, 15].We consider the drift-flux model of two-phase flows in the present paper, which isprincipally developed by Zuber and Findlay (1965), Wallis (1969) and Ishii (1977).The basic idea about drift-flux models is that both phases are well mixed, but therelative motion between the phases is governed by a particular subset of the flowparameters. In the case of liquid-gas fluids, it relates the liquid-gas velocity differenceto the drift-flux (or “drift velocity”) of the vapor relative to the liquid due to buoyancyeffects. In general, the drift-flux models consist of two mass conservation equationscorresponding to each of the two phases, and one equation for the conservation ofthe mixture momentum, and are particularly useful in the analysis of sedimentation,fluidization (batch, cocurrent and countercurrent), and so on ([13, 18, 23, 24]).The Cauchy problem to a simplified version of the viscous compressible liquid-gastwo-phase flow model of drift-flux type in R d ( d > Mathematics Subject Classification.
Key words and phrases.
Compressible liquid-gas two-phase flow model, global well-posedness,local well-posedness, Besov spaces . equal velocity of the liquid and gas flows has been assumed, reads ˜ m t + div( ˜ m u ) = 0 , ˜ n t + div(˜ n u ) = 0 , ( ˜ m u ) t + div( ˜ m u ⊗ u ) + ∇ P ( ˜ m, ˜ n ) = ˜ µ ∆ u + (˜ µ + ˜ λ ) ∇ div u , (1.1)with the initial data ( ˜ m, ˜ n, u ) | t =0 = ( ˜ m , ˜ n , u )( x ) , in R d , (1.2)where ˜ m = α l ρ l and ˜ n = α g ρ g denote the liquid mass and the gas mass, respectively.The unknowns α l , α g ∈ [0 ,
1] denote the liquid and gas volume fractions, satisfying thefundamental relation: α l + α g = 1. The unknown variables ρ l and ρ g denote the liquidand gas densities, satisfying the equations of states ρ l = ρ l, +( P − P l, ) /a l , ρ g = P/a g ,where a l and a g denote the sonic speeds of the liquid and the gas, respectively, and P l, and ρ l, are the reference pressure and density given as constants. u denotes themixed velocity of the liquid and the gas, and P is the common pressure for bothphases, which satisfies P ( ˜ m, ˜ n ) = C (cid:16) − b ( ˜ m, ˜ n ) + p b ( ˜ m, ˜ n ) + c ( ˜ m, ˜ n ) (cid:17) , (1.3)with C = a l / k = ρ l, − P l, /a l > a = a g /a l and b ( ˜ m, ˜ n ) = k − ˜ m − a ˜ n, c ( ˜ m, ˜ n ) = 4 k a ˜ n. ˜ µ and ˜ λ are the viscosity constants, satisfying˜ µ > , µ + d ˜ λ > . (1.4)For the one-dimensional case, the existence and/or uniqueness of the global weaksolution to the free boundary value problem was studied in [9, 11, 21, 22] where theliquid is incompressible and the gas is polytropic, and in [10] where both of two fluidsare compressible. However, there are few results for multi-dimensional cases exceptfor some computational results [17]. As a generalization of the results in [10], theexistence of the global solution to the 2D model was obtained in [20] for small initialenergies. In [19], a blow-up criterion for the 2D model was proved in terms of theupper bound of the liquid mass for the strong solution in a smooth bounded domain.One of the main results of the present paper is the existence and uniqueness ofthe global strong solution to the Cauchy problem (1.1)–(1.2) under the framework ofBesov spaces, for all multi-dimensions d >
2, provided that the initial data are closeto a constant equilibrium state. The other result is the local well-posedness and thecontinuation criterion to the Cauchy problem with general initial data. Because of thesimilarity of the viscous liquid-gas two-phase flow model to the compressible Navier-Stokes equations, we can apply some ideas adopted in the proof of well-posedness forthe compressible Navier-Stokes equations to deal with the two-phase flow model. It isDanchin who first makes important progress in applying the Littlewood-Paley theoryand Besov spaces to sovle the existence and uniqueness for the compressible Navier-Stokes equations or barotropic viscous fluid in [5, 8] and for the flows of compressibleviscous and heat-conductive gases in [6, 7]. However, it is non-trivial to apply di-rectly the ideas used in single-phase models into the two-phase models because themomentum equation is given only for the mixture and that the pressure involves themasses of two phases in a nonlinear way, which makes it rather difficult to obtain theestimates of the masses and the mixed velocity ( ˜ m, ˜ n, u ) in Lebesgue spaces L p withrespect to the time. In addition, it seems impossible to get the estimates of ˜ m and ˜ n ELL-POSEDNESS OF A VISCOUS LIQUID-GAS FLOW MODEL 3 from the system simultaneously due to the strong coupling among the correspondingterms, even if we change the variables ( ˜ m, ˜ n ) linearly.To overcome these difficulties in global well-posedness theory, we make use of anonlinear variable transform so as to separate the two mass variables from each other,which enable us to decompose the original system into a transport equation anda coupled hyperbolic-parabolic system. To be more precisely, we first divide themomentum equation by ˜ m (which supplies additional information) and take a newvariable n = a (˜ n/ ˜ m − ¯ n/ ¯ m ) for some constants ¯ n and ¯ m . This makes the resultedequation for n a homogeneous transport equation with the velocity u , and the expectedestimates of the new variable depend only on the mixed velocity. Then, we removethe linear terms involving n from the momentum equation so as to separate linearlythe equation about n from the others, which can be done by virtue of the variableschanges with a careful choice of coefficient factors. Finally, to establish the a priori estimates for the global existence theory, we deal with the linearized system directlyinstead of separating the velocity into the compressible and incompressible parts.As for the local well-posedness theory for general data, we need to reformulate theoriginal system and deal with the resulted nonlinear system directly, and in termsof the improvement of the a priori estimates on the densities, we can generalize thelocal well-posedness result in [1, 8] to the two-phase flow model (1.1) with the specifiedpressure function.Before stating the main results, we introduce some notations. Throughout thepaper, C (or c ) stands for a harmless constant, and we sometimes use A . B tostand for A CB . B s and B s,t denote usual homogeneous Besov spaces and hybridBesov spaces, respectively; ˜ L ∞ ( B s,t ) and ˜ C ( B s,t ) are mixed time-spatial spaces, seethe appendix for details. Let us now introduce the functional spaces which appear inthe theorems. Definition 1.1.
For
T > s ∈ R , we denote E sT = (cid:8) ( m, n, u ) : n ∈ C ([0 , T ]; B s − ,s ( R d )) m ∈ C ([0 , T ]; B s − ,s ( R d )) ∩ L ([0 , T ]; B s +1 ,s ( R d )) u ∈ (cid:0) C ([0 , T ]; B s − ( R d )) ∩ L ([0 , T ]; B s +1 ( R d )) (cid:1) d (cid:9) , and k ( m, n, u ) k E sT = k n k ˜ L ∞ ([0 ,T ]; B s − ,s ) + k m k ˜ L ∞ ([0 ,T ]; B s − ,s ) + k u k ˜ L ∞ ([0 ,T ]; B s − ) + k m k L ([0 ,T ]; B s +1 ,s ) + k u k L ([0 ,T ]; B s +1 ) . We use the notation E s if T = + ∞ , changing [0 , T ] into [0 , ∞ ) in the definitionabove. Definition 1.2.
Let α ∈ [0 ,
1] and
T >
0, denote F αT :=( ˜ C ([0 , T ]; B d/ ,d/ α )) × ( ˜ C ([0 , T ]; B d/ − ,d/ − α ) ∩ L ([0 , T ]; B d/ ,d/ α )) d . Now, we state the global well-posedness results briefly as follows. For more infor-mation about the solution, one can see Theorem 2.1 in the second section.
Theorem 1.1 (Global well-posedness for small data) . Let d > , ¯ n > , ¯ m > (1 − sgn¯ n ) k , ˜ µ > and µ + d ˜ λ > , in addition, ˜ µ + ˜ λ > if d = 2 . There exist two C. C. HAO AND H.-L. LI positive constants σ and Q such that if ˜ m − ¯ m , ˜ n − ¯ n ∈ B d/ − ,d/ and u ∈ B d/ − satisfying k ˜ m − ¯ m k B d/ − ,d/ + k ˜ n − ¯ n k B d/ − ,d/ + k u k B d/ − σ, (1.5) then the following results hold (i) Existence: The system (1.1) has a solution ( ˜ m, ˜ n, u ) satisfying ˜ m − ¯ m, ˜ n − ¯ n ∈ C (cid:0) R + ; B d/ − ,d/ (cid:1) , u ∈ C (cid:0) R + ; B d/ − (cid:1) , and moreover, k ( a ( ˜ m − ¯ m ) + ba (˜ n/ ˜ m − ¯ n/ ¯ m ) , ˜ n/ ˜ m − ¯ n/ ¯ m, u ) k E d/ Q (cid:0) k ˜ m − ¯ m k B d/ − ,d/ + k ˜ n − ¯ n k B d/ − ,d/ + k u k B d/ − (cid:1) , where the constants a and b are defined by a = 1¯ m a ¯ n + ¯ m + ( ¯ m − a ¯ n )( ¯ m − a ¯ n − k ) p ( ¯ m + a ¯ n − k ) + 4 k a ¯ n ! > ,b =1 + ( ¯ m + a ¯ n + k ) p ( ¯ m + a ¯ n − k ) + 4 k a ¯ n > . (1.6)(ii) Uniqueness: Uniqueness holds in C (cid:0) R + ; ( B d/ − ,d/ ) × ( B d/ ) d (cid:1) if d > . If d = 2 , one should also suppose that ˜ m − ¯ m , ˜ n − ¯ n ∈ B ε, ε and u ∈ B ε for a ε ∈ (0 , , to get uniqueness in C ( R + ; ( B , ) × ( B ) d ) . For the general data bounded away from the infinity and the vacuum, we havethe following local well-posedness theory (one can refer to Theorem 3.1 for the corre-sponding statement in terms of new variables).
Theorem 1.2 (Local well-posedness for general data) . Let d > , ˜ µ > , µ + d ˜ λ > ,the constants ¯ m > and ¯ n > . Assume that ˜ m − − ¯ m − ∈ B d/ ,d/ , ˜ n − ¯ n ∈ B d/ ,d/ and u ∈ B d/ − ,d/ . In addition, sup x ∈ R d ˜ m ( x ) < ∞ . Then there exists apositive time T such that the system (1.1) has a unique solution ( ˜ m, ˜ n, u ) on [0 , T ] × R d and that ( ˜ m − − ¯ m − , ˜ n − ¯ n, u ) belongs to F T and satisfies sup ( t,x ) ∈ [0 ,T ] × R d ˜ m ( t, x ) < ∞ . We also have the following continuation criterion for the local existence of thesolution (see also Proposition 3.6).
Theorem 1.3 (Continuation criterion) . Under the hypotheses of Theorem 1.2, assumethat the system (1.1) has a solution ( ˜ m, ˜ n, u ) on [0 , T ) × R d such that ( ˜ m − − ¯ m − , ˜ n − ¯ n, u ) belongs to F T ′ for all T ′ < T and satisfies ˜ m − − ¯ m − , ˜ n − ¯ n ∈ L ∞ ([0 , T ); B d/ ,d/ ) , sup ( t,x ) ∈ [0 ,T ) × R d ˜ m ( t, x ) < ∞ , Z T k∇ u k ∞ dt < ∞ . Then, there exists some T ∗ > T such that ( ˜ m, ˜ n, u ) may be continued on [0 , T ∗ ] × R d to a solution of (1.1) such that ( ˜ m − − ¯ m − , ˜ n − ¯ n, u ) belongs to F T ∗ .Remark . The results of the present paper are independent of the special structure(1.3) of the nonlinear pressure term P . Indeed, the similar results hold true as longas the term ∇ P/ ˜ m can be decomposed into a linear term involving the modified massand some nonlinear terms, similarly as (2.1) in the next section. ELL-POSEDNESS OF A VISCOUS LIQUID-GAS FLOW MODEL 5
The rest of this paper is organized as follows. In Section 2, we investigate theglobal well-posedness of the Cauchy problem. We first reformulate the system throughchanging variables in order to obtain a priori estimates in the subsection 2.1. In thesubsection 2.2, we are devoted to deriving a priori estimates for the transport equa-tion and the linear coupled parabolic-hyperbolic system with convection terms. Thesubsection 2.3 involves the proof of the existence and uniqueness of the solution. InSection 3, we prove the local well-posedness of the problem through some subsectionssimilar to the global ones. An appendix is devoted to recalling some properties of theLittlewood-Paley decomposition and Besov spaces which we have used in this paper.2.
Global well-posedness for small data
Reformulation of the system.
Let ¯ n > m > (1 − sgn¯ n ) k , we introducenew variables n = a (˜ n/ ˜ m − ¯ n/ ¯ m ) and m = a ( ˜ m − ¯ m ) + bn , i.e. ˜ m = ¯ m + ( m − bn ) /a in order to cancel the linear terms involving one modified mass from the momentumequation, where a and b are positive constants defined in (1.6). We also denote n = a (˜ n / ˜ m − ¯ n/ ¯ m ) and m = a ( ˜ m − ¯ m ) + bn throughout the sections for theglobal well-posedness theory. Then, we have PC = (cid:16) a ¯ n ¯ m + n (cid:17) ˜ m − k + r(cid:16)(cid:16) a ¯ n ¯ m + n (cid:17) ˜ m − k (cid:17) + 4 k (cid:16) n + a ¯ n ¯ m (cid:17) ˜ m. Taking the gradient of both sides, we get ∇ PC ˜ m = ∇ m + H ( m, n ) , (2.1)where the nonlinear term is H ( m, n ) := ∇ m − b ∇ na ¯ m ˜ m (cid:16) − ( a ¯ n + ¯ m ) m + ( a ¯ m + b ( a ¯ n + ¯ m )) n (cid:17) + ( K ( m, n ) − K (0 , n ( n + a ¯ n ¯ m + 1) ˜ m ∇ n + ( n + a ¯ n ¯ m + 1) ∇ ˜ m + k ( n + a ¯ n ¯ m − ∇ m − b ∇ na ˜ m + k ∇ n o + K (0 , (cid:26) ( n + a ¯ n ¯ m + 1) ∇ n ( m − bn ) a + ¯ mn ∇ n + [ n + 2 n ( a ¯ n ¯ m + 1)] ∇ m − b ∇ na + k a ˜ m n ( ∇ m − b ∇ n ) − k ( a ¯ n ¯ m −
1) ( m − bn )( ∇ m − b ∇ n ) a ¯ m ˜ m (cid:27) . Here, K ( m, n ) = 1 q(cid:2)(cid:0) ¯ m + m − bna (cid:1) (cid:0) n + a ¯ n ¯ m + 1 (cid:1) − k (cid:3) + 4 k (cid:0) n + a ¯ n ¯ m (cid:1) (cid:0) ¯ m + m − bna (cid:1) , and K (0 ,
0) = 1 / p ( ¯ m + a ¯ n − k ) + 4 k a ¯ n > C. C. HAO AND H.-L. LI
Therefore, with the new unknowns, we can rewrite the Cauchy problem of thesystem (1.1) as follows n t + u · ∇ n = 0 ,m t + u · ∇ m + a ¯ m div u = F ( m, n, u ) , u t + u · ∇ u − µ ∆ u − ( µ + λ ) ∇ div u + C ∇ m = G ( m, n, u ) , ( m, n, u ) | t =0 = ( m , n , u ) , (2.2)where µ = ˜ µ/ ¯ m, λ = ˜ λ/ ¯ m, F ( m, n, u ) = − ( m − bn )div u , G ( m, n, u ) = − C H ( m, n ) − m − bna ˜ m ( µ ∆ u + ( µ + λ ) ∇ div u ) . Note here that the first equation in (2.2) is a homogeneous transport equation, theestimates of n depend only on those of the velocity u . The second and the third onesin (2.2) consist of a coupled parabolic-hyperbolic system with the modified mass m and mixed velocity u involved. Thus, with the help of the decomposition (2.1), theoriginal system is decoupled into a transport equation for the modified gas flow anda coupled system for the motion of the modified liquid fluid.We can get the following result for the reformulated system. Theorem 2.1.
Let d > , ¯ n > , ¯ m > (1 − sgn¯ n ) k , µ > and µ + dλ > , inaddition, µ + λ > if d = 2 . There exist two positive constants η and Q such that if m , n ∈ B d/ − ,d/ and u ∈ B d/ − satisfying k m k B d/ − ,d/ + k n k B d/ − ,d/ + k u k B d/ − η, (2.3) then the following results hold: (i) Existence: The system (2.2) has a solution ( m, n, u ) in E d/ which satisfies k ( m, n, u ) k E d/ Q (cid:0) k m k B d/ − ,d/ + k n k B d/ − ,d/ + k u k B d/ − (cid:1) . It also belongs to the affine space ( m L , n , u L ) + ( C / ( R + ; B d/ − )) × ( C / ( R + ; B d/ − / )) d , where ( m L , u L ) is the solution of the linear system ∂ t m L + a ¯ m div u L = 0 ,∂ t u L − µ ∆ u L − ( µ + λ ) ∇ div u L + C ∇ m L = 0 , ( m L , u L ) | t =0 = ( m , u ) . (2.4)(ii) Uniqueness: Uniqueness holds in E d/ if d > . If d = 2 , one should alsosuppose that n , m ∈ B ε, ε and u ∈ B ε for a ε ∈ (0 , , to get uniqueness in E . With the help of Theorem 2.1, we can prove Theorem 1.1 as follows.
Proof of Theorem 1.1 . From the conditions, we have n ∈ B d/ − ,d/ . In addition,from ˜ m − ¯ m ∈ B d/ − ,d/ , we can derive m = a ( ˜ m − ¯ m ) / ¯ m + bn ∈ B d/ − ,d/ . Since(1.5) implies (2.3), the conclusion of Theorem 2.1 follows for ( m, n, u ). Changing backto the original variables ( ˜ m, ˜ n, u ), it leads to Theorem 1.1. By Lemma A.3, it is easyto see that ˜ m − ¯ m and ˜ n − ¯ n also belong to C ( R + ; B d/ − ∩ B d/ ). ELL-POSEDNESS OF A VISCOUS LIQUID-GAS FLOW MODEL 7
A priori estimates for linear system with convection terms.
We firstinvestigate some a priori estimates for the linear system with convection terms n t + v · ∇ n = 0 ,m t + v · ∇ m + a ¯ m div u = F, u t + v · ∇ u − µ ∆ u − ( µ + λ ) ∇ div u + C ∇ m = G , ( m, n, u ) | t =0 = ( m , n , u ) . (2.5)We do not need to separate the velocity into the compressible and incompressibleparts. In fact, we can prove the following proposition. Proposition 2.2.
Let a > , ¯ m > , s ∈ (1 − d/ , d/ and s , s ∈ ( − d/ , d/ be constants. Assume v ∈ L ([0 , T ]; B d/ ) and denote V ( t ) = R t k v ( τ ) k B d/ dτ . Let ( m, n, u ) be a solution of (2.5) on [0 , T ] , then the following estimates hold: k n k ˜ L ∞ ([0 ,T ]; B s ,s ) e CV ( T ) k n k B s ,s , (2.6) and k m k ˜ L ∞ ([0 ,T ]; B s − ,s ) + k u k ˜ L ∞ ([0 ,T ]; B s − ) + k m k L ([0 ,T ]; B s +1 ,s ) + k u k L ([0 ,T ]; B s +1 ) . e CV ( T ) (cid:0) k m k B s − ,s + k u k B s − + k F k L ([0 ,T ]; B s − ,s ) + k G k L ([0 ,T ]; B s − ) (cid:1) . (2.7) Proof. Step 1: Estimates for the homogeneous transport equation.
We derive theestimates for the first equation of (2.5) in Besov spaces.Applying the Littlewood-Paley operator △ k to the (2.5) , it yields (cid:26) ∂ t △ k n + △ k ( v · ∇ n ) = 0 , △ k n | t =0 = △ k n . (2.8)Taking the inner product of (2.8) with △ k n , we get for any s , s ∈ ( − d/ , d/ ddt k△ k n k = − ( △ k ( v · ∇ n ) , △ k n ) . γ k − kϕ s ,s ( k ) k v k B d/ k n k B s ,s k△ k n k . It follows X k kϕ s ,s ( k ) k△ k n k k n k B s ,s + C Z t k v k B d/ k n k B s ,s , which implies the desired estimate (2.6) with the help of the Gronwall inequality. Step 2: Estimates for ( m, u ) . Applying the Littlewood-Paley operator △ k to (2.5) and (2.5) , we have (cid:26) ∂ t △ k m + △ k ( v · ∇ m ) + a ¯ m div △ k u = △ k F,∂ t △ k u + △ k ( v · ∇ u ) − µ ∆ △ k u − ( µ + λ ) ∇ div △ k u + C ∇△ k m = △ k G . (2.9)Taking the inner product of (2.9) with △ k m and − ∆ △ k m , and (2.9) with △ k u ,we obtain12 ddt k△ k m k + ( △ k ( v · ∇ m ) , △ k m ) + a ¯ m (div △ k u , △ k m ) = ( △ k F, △ k m ) , (2.10)12 ddt k∇△ k m k + ( △ k ( v · ∇ m ) , − ∆ △ k m ) − a ¯ m (div △ k u , ∆ △ k m ) (2.11)= − ( △ k F, ∆ △ k m ) , ddt k△ k u k + ( △ k ( v · ∇ u ) , △ k u ) + µ k∇△ k u k + ( µ + λ ) k div △ k u k (2.12) C. C. HAO AND H.-L. LI + C ( ∇△ k m, △ k u ) = ( △ k G , △ k u ) . For the intersected term, we have ddt ( △ k u , ∇△ k m ) + ( △ k ( v · ∇ u ) , ∇△ k m ) − ( △ k ( v · ∇ m ) , div △ k u ) − a ¯ m k div △ k u k + C k∇△ k m k + (2 µ + λ )(div △ k u , ∆ △ k m )= − ( △ k F, div △ k u ) + ( △ k G , ∇△ k m ) . (2.13)Let α k := C a ¯ m k△ k m k + k△ k u k + (2 µ + λ ) Aa ¯ m k∇△ k m k + 2 A ( △ k u , ∇△ k m ) . (2.14)For A = ( µ + λ ) / (2 a ¯ m ) >
0, there exist two positive constants c and c such that c α k k△ k m k + k△ k u k + k∇△ k m k c α k , (2.15)since we have, for M ∈ ( a ¯ m/ (2 µ + λ ) , a ¯ m/ ( µ + λ )), that | △ k u , ∇△ k m ) | M k△ k u k + k∇△ k m k /M. Combining (2.10)-(2.13), it yields, with the help of Lemma A.4, that12 ddt α k + µ k∇△ k u k + ( µ + λ − a ¯ mA ) k div △ k u k + C A k∇△ k m k = C a ¯ m ( △ k F, △ k m ) + ( △ k G , △ k u ) − (2 µ + λ ) Aa ¯ m ( △ k F, ∆ △ k m ) − A ( △ k F, div △ k u )+ A ( △ k G , ∇△ k m ) − C a ¯ m ( △ k ( v · ∇ m ) , △ k m ) − ( △ k ( v · ∇ u ) , △ k u )+ (2 µ + λ ) Aa ¯ m ( △ k ( v · ∇ m ) , ∆ △ k m ) + A ( △ k ( v · ∇ u ) , ∇△ k m )+ A ( ∇△ k ( v · ∇ m ) , △ k u ) . ( k△ k F k + k∇△ k F k + k△ k G k )( k△ k m k + k△ k u k + k∇△ k m k )+ γ k − kϕ s − ,s ( k ) k v k B d/ k m k B s − ,s k△ k m k + γ k − k ( s − k v k B d/ k u k B s − k△ k u k + γ k − k ( ϕ s − ,s ( k ) − k v k B d/ k m k B s − ,s k∇△ k m k + γ k k v k B d/ (cid:16) − k ( s − k∇△ k m k k u k B s − + 2 − k ( ϕ s − ,s ( k ) − k m k B s − ,s k△ k u k (cid:17) . (cid:16) k△ k F k + k∇△ k F k + k△ k G k + γ k − k ( s − k v k B d/ ( k m k B s − ,s + k u k B s − ) (cid:17) × ( k△ k m k + k△ k u k + k∇△ k m k ) . Thus, it follows12 ddt α k + c min(2 k , α k . γ k − k ( s − h k F k B s − ,s + k G k B s − + k v k B d/ (cid:16) k m k B s − ,s + k u k B s − (cid:17)i α k , which implies2 k ( s − α k + c Z t min(2 k , k ( s − α k ( τ ) dτ ELL-POSEDNESS OF A VISCOUS LIQUID-GAS FLOW MODEL 9 k ( s − α k (0) + Cγ k Z t " k F k B s − ,s + k G k B s − + k v k B d/ X k k ( s − α k . Thus, by the Gronwall inequality, we have k m k ˜ L ∞ ([0 ,T ]; B s − ,s ) + k u k ˜ L ∞ ([0 ,T ]; B s − ) + k m k L ([0 ,T ]; B s +1 ,s ) + k u k L ([0 ,T ]; B s +1 ,s − ) . e CV ( T ) (cid:0) k m k B s − ,s + k u k B s − + k F k L ([0 ,T ]; B s − ,s ) + k G k L ([0 ,T ]; B s − ) (cid:1) . (2.16) Step 3: The smoothing effect for u . By (2.12), we have12 ddt k△ k u k + C k k△ k u k . k△ k u k (2 k k△ k m k + k△ k G k + γ k − k ( s − k v k B d/ k u k B s − ) . It follows that ddt X k > k ( s − k△ k u k + C X k > k ( s +1) k△ k u k . X k > k ( s − (cid:2) k k△ k m k + k△ k G k + γ k − k ( s − k u k B d/ k u k B s − (cid:3) . X k > ks k△ k m k + k G k B s − + k v k B d/ k u k B s − , which implies, with the help of (2.16), that Z t X k > k ( s +1) k△ k u ( τ ) k . k u k B s − + Z t X k > ks k△ k m ( τ ) k dτ + Z t k G ( τ ) k B s − dτ + sup τ ∈ [0 ,t ] k u ( τ ) k B s − Z t k v ( τ ) k B d/ dτ . e CV ( t ) (cid:18) k m k B s − ,s + k u k B s − Z t [ k F ( τ ) k B s − ,s + k G ( τ ) k B s − ] dτ (cid:19) . Combining with (2.16), we get (2.7). (cid:3)
From the proof of Proposition 2.2, we immediately have
Corollary 2.3.
If a bounded operator B acts on the convection terms in (2.5) , thenthe same estimates hold for the refined system n t + B ( v · ∇ n ) = 0 ,m t + B ( v · ∇ m ) + a ¯ m div u = F, u t + B ( v · ∇ u ) − µ ∆ u − ( µ + λ ) ∇ div u + C ∇ m = G . (2.17) Global existence and uniqueness of the solution.
Step 1: Friedrich’s ap-proximation.
Let L ℓ be the set of L functions spectrally supported in the annulus C ℓ := { ξ ∈ R d : 1 /ℓ | ξ | ℓ } endowed with the standard L topology. In or-der to construct the classical Friedrichs approximation, we first define the Friedrichsprojectors ( F ℓ ) ℓ ∈ N by F ℓ f := F − C ℓ ( ξ ) F f, for any f ∈ L ( R d ) where C ℓ ( ξ ) denotes the characteristic function on the annulus C ℓ . Then, we can define the following approximate system n ℓt + F ℓ ( F ℓ u ℓ · ∇ F ℓ n ℓ ) = 0 ,m ℓt + F ℓ ( F ℓ u ℓ · ∇ F ℓ m ℓ ) + a ¯ m div F ℓ u ℓ = F ℓ , u ℓt + F ℓ ( F ℓ u ℓ · ∇ F ℓ u ℓ ) − µ ∆ F ℓ u ℓ − ( µ + λ ) ∇ div F ℓ u ℓ + C ∇ F ℓ m ℓ = G ℓ , ( m ℓ , n ℓ , u ℓ ) | t =0 = ( m ℓ , n ℓ , u ℓ ) , (2.18)where m ℓ = F ℓ m , n ℓ = F ℓ n , u ℓ = F ℓ u ,F ℓ = F ℓ F ( F ℓ m ℓ , F ℓ n ℓ , F ℓ u ℓ ) , G ℓ = F ℓ G ( F ℓ m ℓ , F ℓ n ℓ , F ℓ u ℓ ) . It is easy to check that it is an ordinary differential equation in L ℓ × L ℓ × ( L ℓ ) d for every ℓ ∈ N . By the usual Cauchy-Lipschitz theorem, there is a strictly positivemaximal time T ∗ ℓ such that a unique solution ( m ℓ , n ℓ , u ℓ ) exists in [0 , T ∗ ℓ ) which iscontinuous in time with value in L ℓ × L ℓ × ( L ℓ ) d , i.e. ( m ℓ , n ℓ , u ℓ ) ∈ C ([0 , T ∗ ℓ ); L ℓ × L ℓ × ( L ℓ ) d ). As F ℓ = F ℓ , we see that F ℓ ( m ℓ , n ℓ , u ℓ ) is also a solution, so the uniquenessimplies that F ℓ ( m ℓ , n ℓ , u ℓ ) = ( m ℓ , n ℓ , u ℓ ). Thus, this system can be rewritten as thefollowing system n ℓt + F ℓ ( u ℓ · ∇ n ℓ ) = 0 ,m ℓt + F ℓ ( u ℓ · ∇ m ℓ ) + a ¯ m div u ℓ = F ℓ , u ℓt + F ℓ ( u ℓ · ∇ u ℓ ) − µ ∆ u ℓ − ( µ + λ ) ∇ div u ℓ + C ∇ m ℓ = G ℓ , ( m ℓ , n ℓ , u ℓ ) | t =0 = ( m ℓ , n ℓ , u ℓ ) , (2.19)where F ℓ = F ℓ F ( m ℓ , n ℓ , u ℓ ) , and G ℓ = F ℓ G ( m ℓ , n ℓ , u ℓ ) . Step 2: Uniform estimates.
Denote E = k m k B d/ − ,d/ + k n k B d/ + k u k B d/ − , and T ℓ := sup { T ∈ [0 , T ∗ ℓ ) : k ( m ℓ , n ℓ , u ℓ ) k E d/ T A ¯ CE } , where ¯ C corresponds to the constant in Proposition 2.2 and A > max(2 , / ¯ C ) is aconstant. Thus, by the continuity, we have T ℓ > M be the continuity modulus of the embedding relation B d/ ( R d ) ֒ → L ∞ ( R d ).We make the assumption 2(1 + b ) A ¯ CM E a ¯ m. Then, it implies k m ℓ k L ∞ ([0 ,T ] × R d ) M k m ℓ k L ∞ ([0 ,T ]; B d/ ) M k m ℓ k L ∞ ([0 ,T ]; B d/ − ,d/ ) ELL-POSEDNESS OF A VISCOUS LIQUID-GAS FLOW MODEL 11 A ¯ CM E a ¯ m b ) . Similarly, we have k n ℓ k L ∞ ([0 ,T ] × R d ) a ¯ m b ) . Then, ˜ m ( m ℓ , n ℓ ) = ¯ m + m − bna ∈ (cid:20) ¯ m , m (cid:21) . By Proposition 2.2, we have k ( m ℓ , n ℓ , u ℓ ) k S T . e C k u ℓ k L ,T ]; Bd/ (cid:16) E + k F ℓ k L ([0 ,T ]; B d/ − ,d/ ) + k G ℓ k L ([0 ,T ]; B d/ − ) (cid:17) . From Lemmas A.1 and A.3, we get k F ℓ k L ([0 ,T ]; B d/ − ,d/ ) . k m ℓ div u ℓ k L ([0 ,T ]; B d/ − ,d/ ) + k n ℓ div u ℓ k L ([0 ,T ]; B d/ − ,d/ ) . ( k m ℓ k L ∞ ([0 ,T ]; B d/ − ,d/ ) + k n ℓ k L ∞ ([0 ,T ]; B d/ − ,d/ ) ) k div u ℓ k L ([0 ,T ]; B d/ ) . ( k m ℓ k L ∞ ([0 ,T ]; B d/ − ,d/ ) + k n ℓ k L ∞ ([0 ,T ]; B d/ − ,d/ ) ) k u ℓ k L ([0 ,T ]; B d/ ) . k ( m ℓ , n ℓ , u ℓ ) k E d/ T . Thus, k m ℓ − bn ℓ a ˜ m ( m ℓ , n ℓ ) ( µ ∆ u ℓ + ( µ + λ ) ∇ div u ℓ ) k L ([0 ,T ]; B d/ − ) . ( k n ℓ k L ∞ ([0 ,T ]; B d/ ) + k m ℓ k L ∞ ([0 ,T ]; B d/ ) ) k u ℓ k L ([0 ,T ]; B d/ ) . k ( m ℓ , n ℓ , u ℓ ) k S T . Similarly, we can get k H ( m ℓ , n ℓ ) k L ([0 ,T ]; B d/ − ) . ( k n ℓ k L ∞ ([0 ,T ]; B d/ ) + k m ℓ k L ∞ ([0 ,T ]; B d/ ) ) . k ( m ℓ , n ℓ , u ℓ ) k E d/ T . Hence k ( m ℓ , n ℓ , u ℓ ) k E d/ T ¯ Ce ¯ C k ( m ℓ ,n ℓ , u ℓ ) k Ed/ T (cid:16) E + C k ( m ℓ , n ℓ , u ℓ ) k E d/ T (cid:17) ¯ Ce ¯ C AE (1 + CA ¯ C E ) E . Thus, we can choose E so small that1 + CA ¯ C E A A + 2 , e ¯ C AE A + 1 A and 2(1 + b ) A ¯ CM E a ¯ m, (2.20)which yields k ( m ℓ , n ℓ , u ℓ ) k E d/ T A +1 A +2 A ¯ CE for any T < T ℓ .We claim that T ℓ = T ∗ ℓ . Indeed, if T ℓ < T ∗ ℓ , we have seen that k ( m ℓ , n ℓ , u ℓ ) k E d/ T A +1 A +2 A ¯ CE . So by the continuity, for a sufficiently small constant s >
0, we can obtain k ( m ℓ , n ℓ , u ℓ ) k E d/ T + s ) A ¯ CE which contradicts with the definition of T ℓ . Now, we show the approximate solution is a global one, i.e. T ∗ ℓ = ∞ . We assume T ∗ ℓ < ∞ , then we have shown k ( m ℓ , n ℓ , u ℓ ) k E d/ T A ¯ CE . As m ℓ ∈ L ∞ ([0 , T ∗ ℓ ); B d/ − ,d/ ) , n ℓ ∈ L ∞ ([0 , T ∗ ℓ ); B d/ ) and u ℓ ∈ L ∞ ([0 , T ∗ ℓ ); B d/ − ) , it implies that k ( m ℓ , n ℓ , u ℓ ) k L ∞ ([0 ,T ∗ ℓ ); L ℓ ) < ∞ . Thus, we may extend the solution continuously beyond the time T ∗ ℓ by the Cauchy-Lipschitz theorem. This contradicts the definition of T ∗ ℓ . Therefore, the solution( m ℓ , n ℓ , u ℓ ) ℓ ∈ N exists global in time. Step 3: Time derivatives.
For convenience, we split the approximate solution( m ℓ , n ℓ , u ℓ ) into a solution of the linear system with initial data ( m ℓ , n ℓ , u ℓ ), andthe discrepancy to that linear solution. More precisely, we denote by ( m ℓL , u ℓL ) thesolution to the linear system m ℓt + a ¯ m div u ℓ = 0 , u ℓt − µ ∆ u ℓ − ( µ + λ ) ∇ div u ℓ + C ∇ m ℓ = 0 , ( m ℓ , u ℓ ) | t =0 = ( m ℓ , u ℓ ) , (2.21)and ( m ℓD , n ℓD , u ℓD ) = ( m ℓ − m ℓL , n ℓ − n ℓ , u ℓ − u ℓL ).It is clear that the definition of ( m ℓ , n ℓ , u ℓ ) implies m ℓ → m in B d/ − ,d/ , n ℓ → n in B d/ − ,d/ , u ℓ → u in B d/ − . From Corollary 2.3, we have( m ℓL , n ℓ , u ℓL ) → ( m L , n , u L ) in E d/ , where m L and u L satisfy the linear system (2.4).Now, we derive the uniform boundedness of the time derivatives of the discrepancy( m ℓD , n ℓD , u ℓD ). Lemma 2.4. (( m ℓD , n ℓD , u ℓD )) ℓ ∈ N is uniformly bounded in ( C / ( R + ; B d/ − )) × ( C / ( R + ; B d/ − / )) d . Proof.
Since ∂ t n ℓD = − F ℓ ( u ℓ · ∇ n ℓ ) , we have ∂ t n ℓD ∈ L ( R + ; B d/ − ) since n ℓ ∈ L ∞ ( R + ; B d/ ) and u ℓ ∈ L ( R + ; B d/ ) withthe help of the interpolation theorem.From the equation ∂ t m ℓD = − F ℓ ( u ℓ · ∇ m ℓ ) − a ¯ m div u ℓ + a ¯ m div u ℓL − F ℓ (( m ℓ − bn ℓ )div u ℓ ) , it follows that ∂ t m ℓD ∈ L ( R + ; B d/ − ).Recall that ∂ t u ℓD = − F ℓ ( u ℓ · ∇ u ℓ ) + µ ∆ u ℓ + µ ∆ u ℓL + ( µ + λ ) ∇ div u ℓ − ( µ + λ ) ∇ div u ℓL − C ∇ m ℓ − C ∇ m ℓL + G ℓ , we can obtain ∂ t u ℓD ∈ ( L ∞ + L / + L / )( R + ; B d/ − / ) through easy but tediouscomputations with the help of Lemmas A.1, A.3 and A.5.Applying the Morrey embedding relation W ,p ( R ) ⊂ C − /p ( R ) to the time variablefor 1 < p ∞ , we obtain the desired result. (cid:3) ELL-POSEDNESS OF A VISCOUS LIQUID-GAS FLOW MODEL 13
Step 4: Compactness and convergence.
The proof of the existence of a solution isnow standard. Indeed, we can use Arzel`a-Ascoli theorem to get strong convergenceof the approximate solutions. We need to localize the spatial space in order to utilizesome compactness results of local Besov spaces (see [1, Chapter 2]). Let ( χ p ) p ∈ N be asequence of D ( R d ) cut-off functions supported in the ball B (0 , p +1) of R d and equal to1 in a neighborhood of B (0 , p ). In view of Lemma 2.4 and uniform estimates obtainedin Step 2, we see that (( χ p m ℓD , χ p n ℓD , χ p u ℓD )) ℓ ∈ N is bounded in E d/ and uniformlyequi-continuous in C (cid:0) [0 , T ]; ( B d/ − ) × ( B d/ − / ) d (cid:1) for any p ∈ N and T >
0. Moreover, the mapping f χ p f is compact from B d/ − ,d/ into B d/ − and from B d/ − into B d/ − / .Applying the Arzel`a-Ascoli theorem to the family (( χ p m ℓD , χ p n ℓD , χ p u ℓD )) ℓ ∈ N on thetime interval [0 , p ], then we use the Cantor diagonal process. This finally providesus with a distribution ( m D , n D , u D ) continuous in time with values in ( B d/ − ) × ( B d/ − / ) d and a subsequence (which we still denote by the same notation) such thatwe have for all p ∈ N ( χ p m ℓD , χ p n ℓD , χ p u ℓD ) → ( χ p m D , χ p n D , χ p u D ) , as ℓ → ∞ in C ([0 , p ]; ( B d/ − ) × ( B d/ − / ) d ). This obviously implies that ( m ℓD , n ℓD , u ℓD ) tendsto ( m D , n D , u D ) in D ′ ( R + × R d ).Coming back to the uniform estimates and Lemma 2.4, we further obtain that( m D , n D , u D ) belongs to E d/ and to ( C / ( R + ; B d/ − )) × ( C / ( R + ; B d/ − / )) d .The convergence results stemming from this last result and the interpolation argumentenable us to pass to the limit in D ′ ( R + × R d ) in the system (2.18) and to prove that( m, n, u ) := ( m L , n L , u L ) + ( m D , n D , u D ) is indeed a solution of (2.2) with the initialdata. Since it is just a matter of doing tedious verifications, we omit the details. Step 5: Continuities in time.
The continuity of u is straightforward. Indeed, fromthe third equation of (2.2), we have u t ∈ ( L + L )( R + ; ( B d/ − ) d ) which implies u ∈ C ( R + ; ( B d/ − ) d ) in view of the Morrey embedding and the embedding relation W , ( R ) ⊂ C ( R ). Consequently, the continuity of n in time is obtained from (2.6).For m , it is easily to see that m t ∈ L ( R + ; B d/ − ) ∩ L ( R + ; B d/ ) from the secondequation of (2.2) which yields m ∈ C ( R + ; B d/ − ,d/ ) by the embeddings mentionedabove. Step 6: Uniqueness.
Next, we prove the uniqueness of solutions. Let ( m , n , u )and ( m , n , u ) be two solutions of (2.2) in E d/ T with the same initial data. Denote( δm, δn, δ u ) = ( m − m , n − n , u − u ). Then they satisfy ∂ t δn + u · ∇ δn = − δ u · ∇ n ,∂ t δm + u · ∇ δm + a ¯ m div δ u = − δ u · ∇ m + δF,∂ t δ u + u · ∇ δ u − µ ∆ δ u − ( µ + λ ) ∇ div δ u + C ∇ δm = − δ u · ∇ u + δ G , ( δm, δn, δ u ) | t =0 = (0 , , ) , (2.22)where δF = F ( m , n , u ) − F ( m , n , u ) and δ G = G ( m , n , u ) − G ( m , n , u ).We first consider the case d >
3. Similar to the derivation of (2.6), we can get for t ∈ [0 , T ] with the help of (2.6) k δn k ˜ L ∞ ([0 ,T ]; B d/ − ,d/ − ) e C R T k u k Bd/ dτ k δ u k L ([0 ,T ]; B d/ ) k n k L ∞ ([0 ,T ]; B d/ − ,d/ ) . e C R T k ( u , u ) k Bd/ dτ k n k B d/ − ,d/ k δ u k L ([0 ,T ]; B d/ ) . (2.23) By Lemma A.1 and A.3, we have k δm k ˜ L ∞ ([0 ,T ]; B d/ − ,d/ − ) + k δ u k ˜ L ∞ ([0 ,T ]; B d/ − ) + k δm k L ([0 ,T ]; B d/ ,d/ − ) + k δ u k L ([0 ,T ]; B d/ ) . e C R T k u k Bd/ dτ (cid:2) k δ u · ∇ m k L ([0 ,T ]; B d/ − ,d/ − ) + k δF ( τ ) k L ([0 ,T ]; B d/ − ,d/ − ) + k δ u · ∇ u k L ([0 ,T ]; B d/ − ) + k δ G ( τ ) k L ([0 ,T ]; B d/ − ) (cid:3) . e C R T k u k Bd/ dτ ( k δ u k L ([0 ,T ]; B d/ ) k m k ˜ L ∞ ([0 ,T ]; B d/ − ,d/ ) + k ( δm, δn ) k ˜ L ∞ ([0 ,T ]; B d/ − ,d/ − ) k u k L ([0 ,T ]; B d/ ) + k ( m , n ) k ˜ L ∞ ([0 ,T ]; B d/ − ,d/ ) k δ u k L ([0 ,T ]; B d/ ) + k δ u k ˜ L ∞ ([0 ,T ]; B d/ − ) k u k L ([0 ,T ]; B d/ ) + k δ G ( τ ) k L ([0 ,T ]; B d/ − ) . e C k u k L ,T ]; Bd/ (cid:16)(cid:0) k ( m , n ) k ˜ L ∞ ([0 ,T ]; B d/ − ,d/ ) (cid:1) × k ( m , n ) k ˜ L ∞ ([0 ,T ]; B d/ − ,d/ ) + Z ( T ) (cid:17) k ( δm, δn, δ u ) k S d/ − T , where lim sup T → + Z ( T ) = 0. Thus, k ( δm, δn, δ u ) k E d/ − T Ce C k ( u , u ) k L ,T ]; Bd/ (cid:16) (1 + k ( m , n ) k ˜ L ∞ ([0 ,T ]; B d/ − ,d/ ) ) × k ( m , n ) k ˜ L ∞ ([0 ,T ]; B d/ − ,d/ ) + E + Z ( T ) (cid:17) k ( δm, δn, δ u ) k E d/ − T . We take E small enough such that it satisfies the condition 2 C (1 + CA ¯ CE ) A ¯ CE + E < / T > C k ( u , u ) k L ([0 ,T ]; B d/ ) ln 2and Z ( T ) < /
2, then it follows k ( δm, δn, δ u ) k E d/ − T ≡
0. Hence, ( m , n , u )( t ) =( m , n , u )( t ) on [0 , T ]. By a standard argument (e.g. [5]), we can conclude that( m , n , u )( t ) = ( m , n , u )( t ) on R + .For the case d = 2, we have to raise the regularity of the spaces. Thus, we alsosuppose that m , n ∈ B ε, ε and u ∈ B ε for a ε ∈ (0 , m, n, u ) in the space E ε provided the norms ofinitial data is sufficiently small. Then, in the same way as in the case d >
3, we mayprove the uniqueness of solutions in the space E ε (of course, holds in E ). We omitthe details. 3. Local well-posedness for large data
Reformulation of the system.
We change variables to ρ = ¯ m ( ˜ m − − ¯ m − )and g = ˜ n − ¯ n . Then we can reformulate the system (1.1)-(1.2) as ρ t + u · ∇ ρ = ( ρ + 1)div u ,g t + u · ∇ g = − ( g + ¯ n )div u , u t + u · ∇ u − (1 + ρ )( µ ∆ u + ( µ + λ ) ∇ div u ) + Q ( ρ, g ) = 0 , ( ρ, g, u ) | t =0 = ( ρ , g , u ) , (3.1)where ρ = ¯ m ( ˜ m − − ¯ m − ), g = ˜ n − ¯ n and Q ( ρ, g ) := ¯ m − (1 + ρ ) ∇ P ( ¯ m/ (1 + ρ ) , g + ¯ n ) ELL-POSEDNESS OF A VISCOUS LIQUID-GAS FLOW MODEL 15 = ρ ∇ ρρ + 1 − ∇ ρ + a ¯ m ( ρ + 1) ∇ g + B ( ρ, g ) h − ¯ m ( ρ + 1) ∇ ρ + k − a ¯ nρ + 1 ∇ ρ + a g ∇ ρρ + 1 + a ¯ m ( g + gρ ) ∇ g + a ( k + ¯ m + a ¯ n )¯ m ∇ g + a k + a ¯ n ¯ m ρ ∇ g i , with B ( ρ, g ) := h(cid:0) ¯ mρ + 1 + a ( g + ¯ n ) − k (cid:1) + 4 k a ( g + ¯ n ) i − / . We now state the result for the local theory for general data bounded away fromthe vacuum as follows.
Theorem 3.1.
Let d > , µ > , µ + dλ > , the constants ¯ m > and ¯ n > . Assume that ρ ∈ B d/ ,d/ , g ∈ B d/ ,d/ and u ∈ B d/ − ,d/ . In addition, inf x ∈ R d ρ ( x ) > − . Then there exists a positive time T such that the system (3.1) has aunique solution ( ρ, g, u ) on [0 , T ] × R d which belongs to ( ˜ C ([0 , T ]; B d/ ,d/ )) × ( ˜ C ([0 , T ]; B d/ − ,d/ ) ∩ L ([0 , T ]; B d/ ,d/ )) d , and satisfies inf ( t,x ) ∈ [0 ,T ] × R d ρ ( t, x ) > − . A priori Estimates.
Now, let us recall some estimates for the following para-bolic system which is obtained by linearizing the momentum equation (cid:26) u t + v · ∇ u + u · ∇ w − b ( t, x )( µ ∆ u + ( µ + λ ) ∇ div u ) = f, u | t =0 = u , which had been studied in [1, 8]. Precisely, we have the following lemma (cf. [1,Proposition 10.12]). Lemma 3.2.
Let α ∈ (0 , , s ∈ ( − d/ , d/ , ν = min( µ, λ + 2 µ ) and ¯ ν = µ + | µ + λ | .Assume that b = 1 + ρ with ρ ∈ L ∞ ([0 , T ]; B d/ α ) and that b ∗ := inf ( t,x ) ∈ [0 ,T ] × R d b ( t, x ) > . There exist a universal constant κ , and a constant C depending only on d , α and s ,such that for all t ∈ [0 , T ] , k u k ˜ L ∞ ([0 ,t ]; B s ) + κb ∗ ν k u k L ([0 ,t ]; B s +2 ) (cid:0) k u k B s + k f k L ([0 ,t ]; B s ) (cid:1) × exp (cid:18) C Z t (cid:16) k v k B d/ + k w k B d/ + ( b ∗ ν ) − /α ¯ ν /α k ρ k /αB d/ α (cid:17) dτ (cid:19) . If v and w depend linearly on u , then the above inequality is true for all s ∈ (0 , d/ α ] , and the argument of the exponential term may be replaced with C Z t (cid:16) k∇ u k ∞ + ( b ∗ ν ) − /α ¯ ν /α k ρ k /αB d/ α (cid:17) dτ. For the mass equations, we only need to study the following equation with twoconstants θ ∈ R and β > (cid:26) h t + v · ∇ h = θ ( h + β )div v ,h | t =0 = h . (3.2) Proposition 3.3.
Let s ∈ ( − d/ , d/ , T > , θ ∈ R and β > be constants.Assume that h ∈ B d/ , v ∈ L ([0 , T ); B d/ ) and a satisfies (3.2) . There exists aconstant C depending only on d such that for all t ∈ [0 , T ] , we have k h k ˜ L ∞ ([0 ,t ]; B d/ ) e C (1+2 | θ | ) R t k v k Bd/ dτ (cid:18) k h k B d/ + β | θ | (cid:19) − β | θ | , (3.3) and k h k ˜ L ∞ ([0 ,t ]; B s ) e C (1+ | θ | ) R t k v k Bd/ dτ (cid:16) k h k B s + C | θ | h e C (1+2 | θ | ) R t k v k Bd/ dτ (cid:16) k h k B d/ + β | θ | (cid:17) + 2 | θ | β | θ | i Z t k v k B s +1 dτ (cid:17) . (3.4) Proof.
Applying the operator △ k to (3.2) yields ∂ t △ k h + △ k ( v · ∇ h ) = θ △ k (( h + β )div v ) . Taking the L inner product with △ k h , we get, with the help of Lemmas A.3 and A.4,that 12 ddt k△ k h k = − ( △ k ( v · ∇ a ) , △ k h ) + θ ( △ k (( h + β )div v ) , △ k h ) . γ k − ks k v k B d/ k h k B s k△ k h k + | θ | γ k − ks ( k h div v k B s + β k div v k B s ) k△ k h k . γ k − ks ((1 + | θ | ) k h k B s k v k B d/ + ( k h k B d/ + β ) | θ |k v k B s +1 ) k△ k h k . Eliminating the factor k△ k h k from both sides and integrating in the time, we have k△ k h k k△ k h k + Cγ k Z t − ks ((1 + | θ | ) k h k B s k v k B d/ + ( k h k B d/ + β ) | θ |k v k B s +1 ) dτ. It follows, for any k ∈ Z and any t ∈ [0 , T ], that2 ks k△ k h k ks k△ k h k + Cγ k Z t ((1 + | θ | ) k h k B s k v k B d/ + ( k h k B d/ + β ) | θ |k v k B s +1 ) dτ. (3.5)Summing up on k ∈ Z , it yields k h k ˜ L ∞ ([0 ,T ]; B s ) k h k B s + Z t C [(1 + | θ | ) k v k B d/ k h k B s + ( k h k B d/ + β ) | θ |k v k B s +1 ] dτ. By the Gronwall inequality, we have (3.3) for s = d/ s ∈ ( − d/ , d/ k h k ˜ L ∞ ([0 ,t ]; B s ) e C (1+ | θ | ) R t k v k Bd/ dτ (cid:18) k h k B s + C Z t | θ | ( k h k B d/ + β ) k v k B s +1 dτ (cid:19) e C (1+ | θ | ) R t k v k Bd/ dτ (cid:16) k h k B s + C | θ | (cid:20) e C (1+2 | θ | ) R t k v k Bd/ dτ (cid:16) k h k B d/ + β | θ | (cid:17) + 2 | θ | β | θ | (cid:21) Z t k v k B s +1 dτ (cid:17) . ELL-POSEDNESS OF A VISCOUS LIQUID-GAS FLOW MODEL 17
This completes the proofs. (cid:3)
In general, for the transport equation (cid:26) h t + v · ∇ h = f,h (0) = h , we can get, in a similar way with Proposition 3.3, that Proposition 3.4.
Let s , s ∈ ( − d/ , d/ and T > . Then it holds for t ∈ [0 , T ] k h k ˜ L ∞ ([0 ,t ]; B s ,s ) e C R t k v k Bd/ dτ (cid:18) k h k B s ,s + Z t k f k B s ,s dτ (cid:19) . Existence of local solution.
Step 1: The Friedrich’s approximation.
For con-venience, we introduce the solution u ls to the linear system ∂ t u ls − µ ∆ u ls − ( µ + λ ) ∇ div u ls = 0 , u ls (0) = u . (3.6)Denote u ℓ ls := F ℓ u ls and ˜ u ℓ := u ℓ − u ℓ ls . Then we can construct the following approxi-mation ( ρ ℓ , g ℓ , ˜ u ℓ ) satisfying ρ ℓt + F ℓ ( u ℓ · ∇ ρ ℓ ) = F ℓ (( ρ ℓ + 1)div u ℓ ) ,g ℓt + F ℓ ( u ℓ · ∇ g ℓ ) = − F ℓ (( g ℓ + ¯ n )div u ℓ ) ,∂ t ˜ u ℓ + F ℓ ( u ℓ ls · ∇ ˜ u ℓ ) + F ℓ (˜ u ℓ · ∇ u ℓ ) − F ℓ [(1 + ρ ℓ )( µ ∆˜ u ℓ + ( µ + λ ) ∇ div ˜ u ℓ )]= F ℓ [ ρ ℓ ( µ ∆ u ℓ ls + ( µ + λ ) ∇ div u ℓ ls )] − F ℓ ( u ℓ ls · ∇ u ℓ ls ) − F ℓ Q ( ρ ℓ , g ℓ ) , ( ρ ℓ , g ℓ , ˜ u ℓ ) | t =0 = ( ρ ℓ , g ℓ , ) , (3.7)where ρ ℓ := F ℓ ρ , g ℓ := F ℓ g and u ℓ := u ℓ ls + ˜ u ℓ .Note that if 1+ ρ is bounded away from zero, then so is 1+ F ℓ ρ for sufficiently large ℓ . It is easy to check that (3.7) is an ordinary differential equation in L ℓ × L ℓ × ( L ℓ ) d for every ℓ ∈ N . By the usual Cauchy-Lipschitz theorem, there is a strictly positivemaximal time T ∗ ℓ such that a unique solution ( ρ ℓ , g ℓ , ˜ u ℓ ) exists in [0 , T ∗ ℓ ) which iscontinuous in time with value in L ℓ × L ℓ × ( L ℓ ) d , i.e. ( ρ ℓ , g ℓ , ˜ u ℓ ) ∈ C ([0 , T ∗ ℓ ); L ℓ × L ℓ × ( L ℓ ) d ), and 1 + ρ ℓ is bounded away from zero. Step 2: Lower bound for lifespan and uniform estimates.
We introduce the followingnotations M := k ρ k B d/ ,d/ α , M ℓ ( t ) := k ρ ℓ k ˜ L ∞ ([0 ,t ]; B d/ ,d/ α ) ,N := k g k B d/ ,d/ α , N ℓ ( t ) := k g ℓ k ˜ L ∞ ([0 ,t ]; B d/ ,d/ α ) ,U := k u k B d/ − ,d/ − α , U ℓ ls ( t ) := k u ℓ ls k L ([0 ,t ]; B d/ ,d/ α ) , ˜ U ℓ ( t ) := k ˜ u ℓ k ˜ L ∞ ([0 ,t ]; B d/ − ,d/ − α ) + b ∗ ν k ˜ u ℓ k L ([0 ,t ]; B d/ ,d/ α ) . In view of Lemma 3.2, we take v = w = f = and ρ = 0 there to get k u ℓ ls k ˜ L ∞ ([0 ,t ]; B d/ − ,d/ − α ) . U . Note that for all k ∈ Z , we have k△ k ρ ℓ k k△ k ρ k , k ρ ℓ k B s k ρ k B s and similar properties for g ℓ because of the boundedness of the operators △ k .From (3.3) and (3.4), we get k ρ ℓ k ˜ L ∞ ([0 ,t ]; B d/ ) e C R t k u ℓ k Bd/ dτ (cid:18) k ρ k B d/ + 13 (cid:19) − , (3.8) k ρ ℓ k ˜ L ∞ ([0 ,t ]; B d/ α ) e C R t k u ℓ k Bd/ dτ (cid:16) k ρ k B d/ α + C (cid:20) e C R t k u ℓ k Bd/ dτ (cid:16) k ρ k B d/ + 13 (cid:17) + 23 (cid:21) Z t k u ℓ k B d/ α dτ (cid:17) , (3.9) k g ℓ k ˜ L ∞ ([0 ,t ]; B d/ ) e C R t k u ℓ k Bd/ dτ (cid:16) k g k B d/ + ¯ n (cid:17) − ¯ n , (3.10)and k g ℓ k ˜ L ∞ ([0 ,t ]; B d/ α ) e C R t k u ℓ k Bd/ dτ (cid:16) k g k B d/ α + C (cid:20) e C R t k u ℓ k Bd/ dτ (cid:16) k g k B d/ + ¯ n (cid:17) + 2¯ n (cid:21) Z t k u ℓ k B d/ α dτ (cid:17) . (3.11)Let b ( t, x ) = 1 + ρ ℓ . From Step 1, we know b ∗ >
0. Thus, by Lemma 3.2, A.1 andA.3, we have k ˜ u ℓ k ˜ L ∞ ([0 ,t ]; B d/ − ) + κb ∗ ν k ˜ u ℓ k L ([0 ,t ]; B d/ ) Z t (cid:16) k ρ ℓ ( µ ∆ u ℓ ls + ( µ + λ ) ∇ div u ℓ ls ) k B d/ − + k u ℓ ls · ∇ u ℓ ls k B d/ − + k Q ( ρ ℓ , g ℓ ) k B d/ − (cid:17) dτ × exp (cid:18) C Z t (cid:16) k u ℓ k B d/ + k u ℓ ls k B d/ + ( b ∗ ν ) − /α ¯ ν /α k ρ ℓ k /αB d/ α (cid:17) dτ (cid:19) , (3.12)and similarly k ˜ u ℓ k ˜ L ∞ ([0 ,t ]; B d/ − α ) + κb ∗ ν k ˜ u ℓ k L ([0 ,t ]; B d/ α ) Z t (cid:16) k ρ ℓ ( µ ∆ u ℓ ls + ( µ + λ ) ∇ div u ℓ ls ) k B d/ − α + k u ℓ ls · ∇ u ℓ ls k B d/ − α + k Q ( ρ ℓ , g ℓ ) k B d/ − α (cid:17) dτ × exp (cid:18) C Z t (cid:16) k u ℓ k B d/ + k u ℓ ls k B d/ + ( b ∗ ν ) − /α ¯ ν /α k ρ ℓ k /αB d/ α (cid:17) dτ (cid:19) . (3.13)By Lemma A.3, we get, for all σ ∈ { , α } , that k ρ ℓ ∆ u ℓ ls k B d/ − σ . k ρ ℓ k B d/ k ∆ u ℓ ls k B d/ − σ . k ρ ℓ k B d/ k u ℓ ls k B d/ σ , k ρ ℓ ∇ div u ℓ ls k B d/ − σ . k ρ ℓ k B d/ k u ℓ ls k B d/ σ , k u ℓ ls · ∇ u ℓ ls k B d/ − σ . k u ℓ ls k B d/ k∇ u ℓ ls k B d/ − σ . k u ℓ ls k B d/ k u ℓ ls k B d/ σ . Recall that Q ( ρ ℓ , g ℓ ) = ρ ℓ ∇ ρ ℓ ρ ℓ + 1 − ∇ ρ ℓ + a ¯ m ( ρ ℓ + 1) ∇ g ℓ + B ( ρ ℓ , g ℓ ) h − ¯ m ( ρ ℓ + 1) ∇ ρ ℓ + k − a ¯ nρ ℓ + 1 ∇ ρ ℓ + a g ℓ ∇ ρ ℓ ρ ℓ + 1 + a ¯ m ( g ℓ + g ℓ ρ ℓ ) ∇ g ℓ + a ( k + ¯ m + a ¯ n )¯ m ∇ g ℓ + a k + a ¯ n ¯ m ρ ℓ ∇ g ℓ i , where B ( ρ ℓ , g ℓ ) := h(cid:16) ¯ mρ ℓ + 1 + a ( g ℓ + ¯ n ) − k (cid:17) + 4 k a ( g ℓ + ¯ n ) i − / . ELL-POSEDNESS OF A VISCOUS LIQUID-GAS FLOW MODEL 19
Similarly, for the third term of Q k ( ρ ℓ + 1) ∇ g ℓ k B d/ − σ . ( k ρ ℓ k B d/ + 1) k∇ g ℓ k B d/ − σ . ( k ρ ℓ k B d/ + 1) k g ℓ k B d/ σ . By Lemmas A.3 and A.1, we get for the first two terms of Q (cid:13)(cid:13)(cid:13)(cid:13) ρ ℓ ∇ ρ ℓ ρ ℓ + 1 − ∇ ρ ℓ (cid:13)(cid:13)(cid:13)(cid:13) B d/ − σ . (cid:18)(cid:13)(cid:13)(cid:13)(cid:13) ρ ℓ ρ ℓ + 1 (cid:13)(cid:13)(cid:13)(cid:13) B d/ + 1 (cid:19) k∇ ρ ℓ k B d/ − σ . ( k ρ ℓ k B d/ + 1) k ρ ℓ k B d/ σ , and (cid:13)(cid:13)(cid:13)(cid:13) B ( ρ ℓ , g ℓ ) ∇ ρ ℓ ( ρ ℓ + 1) (cid:13)(cid:13)(cid:13)(cid:13) B d/ − σ . (cid:13)(cid:13)(cid:13)(cid:13) [ B ( ρ ℓ , g ℓ ) − B (0 , ∇ ρ ℓ ( ρ ℓ + 1) (cid:13)(cid:13)(cid:13)(cid:13) B d/ − σ + B (0 , (cid:13)(cid:13)(cid:13)(cid:13) ∇ ρ ℓ ( ρ ℓ + 1) (cid:13)(cid:13)(cid:13)(cid:13) B d/ − σ . (cid:0) k B ( ρ ℓ , g ℓ ) − B (0 , k B d/ + 1 (cid:1) (cid:18)(cid:13)(cid:13)(cid:13)(cid:13) ρ ℓ ρ ℓ + 2 ρ ℓ ( ρ ℓ + 1) (cid:13)(cid:13)(cid:13)(cid:13) B d/ + 1 (cid:19) k∇ ρ ℓ k B d/ − σ . (cid:0) k ρ ℓ k B d/ + k g ℓ k B d/ + 1 (cid:1) k ρ ℓ k B d/ σ . Similarly, (cid:13)(cid:13)(cid:13)(cid:13) B ( ρ ℓ , g ℓ ) ∇ ρ ℓ ρ ℓ + 1 (cid:13)(cid:13)(cid:13)(cid:13) B d/ − σ . (cid:0) k ρ ℓ k B d/ + k g ℓ k B d/ + 1 (cid:1) k ρ ℓ k B d/ σ , (cid:13)(cid:13)(cid:13)(cid:13) B ( ρ ℓ , g ℓ ) g ℓ ∇ ρ ℓ ρ ℓ + 1 (cid:13)(cid:13)(cid:13)(cid:13) B d/ − σ . (cid:0) k ρ ℓ k B d/ + k g ℓ k B d/ + 1 (cid:1) k g ℓ k B d/ k ρ ℓ k B d/ σ , k B ( ρ ℓ , g ℓ )(1 + ρ ℓ ) g ℓ ∇ g ℓ k B d/ − σ . (cid:0) k ρ ℓ k B d/ + k g ℓ k B d/ + 1 (cid:1) k g ℓ k B d/ k g ℓ k B d/ σ , k B ( ρ ℓ , g ℓ ) ∇ g ℓ k B d/ − σ . (cid:0) k ρ ℓ k B d/ + k g ℓ k B d/ + 1 (cid:1) k g ℓ k B d/ σ , k B ( ρ ℓ , g ℓ ) ρ ℓ ∇ g ℓ k B d/ − σ . (cid:0) k ρ ℓ k B d/ + k g ℓ k B d/ + 1 (cid:1) k ρ ℓ k B d/ k g ℓ k B d/ σ . Thus, we get k Q ( ρ ℓ , g ℓ ) k B d/ − σ . (cid:0) k ρ ℓ k B d/ + k g ℓ k B d/ + 1 (cid:1) (cid:0) k ρ ℓ k B d/ σ + k g ℓ k B d/ σ (cid:1) . Therefore, from (3.8)-(3.13), we conclude that M ℓ ( T ) . e C ( U ℓ ls ( T )+ ˜ U ℓ ( T ) / ( b ∗ ν )) ( M + 13 )+ e C ( U ℓ ls ( T )+ ˜ U ℓ ( T ) / ( b ∗ ν )) ( M + 23 )( U ℓ ls ( T ) + ˜ U ℓ ( T ) / ( b ∗ ν )) − ,N ℓ ( T ) . e C ( U ℓ ls ( T )+ ˜ U ℓ ( T ) / ( b ∗ ν )) ( N + ¯ n e C ( U ℓ ls ( T )+ ˜ U ℓ ( T ) / ( b ∗ ν )) ( N + 2¯ n U ℓ ls ( T ) + ˜ U ℓ ( T ) / ( b ∗ ν )) − ¯ n , ˜ U ℓ ( T ) . (cid:0) (¯ νM ℓ ( T ) + U ) U ℓ ls ( T ) + ( M ℓ ( T ) + N ℓ ( T ) + 1) ( M ℓ ( T ) + N ℓ ( T )) T (cid:1) × e C [ U ℓ ls ( T )+ ˜ U ℓ ( T ) / ( b ∗ ν )+( b ∗ ν ) − /α ¯ ν /α ( M ℓ ( T )) /α T ] . Now, if we take T so small thatexp (cid:0) CU ℓ ls ( T ) (cid:1) √ , exp C ˜ U ℓ ( T ) b ∗ ν ! √ , and exp (cid:0) C ( b ∗ ν ) − /α ¯ ν /α ( M ℓ ( T )) /α T (cid:1) , then we have M ℓ ( T ) M + 53 , N ℓ ( T ) N + 5¯ n , ˜ U ℓ ( T ) C (cid:0) ( M + N + 1) ( T + ¯ νU ℓ ls ( T )) + U U ℓ ls ( T ) (cid:1) . (3.14)Noticing that ( F ℓ ρ ℓ , F ℓ g ℓ , F ℓ ˜ u ℓ ) = ( ρ ℓ , g ℓ , ˜ u ℓ ) by the construction of the approxi-mated system. Thus, we have ∂ t (1 + ρ ℓ ) ± + F ℓ ( u ℓ · ∇ (1 + ρ ℓ ) ± ) ± F ℓ ((1 + ρ ℓ ) ± div u ℓ ) = 0 . It follows, by noticing that | ∂ t | f || = | ∂ t f | , that k (1 + ρ ℓ ) ± ( t ) k ∞ k (1 + ρ ℓ ) ± k ∞ + Z t [ k u ℓ · ∇ (1 + ρ ℓ ) ± k ∞ + k (1 + ρ ℓ ) ± div u ℓ k ∞ ] dτ, which yields, by the Gronwall inequality, that k (1 + ρ ℓ ) ± ( t ) k ∞ e R t k div u ℓ k ∞ dτ (cid:18) k (1 + ρ ℓ ) ± k ∞ + Z t k u ℓ k ∞ k∇ ρ ℓ k ∞ dτ (cid:19) e R t k div u ℓ k ∞ dτ (cid:18) k (1 + ρ ℓ ) ± k ∞ + C Z t ( k u ℓ ls k B d/ + k ˜ u ℓ k B d/ ) k ρ ℓ k B d/ dτ (cid:19) e R t k div u ℓ k ∞ dτ (cid:16) k (1 + ρ ℓ ) ± k ∞ + CT ( U + ˜ U ℓ ( T )) M ℓ ( T ) (cid:17) , where we have to choose α = 1 in the previous estimates. Hence, if we assume thatthere exist two positive constants b ∗ and b ∗ such that b ∗ ρ b ∗ , then we can take T small enough such that Z T k div u ℓ k ∞ dτ ln 2 , and CT ( U + ˜ U ℓ ( T )) M ℓ ( T ) , and so b ∗ b ∗ ) ρ ℓ b ∗ + 1) . (3.15)Now, by means of a bootstrap argument, we can get that there exist two constants η and C depending only on d such that if ( ( b ∗ ν ) − /α ¯ ν /α ( M ℓ ( T )) /α T η, ( M + N + 1) ( T + ¯ νU ℓ ls ( T )) + U U ℓ ls ( T ) ηb ∗ ν, (3.16)then we have (3.14) and (3.15).Therefore, T ∗ ℓ may be bounded from below by any time T satisfying (3.16), and theinequalities (3.14) and (3.15) are satisfied by ( ρ ℓ , g ℓ , u ℓ ). In particular, ( ρ ℓ , g ℓ , u ℓ ) ℓ ∈ N is bounded in F T . Step 3: Time derivatives.
In order to pass to the limit in the approximated system,we first give the following lemma.
ELL-POSEDNESS OF A VISCOUS LIQUID-GAS FLOW MODEL 21
Lemma 3.5.
Let ˜ ρ ℓ := ρ ℓ − F ℓ ρ , ˜ g ℓ := g ℓ − F ℓ g . Then the sequences ( ˜ ρ ℓ ) ℓ ∈ N and (˜ g ℓ ) ℓ ∈ N are uniformly bounded in C ([0 , T ]; B d/ ,d/ ) ∩ C / ([0 , T ]; B d/ − ,d/ ) , and the sequence (˜ u ℓ ) ℓ ∈ N is uniformly bounded in ( C ([0 , T ]; B d/ − ,d/ ) ∩ C / ([0 , T ]; B d/ − ,d/ + B d/ − / ,d/ − / )) d . Proof.
From ∂ t ˜ ρ ℓ = − F ℓ ( u ℓ · ∇ ρ ℓ ) + F ℓ (( ρ ℓ + 1)div u ℓ ), we have k ∂ t ˜ ρ ℓ k L ([0 ,T ]; B d/ − ,d/ ) k ∂ t ˜ ρ ℓ k ˜ L ([0 ,T ]; B d/ − ,d/ ) k u ℓ · ∇ ρ ℓ k ˜ L ([0 ,T ]; B d/ − ,d/ ) + k ( ρ ℓ + 1)div u ℓ k ˜ L ([0 ,T ]; B d/ − ,d/ ) . k u ℓ k ˜ L ([0 ,T ]; B d/ ) k ρ ℓ k ˜ L ∞ ([0 ,T ]; B d/ ,d/ ) + ( k ρ ℓ k ˜ L ∞ ([0 ,T ]; B d/ ) + 1) k u ℓ k ˜ L ([0 ,T ]; B d/ ,d/ ) . Since ( u ℓ ) ℓ ∈ N is uniformly bounded in˜ L ∞ ([0 , T ]; B d/ − ,d/ ) ∩ L ([0 , T ]; B d/ ,d/ ) , it is also bounded in ˜ L ([0 , T ]; B d/ ,d/ ) by Lemma A.5. Recall that ( ρ ℓ ) ℓ ∈ N is uni-formly bounded in ˜ L ∞ ([0 , T ]; B d/ ,d/ ), then ( ∂ t ˜ ρ ℓ ) ℓ ∈ N is uniformly bounded in L ([0 , T ]; B d/ − ,d/ ) , and so ( ˜ ρ ℓ ) ℓ ∈ N is uniformly bounded in C / ([0 , T ]; B d/ − ,d/ ) and in C ([0 , T ]; B d/ ,d/ − ) . Similarly, we have the same arguments for ˜ g ℓ .Recall that ∂ t ˜ u ℓ = − F ℓ ( u ℓ · ∇ u ℓ ) + F ℓ [(1 + ρ ℓ )( µ ∆˜ u ℓ + ( µ + λ ) ∇ div ˜ u ℓ )]+ F ℓ [ ρ ℓ ( µ ∆ u ℓ ls + ( µ + λ ) ∇ div u ℓ ls )] − F ℓ Q ( ρ ℓ , g ℓ ) . Since k u ℓ · ∇ u ℓ k L ([0 ,T ]; B d/ − ,d/ ) . k u ℓ k ˜ L ∞ ([0 ,T ]; B d/ − ,d/ ) k u ℓ k L ([0 ,T ]; B d/ ) , k (1 + ρ ℓ )( µ ∆˜ u ℓ + ( µ + λ ) ∇ div ˜ u ℓ ) k L / ([0 ,T ]; B d/ − / ,d/ − / ) . (1 + k ρ ℓ k ˜ L ∞ ([0 ,T ]; B d/ ) ) k ˜ u ℓ k L / ([0 ,T ]; B d/ / ,d/ / ) , k ρ ℓ ( µ ∆ u ℓ ls + ( µ + λ ) ∇ div u ℓ ls ) k L / ([0 ,T ]; B d/ − / ,d/ − / ) . k ρ ℓ k ˜ L ∞ ([0 ,T ]; B d/ ) k u ℓ ls k L / ([0 ,T ]; B d/ / ,d/ / ) , k Q ( ρ ℓ , g ℓ ) k ˜ L ∞ ([0 ,T ]; B d/ − ,d/ ) . (cid:16) k ρ ℓ k ˜ L ∞ ([0 ,T ]; B d/ + k g ℓ k ˜ L ∞ ([0 ,T ]; B d/ ) + 1 (cid:17) (cid:16) k ρ ℓ k ˜ L ∞ ([0 ,T ]; B d/ ,d/ ) + k g ℓ k ˜ L ∞ ([0 ,T ]; B d/ ,d/ ) (cid:17) , by Lemma A.5, we can conclude that (˜ u ℓ ) ℓ ∈ N is uniformly bounded in C / ([0 , T ]; B d/ − ,d/ + B d/ − / ,d/ − / ) and in C ([0 , T ]; B d/ − ,d/ ) . This completes the proof of the lemma. (cid:3)
Step 4: Compactness and convergence.
The proof is based on the Arzel`a-Ascoli the-orem and compact embeddings for Besov spaces. Since it is similar to the argumentsfor global well-posedness, we only give the outlines of the proof.From Lemma 3.5, ( ˜ ρ ℓ ) ℓ ∈ N is uniformly bounded in the space˜ L ∞ ([0 , T ]; B d/ ,d/ )and equicontinuous on [0 , T ] with values in B d/ − ,d/ . Since the embedding B d/ − ,d/ ֒ → B d/ − is (locally) compact, and ( ρ ℓ ) ℓ ∈ N tends to ρ in B d/ ,d/ , we conclude that ( ρ ℓ ) ℓ ∈ N tends (up to an extraction) to some distribution ρ . Given that ( ρ ℓ ) ℓ ∈ N is uniformlybounded in ˜ L ∞ ([0 , T ]; B d/ ), we actually have ρ ∈ ˜ L ∞ ([0 , T ]; B d/ ) . The same arguments are valid for the sequence ( g ℓ ) ℓ ∈ N .From the definition of ( u ℓ ls ) ℓ ∈ N , it is clear that ( u ℓ ls ) ℓ ∈ N tends to the solution u ls of(3.6) in ˜ L ∞ ([0 , t ]; B d/ − ,d/ ) ∩ L ([0 , T ]; B d/ ,d/ ).Since (˜ u ℓ ) ℓ ∈ N is uniformly bounded in ˜ L ∞ ([0 , T ]; B d/ − ,d/ ) and equicontinuouson [0 , T ] with values in B d/ − ,d/ + B d/ − / ,d/ − / , it enable us to conclude that(˜ u ℓ ) ℓ ∈ N converges, up to an extraction, to some function ˜ u ∈ ˜ L ∞ ([0 , T ]; B d/ − ) ∩ L ([0 , T ]; B d/ ).Thus, we can pass to the limit in the system (3.7) and setting u := ˜ u + u ls . Then,( ρ, g, u ) satisfies the system (3.1). Step 5: Continuities in time.
From the first equation of (3.1), we get ∂ t ρ ∈ L ([0 , T ]; B d/ − ,d/ ) which implies ρ ∈ C ([0 , T ]; B d/ − ,d/ ). So does g in the same space. For u , we can derive, fromthe third equation of (3.1), that ∂ t u ∈ ( L + L )([0 , T ]; B d/ − ,d/ ) which yields u ∈C ([0 , T ]; B d/ − ,d/ ).3.4. Uniqueness.
Let ( ρ , g , u ) and ( ρ , g , u ) be two solutions in F T of (3.1) withthe same initial data. Without loss of generality, we can assume that ( ρ , g , u ) isthe solution constructed in the previous subsection such that1 + inf ( t,x ) ∈ [0 ,T ] × R d ρ ( t, x ) > . We want to prove that ( ρ , g , u ) ≡ ( ρ , g , u ) on [0 , T ] × R d . To this goal, we shallestimate the discrepancy ( δρ, δg, δ u ) := ( ρ − ρ , g − g , u − u ) with respect to asuitable norm, satisfying ∂ t δρ + u · ∇ δρ + δ u · ∇ ρ = δρ div u + ( ρ + 1)div δ u ,∂ t δg + u · ∇ δg + δ u · ∇ g = − δg div u − ( g + ¯ n )div δ u ,∂ t δ u + u · ∇ δ u + δ u · ∇ u − (1 + ρ )( µ ∆ δ u ( µ + λ ) ∇ div u ) − δρ ( µ ∆ u + ( µ + λ ) ∇ div u ) + Q ( ρ , g ) − Q ( ρ , g ) = 0 , ( δρ, δg, δ u ) | t =0 = (0 , , ) . (3.17)We shall prove the uniqueness in a larger function space F T := ( C ([0 , T ]; B d/ )) × ( C ([0 , T ]; B d/ ) ∩ L ([0 , T ]; B d/ )) d . By Proposition 3.4, we get for all T ′ ∈ [0 , T ] k δρ k ˜ L ∞ ([0 ,T ′ ]; B d/ ) ELL-POSEDNESS OF A VISCOUS LIQUID-GAS FLOW MODEL 23 e C k u k L ,T ′ ]; Bd/ Z T ′ (cid:16) k δ u · ∇ ρ k B d/ + k δρ div u k B d/ + k ( ρ + 1)div δ u ) k B d/ (cid:17) dτ . e C k u k L ,T ′ ]; Bd/ Z T ′ h k δ u k B d/ k ρ k B d/ + k δρ k B d/ k u k B d/ + (1 + k ρ k B d/ ) k δ u k B d/ i dτ. Using the Gronwall inequality, it yields k δρ k ˜ L ∞ ([0 ,T ′ ]; B d/ ) . e C k u k L ,T ′ ]; Bd/ Z T ′ (cid:2) k δ u k B d/ k ρ k B d/ + (1 + k ρ k B d/ ) k δ u k B d/ (cid:3) dτ C T (cid:0) k δ u k L ([0 ,T ′ ]; B d/ ) + k δ u k L ([0 ,T ′ ]; B d/ ) (cid:1) , (3.18)where C T is independent of T ′ .Similarly, we have k δg k ˜ L ∞ ([0 ,T ′ ]; B d/ ) C T (cid:0) k δ u k L ([0 ,T ′ ]; B d/ ) + k δ u k L ([0 ,T ′ ]; B d/ ) (cid:1) . (3.19)Applying Lemmas 3.2 and A.3 to the third equation of (3.17), it yields k δ u k ˜ L ∞ ([0 ,T ′ ]; B d/ − ) + k δ u k L ([0 ,T ′ ]; B d/ ) Ce C R T ′ [ k u k Bd/ + k u k Bd/ ] dτ Z T ′ (cid:16) k δρ k B d/ k u k B d/ + k Q ( ρ , g ) − Q ( ρ , g ) k B d/ − (cid:17) dτ. By Lemma A.1, we get k Q ( ρ , g ) − Q ( ρ , g ) k B d/ − . (1 + k ( ρ , ρ , g , g ) k B d/ ) ( k δρ k B d/ + k δg k B d/ ) . Thus, it follows that k δ u k ˜ L ∞ ([0 ,T ′ ]; B d/ − ) + k δ u k L ([0 ,T ′ ]; B d/ ) C T ( T ′ + T ′ / )( k δρ k L ∞ ([0 ,T ′ ]; B d/ ) + k δg k L ∞ ([0 ,T ′ ]; B d/ ) ) , (3.20)since u ∈ L ([0 , T ]; B d/ ) by Lemma A.5.From (3.18)-(3.20), it yields, with the help of Lemma A.5, that k δ u k ˜ L ∞ ([0 ,T ′ ]; B d/ − ) + k δ u k L ([0 ,T ′ ]; B d/ ) C T ( T ′ + T ′ / )( k δ u k L ([0 ,T ′ ]; B d/ ) + k δ u k L ([0 ,T ′ ]; B d/ ) ) C T ( T ′ + T ′ / )( k δ u k ˜ L ∞ ([0 ,T ′ ]; B d/ − ) + k δ u k L ([0 ,T ′ ]; B d/ ) ) . Therefore, if we choose T ′ so small that C T ( T ′ + T ′ / ) <
1, then we obtain that( δρ, δg, δ u ) = (0 , , ) on the time interval [0 , T ′ ]. As in the proof of uniqueness forglobal well-posedness, we can extend T ′ to T by the translation with respect to thetime variable, i.e. ( δρ, δg, δ u ) = (0 , , ) on the time interval [0 , T ]. A continuation criterion.Proposition 3.6.
Under the hypotheses of Theorem 3.1, assume that the system (3.1) has a solution ( ρ, g, u ) on [0 , T ) × R d which belongs to F T ′ for all T ′ < T and satisfies ρ, g ∈ L ∞ ([0 , T ); B d/ ,d/ ) , inf ( t,x ) ∈ [0 ,T ) × R d ρ ( t, x ) > − , Z T k∇ u k ∞ dt < ∞ . Then, there exists some T ∗ > T such that ( ρ, g, u ) may be continued on [0 , T ∗ ] × R d to a solution of (3.1) which belongs to F T ∗ .Proof. Recall that u satisfies u t + u · ∇ u − (1 + ρ )( µ ∆ u + ( µ + λ ) ∇ div u ) + Q ( ρ, g ) = 0 , u | t =0 = u . By Lemma 3.2, we get, for T ′ < T , that k u k ˜ L ∞ ([0 ,T ′ ]; B d/ − ,d/ ) + ν k u k L ([0 ,T ′ ]; B d/ ,d/ ) Ce C R T ′ (cid:16) k∇ u k ∞ + k ρ k Bd/ (cid:17) dt k u k B d/ − ,d/ + Z T ′ k ρ k B d/ ,d/ dt ! for some constant C depending only on d and viscosity coefficients. Thus, there existsa constant ε > ρ ( T − ε ) , g ( T − ε ) , u ( T − ε )) yieldsa solution on [0 , ε ]. Since the solution ( ρ, g, u ) is unique on [0 , T ), this provides acontinuation of ( ρ, g, u ) beyond T . (cid:3) Appendix A. Littlewood-Paley theory and Besov spaces
This section is devoted to recall some properties of Littlewood-Paley theory andBesov spaces which will be used in this paper. For more details, one can see [6, 12]and references therein.Let ψ : R d → [0 ,
1] be a radial smooth cut-off function valued in [0 ,
1] such that ψ ( ξ ) = , | ξ | / , smooth , / < | ξ | < / , , | ξ | > / . Let ϕ ( ξ ) be the function ϕ ( ξ ) := ψ ( ξ/ − ψ ( ξ ) . Thus, ψ is supported in the ball { ξ ∈ R d : | ξ | / } , and ϕ is also a smooth cut-offfunction valued in [0 ,
1] and supported in the annulus { ξ ∈ R d : 3 / | ξ | / } . Byconstruction, we have X k ∈ Z ϕ (2 − k ξ ) = 1 , ∀ ξ = 0 . One can define the dyadic blocks as follows. For k ∈ Z , let △ k f := F − ϕ (2 − k ξ ) F f. The formal decomposition f = X k ∈ Z △ k f (A.1)is called homogeneous Littlewood-Paley decomposition. Nevertheless, (A.1) is truemodulo polynomials, in other words (cf.[16]), if f ∈ S ′ ( R d ), then P k ∈ Z △ k f convergesmodulo P [ R d ] and (A.1) holds in S ′ ( R d ) / P [ R d ]. ELL-POSEDNESS OF A VISCOUS LIQUID-GAS FLOW MODEL 25
Definition A.1.
Let s ∈ R . For f ∈ S ′ ( R d ), we write k f k B s = X k ∈ Z ks k△ k f k . A difficulty comes from the choice of homogeneous spaces at this point. Indeed, k·k B s cannot be a norm on { f ∈ S ′ ( R d ) : k f k B s < ∞} because k f k B s = 0 means that f is a polynomial. This enforces us to adopt the following definition for homogeneousBesov spaces (cf. [6]). Definition A.2.
Let s ∈ R and m = − [ d/ − s ]. If m <
0, then we define B s ( R d )as B s = n f ∈ S ′ ( R d ) : k f k B s < ∞ and f = X k ∈ Z △ k f in S ′ ( R d ) o . If m >
0, we denote by P m the set of d variables polynomials of degree less than orequal to m and define B s = n f ∈ S ′ ( R d ) / P m : k f k B s < ∞ and f = X k ∈ Z △ k f in S ′ ( R d ) / P m o . For the composition of functions, we have the following estimates.
Lemma A.1.
Let s > and u ∈ B s ∩ L ∞ . Then, it holds (i) Let F ∈ W [ s ]+2 , ∞ loc ( R d ) with F (0) = 0 . Then F ( u ) ∈ B s . Moreover, there existsa function of one variable C depending only on s and F , and such that k F ( u ) k B s C ( k u k L ∞ ) k u k B s . (ii) If u, v ∈ B d/ , ( v − u ) ∈ B s for s ∈ ( − d/ , d/ and G ∈ W [ d/ , ∞ loc ( R d ) satisfies G ′ (0) = 0 , then G ( v ) − G ( u ) ∈ B s and there exists a function of two variables C depending only on s , N and G , and such that k G ( v ) − G ( u ) k B s C ( k u k L ∞ , k v k L ∞ ) ( k u k B d/ + k v k B d/ ) k v − u k B s . We also need hybrid Besov spaces for which regularity assumptions are different inlow frequencies and high frequencies [6]. We are going to recall the definition of thesenew spaces and some of their main properties.
Definition A.3.
Let s, t ∈ R . We define k f k B s,t = X k ks k△ k f k + X k> kt k△ k f k . Let m = − [ d/ − s ], we then define B s,t ( R d ) = { f ∈ S ′ ( R d ) : k f k B s,t < ∞} , if m < ,B s,t ( R d ) = { f ∈ S ′ ( R d ) / P m : k f k B s,t < ∞} , if m > . Lemma A.2.
We have the following inclusions for hybrid Besov spaces. (i)
We have B s,s = B s . (ii) If s t then B s,t = B s ∩ B t . Otherwise, B s,t = B s + B t . (iii) The space B ,s coincides with the usual inhomogeneous Besov space B s , . (iv) If s s and t > t , then B s ,t ֒ → B s ,t . Let us now recall some useful estimates for the product in hybrid Besov spaces.
Lemma A.3.
Let s , s > and f, g ∈ L ∞ ∩ B s ,s . Then f g ∈ B s ,s and k f g k B s ,s . k f k L ∞ k g k B s ,s + k f k B s ,s k g k L ∞ . Let s ∈ ( − d/ , d/ , f ∈ B d/ and g ∈ B s , then f g ∈ B s and k f g k B s . k f k B d/ k g k B s . Let s , s , t , t d/ such that min( s + s , t + t ) > , f ∈ B s ,t and g ∈ B s ,t .Then f g ∈ B s + s − ,t + t − and k f g k B s s − d/ ,t t − d/ . k f k B s ,t k g k B s ,t . For α, β ∈ R , let us define the following characteristic function on Z :˜ ϕ α,β ( k ) = (cid:26) α, if k ,β, if k > . Then, we can recall the following lemma.
Lemma A.4.
Let F be an homogeneous smooth function of degree m . Suppose that − d/ < s , t , s , t d/ . The following two estimates hold: | ( F ( D ) △ k ( v · ∇ a ) , F ( D ) △ k a ) | . γ k − k ( ˜ ϕ s ,s ( k ) − m ) k v k B d/ k a k B s ,s k F ( D ) △ k a k , | ( F ( D ) △ k ( v · ∇ a ) , △ k b ) + ( △ k ( v · ∇ b ) , F ( D ) △ k a ) | . γ k k v k B d/ × (cid:0) − k ˜ ϕ t ,t ( k ) k F ( D ) △ k a k k b k B t ,t + 2 − k ( ˜ ϕ s ,s ( k ) − m ) k a k B s ,s k△ k b k (cid:1) , where ( · , · ) denotes the -inner product, P k ∈ Z γ k and the operator F ( D ) is definedby F ( D ) f := F − F ( ξ ) F f . In the context of this paper, we also need to use the interpolation spaces of hybridBesov spaces together with a time space such as L p ([0 , T ); B s,t ). Thus, we have tointroduce the Chemin-Lerner type space (cf. [4]) which is a refinement of the space L p ([0 , T ); B s,t ). Definition A.4.
Let p ∈ [1 , ∞ ], T ∈ (0 , ∞ ] and s , s ∈ R . Then we define k f k ˜ L p ([0 ,T ); B s,t ) = X k ks k△ k f k L p ([0 ,T ); L ) + X k> kt k△ k f k L p ([0 ,T ); L ) . Noting that Minkowski’s inequality yields k f k L p ([0 ,T ); B s,t ) k f k ˜ L p ([0 ,T ); B s,t ) , we de-fine spaces ˜ L p ([0 , T ); B s,t ) as follows˜ L p ([0 , T ); B s,t ) = { f ∈ L p ([0 , T ); B s,t ) : k f k ˜ L p ([0 ,T ); B s,t ) < ∞} . If T = ∞ , then we omit the subscript T from the notation ˜ L p ([0 , T ); B s,t ), that is,˜ L p ( B s,t ) for simplicity. We will denote by ˜ C ([0 , T ); B s,t ) the subset of functions of˜ L ∞ ([0 , T ); B s,t ) which are continuous on [0 , T ) with values in B s,t .Let us observe that L ([0 , T ); B s,t ) = ˜ L ([0 , T ); B s,t ), but the embedding˜ L p ([0 , T ); B s,t ) ⊂ L p ([0 , T ); B s,t )is strict if p > ELL-POSEDNESS OF A VISCOUS LIQUID-GAS FLOW MODEL 27
Lemma A.5.
Let s, t, s , t , s , t ∈ R and p, p , p ∈ [1 , ∞ ] . We have k f k ˜ L p ([0 ,T ); B s,t ) k f k θ ˜ L p ([0 ,T ); B s ,t ) k f k − θ ˜ L p ([0 ,T ); B s ,t ) , where p = θp + − θp , s = θs + (1 − θ ) s and t = θt + (1 − θ ) t . Acknowledgments
The authors would like to thank the referees for their valuable comments and Dr.L. Yao for helpful comments on the original version of the manuscript. Hao’s workwas partially supported by the National Natural Science Foundation of China (grant11171327), and the Youth Innovation Promotion Association, Chinese Academy of Sci-ences. Li’s work was partially supported by NSFC grant 11171228 and 11011130029,and the AHRDIHL Project of Beijing Municipality (No. PHR 201006107).
References [1]
H. Bahouri, J. Y. Chemin, and R. Danchin , Fourier Analysis and Nonlinear PartialDifferential Equations , GMW 343, Springer-Verlag, Berlin Heidelberg, 2011.[2]
J. Bergh and J. L¨ofstr¨om , Interpolation Spaces, An Introduction , GMW 223, Springer-Verlag, Berlin Heidelberg, 1976.[3]
C. E. Brennen , Fundamental of Multipule Flow , Cambridge University Press, New York,2005.[4]
J.-Y. Chemin and N. Lerner , Flot de champs de vecteurs non lipschitziens et ´equations deNavier-Stokes , J. Differential Equations, 121 (1992), pp. 314–328.[5]
R. Danchin , Global existence in critical spaces for compressible Navier-Stokes equations , In-vent. Math., 141 (2000), pp. 579–614.[6] ——,
Global existence in critical spaces for flows of compressible viscous and heat-conductivegases , Arch. Ration. Mech. Anal., 160 (2001), pp. 1–39.[7] ——,
Local theory in critical spaces for compressible viscous and heat-conductive gases , Com-mun. Partial Differential Equations, 26 (2001), pp. 1183–1233.[8] ——,
Well-Posedness in critical spaces for barotropic viscous fluids with truly not constantdensity , Commun. Partial Differential Equations, 32 (2007), pp. 1373–1397.[9]
S. Evje, T. Fl˚atten, and H. A. Friis , Global weak solutions for a viscous liquid-gas modelwith transition to single-phase gas flow and vacuum , Nonlinear Anal., 70 (2009), pp. 3864–3886.[10]
S. Evje and K. H. Karlsen , Global existence of weak solutions for a viscous two-phase model ,J. Differential Equations, 245 (2008), pp. 2660–2703.[11] ——,
Global weak solutions for a viscous liquid-gas model with singular pressure law , Commun.Pure Appl. Anal., 8 (2009), pp. 1867–1894.[12]
C. C. Hao, L. Hsiao, and H.-L. Li , Cauchy problem for viscous rotating shallow waterequations , J. Differential Equations, 247 (2009), pp. 3234–3257.[13]
M. Ishii , One-dimensional drift-flux model and constitutive equations for relative motion be-tween phases in various two-phase flow regimes , Argonne National Lab Report, ANL 77–47,October 1977.[14]
M. Ishii and T. Hibiki , Thermo-fluid Dynamics of Two-Phase Flow , Springer, New York,2006.[15]
N. I. Kolev , Multiphase flow dynamics, Vol. 1. Fundamentals , Springer-Verlag, Berlin, 2005;
Vol. 2. Thermal and mechanical interactions , Springer-Verlag, Berlin, 2005.[16]
J. Peetre , New thoughts on Besov spaces , Duke University Mathematical Series 1, DurhamN. C., 1976.[17]
A. Prosperetti and G. Tryggvason , Computational Methods for Multiphase Flow , Cam-bridge University Press, 2007.[18]
G. B. Wallis , One-Dimensional Two-Phase Flow , McGraw Hill, New York, 1979.[19]
L. Yao, T. Zhang, and C. J. Zhu , A blow-up criterion for a 2D viscous liquid-gas two-phaseflow model , J. Differential Equations, 250 (2011), pp. 3362–3378.[20] ——,
Existence and asymptotic behavior of global weak solutions to a 2D viscous liquid-gastwo-phase flow model , SIAM J. Math. Anal., 42 (2010), pp. 1874–1897. [21]
L. Yao and C. J. Zhu , Free boundary value problem for a viscous two-phase model withmass-dependent viscosity , J. Differential Equations, 247 (2009), pp. 2705–2739.[22] ——,
Existence and uniqueness of global weak solution to a two-phase flow model with vacuum ,Math. Ann., 349 (2010), pp. 903–928.[23]
N. Zuber , On the dispersed two-phase flow in the laminar flow regime , Chemical EngineeringScience, 19 (1964), pp. 897–917.[24]
N. Zuber and J. Findlay , Average volumetric concentration in two-phase systems , J. HeatTransfer, 87 (1965), pp. 453–468.
Institute of Mathematics, Academy of Mathematics & Systems Science, and HuaLoo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing100190, China
E-mail address : [email protected] Department of Mathematics, Capital Normal University, Beijing 100048, China
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