Global well-posedness and exponential decay of 2D nonhomogeneous Navier-Stokes and magnetohydrodynamic equations with density-dependent viscosity and vacuum
aa r X i v : . [ m a t h . A P ] F e b Global well-posedness and exponential decay of 2D nonhomogeneousNavier-Stokes and magnetohydrodynamic equations withdensity-dependent viscosity and vacuum ∗ Xin Zhong † Abstract
We establish global well-posedness of strong solutions for the nonhomogeneous magnetohy-drodynamic equations with density-dependent viscosity and initial density allowing vanish intwo-dimensional (2D) bounded domains. Applying delicate energy estimates and Desjardins’ in-terpolation inequality, we derive the global existence of a unique strong solution provided that k∇ µ ( ρ ) k L q is suitably small. Moreover, we also obtain exponential decay rates of the solution.In particular, there is no need to impose some compatibility condition on the initial data despitethe presence of vacuum. As a direct application, it is shown that the similar result also holds forthe nonhomogeneous Navier-Stokes equations with density-dependent viscosity. Key words and phrases . Nonhomogeneous magnetohydrodynamic equations; global well-posedness;exponential decay; density-dependent viscosity; vacuum.2020
Mathematics Subject Classification . 76D05; 76D03.
Magnetohydrodynamics studies the dynamics of electrically conducting fluids and the theoryof the macroscopic interaction of electrically conducting fluids with a magnetic field. The dynamicmotion of the fluid and the magnetic field interact strongly with each other, so the hydrodynamic andelectrodynamic effects are coupled. In the present paper, let Ω ⊂ R be a bounded smooth domain,we are concerned with nonhomogeneous magnetohydrodynamic equations with density-dependentviscosity in Ω: ρ t + div( ρ u ) = 0 , ( ρ u ) t + div( ρ u ⊗ u ) − div(2 µ ( ρ ) D ( u )) + ∇ P = b · ∇ b , b t − ν ∆ b + u · ∇ b − b · ∇ u = , div u = div b = 0 , (1.1)with the initial condition ( ρ, ρ u , b )( x,
0) = ( ρ , ρ u , b )( x ) , x ∈ Ω , (1.2)and the Dirichlet boundary condition u = , b = , x ∈ ∂ Ω , t > . (1.3) ∗ This research was partially supported by National Natural Science Foundation of China (Nos. 11901474, 12071359)and the Innovation Support Program for Chongqing Overseas Returnees (No. cx2020082). † School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China( [email protected] ). ρ, u , b , P are the fluid density, velocity, magnetic field, and pressure, respectively. D ( u ) denotesthe deformation tensor given by D ( u ) = 12 ( ∇ u + ( ∇ u ) tr ) . The viscosity coefficient µ ( ρ ) is a function of the density satisfying µ ∈ C [0 , ∞ ) , µ ≥ µ > µ , while the constant ν > b = ), (1.1) reduces to the nonhomo-geneous Navier-Stokes equations with variable viscosity: ρ t + div( ρ u ) = 0 , ( ρ u ) t + div( ρ u ⊗ u ) − div(2 µ ( ρ ) D ( u )) + ∇ P = , div u = 0 . (1.5)Many authors dealt with the above system. Lions [15, Chapter 2] derived the global weak solutions,yet the uniqueness and regularities of such weak solutions are big open questions. Later, Desjardins [6]introduced the so-called pesudo-energy method and established global weak solutions with higherregularity for 2D case provided that k µ ( ρ ) − k L ∞ is suitably small. It should be noted that thesolution obtained by Desjardins [6] still does not have uniqueness. The main difficulty is due to thefact that Riesz transform does not map continuously from L ∞ to L ∞ (see [6] for details). Meanwhile,if the initial density belongs to some Besov spaces with positive index which guarantee that theinitial density is at least a continuous function, Abidi and Zhang [2] can show the uniqueness of thesolution in the whole plane. Recently, some attention was focused on the well-posedness of strongsolutions to (1.5). For the initial density strictly away from vacuum, Abidi and Zhang [3] proved theglobal well-posedness to the 3D Cauchy problem of (1.5) under the smallness assumptions on both k u k L k∇ u k L and k µ ( ρ ) − k L ∞ . On the other hand, when the initial density allows vacuum,under the compatibility condition − div(2 µ ( ρ ) D ( u )) + ∇ P = √ ρ g for some ( P , g ) ∈ H × L , (1.6)which is proposed by Cho and Kim [4] in order to obtain the local existence of solutions solutions (seealso [21] for an improved result), Huang and Wang [13] and Zhang [28] proved the global existence ofstrong solutions of (1.5) in 3D bounded domains provided the initial velocity is suitably small in somesense. Huang and Wang [12] also obtained the global strong solutions in 2D bounded domains. Veryrecently, by time weighted techniques and energy methods, He et al. [10] and Liu [17] establishedglobal well-posedness of strong solutions to the 3D Cauchy problem without using the compatibilitycondition (1.6) under suitable smallness conditions. Moreover, they also obtained exponential decayrates of the solution. Other interesting results concerning nonhomogeneous Navier-Stokes equationswith constant viscosity can be found in [5, 20, 23].Let’s turn our attention to the study of nonhomogeneous magnetohydrodynamic equations withvariable viscosity coefficient. On one hand, in the absence of vacuum, Abidi and Paicu [1] obtainedthe global wellposedness of strong solutions to the 3D Cauchy problem in the critical Besov spaceunder the assumptions that the initial velocity and magnetic field are small enough and the initialdensity ρ approaches a positive constant. Sokrani [26] proved global existence of strong solutionswhen the initial data are small in some Sobolev spaces (see also related work [25]), which generalizedthe result for nonhomogeneous Navier-Stokes equations with variable viscosity obtained by Abidi andZhang [3]. On the other hand, for the initial density allowing vacuum states, under the compatibilitycondition − div(2 µ ( ρ ) D ( u )) + ∇ P − b · ∇ b = √ ρ g for some ( P , g ) ∈ H × L , (1.7)Li [14] showed global-in-time unique strong solution with density-dependent viscosity and resistivitycoefficients to the 3D case under the condition that k∇ u k L + k∇ b k L is small enough. This result2as later improved by Liu [16] without using (1.7) provided that k ρ k L ∞ + k b k L is suitably small(see related work [19] for 2D case). Very recently, Zhang [29] investigated the global existence andlarge time asymptotic behavior of strong solutions to the 3D Cauchy problem provided that theinitial velocity and magnetic field are suitable small in the ˙ H β -norm for some β ∈ ( , ≤ p ≤ ∞ and integer k >
0, we use L p = L p (Ω) and W k,p = W k,p (Ω) to denotethe standard Lebesgue and Sobolev spaces, respectively. When p = 2, we use H k = W k, (Ω). Thespace H ,σ stands for the closure in H of the space C ∞ ,σ := { φφφ ∈ C ∞ (Ω) | div φφφ = 0 } .Our main result reads as follows: Theorem 1.1
For constant q ∈ (2 , ∞ ) , assume that the initial data ( ρ ≥ , u , b ) satisfies ρ ∈ W ,q (Ω) , ( u , b ) ∈ H ,σ (Ω) , (1.8) then there exists a small positive constant ε depending only on Ω , µ, ¯ µ := sup [0 , k ρ k L ∞ ] µ ( ρ ) , ν, q, k ρ k L ∞ , k∇ u k L , and k∇ b k L such that if k∇ µ ( ρ ) k L q ≤ ε , (1.9) the problem (1.1) – (1.3) has a unique global strong solution ( ρ ≥ , u , b ) satisfying for τ > and r ∈ (2 , q ) , ρ ∈ L ∞ (0 , ∞ ; W ,q ) ∩ C ([0 , ∞ ); W ,q ) , u ∈ L ∞ (0 , ∞ ; H ) ∩ L ∞ ( τ, ∞ ; H ) ∩ L ( τ, ∞ ; W ,r ) , ∇ P ∈ L ∞ ( τ, ∞ ; L ) ∩ L ( τ, ∞ ; L r ) , b ∈ L ∞ (0 , ∞ ; H ) ∩ L ∞ ( τ, ∞ ; H ) ∩ L ( τ, ∞ ; H ) , ∇ u , ∇ b ∈ C ([ τ, ∞ ); L ) , ρ u , b ∈ C ([0 , ∞ ); L ) ,t √ ρ u t , t b t ∈ L ∞ (0 , ∞ ; L ) ,e σ t ∇ u , e σ t ∇ b , e σ t √ ρ u t , e σ t ∆ b ∈ L (0 , ∞ ; L ) ,t ∇ u t , t ∇ b t ∈ L (0 , ∞ ; L ) , (1.10) where σ := min n µd k ρ k L ∞ , νd o with d the diameter of Ω . Moreover, there exists a positive constant C depends only on Ω , µ, ¯ µ, ν, q, k ρ k L ∞ , k∇ u k L , and k∇ b k L such that for t ≥ , k u ( · , t ) k H + k∇ P ( · , t ) k L + k b ( · , t ) k H + k√ ρ u t k L + k b t k L ≤ Ce − σt . (1.11) Remark 1.1
The conclusion in Theorem 1.1 is somewhat surprising since the smallness condition (1.9) is independent of the initial magnetic field explicitly and just the same as that of nonhomoge-neous Navier-Stokes equations ( [12]), which is in sharp contrast to the recent works [14, 16, 19, 29],where the authors considered global strong solution to the nonhomogeneous magnetohydrodynamicequations with density-dependent viscosity with the help of small initial magnetic field in some sense.
Remark 1.2
For our case that the viscosity µ ( ρ ) depends on ρ , in order to bound the L -norm ofthe gradients of the velocity and magnetic field, we need the smallness condition on the L q -norm of he gradient of the viscosity (see Lemma 3.3). In the special case that µ is a positive constant, it isclear that (1.9) holds true. Hence Theorem 1.1 implies that for any given (large) initial data ( ρ , u ) satisfying (1.8) , there exists a unique global strong solution to the problem (1.1) – (1.3) with constantviscosity µ . We now make some comments on the key ingredients of the proof of Theorem 1.1. The localexistence and uniqueness of strong solutions to the problem (1.1)–(1.3) follows from [27] (see Lemma2.1). Thus our efforts are devoted to establishing global a priori estimates on solutions in suitablehigher-order norms. We will adapt some basic idea used in Huang and Wang [12], where they inves-tigated the global existence of strong solutions to the 2D nonhomogeneous Navier-Stokes equationswith density-dependent viscosity and vacuum. However, compared with [12], the proof of Theorem1.1 is much more involved due to the strong coupling between the velocity and the magnetic fieldand the absence of the compatibility condition (1.7). Consequently, some new ideas are needed toovercome these difficulties.As mentioned by [13, 28], the key ingredient here is to get the time-independent bounds on the L (0 , T ; L ∞ )-norm of ∇ u and then the L ∞ (0 , T ; L q )-norm of ∇ µ ( ρ ). First, applying the upper boundson the density (see (3.2)) and the Poincar´e inequality, we derive that k√ ρ u k L + k b k L decays withthe rate of e − σt for some σ > µ, ν, k ρ k L ∞ , and the diameter of the Ω (see (3.5)).Next, we need to obtain time-weighted estimates of k∇ u k L + k∇ b k L . To this end, we assume k∇ µ ( ρ ) k L q ≤ , T ]. Motivated by [12], we make use of Desjardins’ interpolation inequality (seeLemma 2.4) to control k√ ρ u k L , and we find the key point is to control the term R b · ∇ b · u t dx (see (3.20)). Inspired by [22], multiplying (3.1) by ∆ b instead of the usual b t (see [11]), the term R b ·∇ b · u t dx can be controlled after using delicate Gagliardo-Nirenberg inequality (see (3.23)). Next,using the structure of the 2D magnetic equation, we multiply (3.1) by | b | b and thus obtain someuseful a priori estimates on k| b ||∇ b |k L , which is crucial in deriving the time-independent estimateson both the L ∞ (0 , T ; L )-norm of t √ ρ u t and the L (0 , T ; L )-norm of t ∇ u (see (3.38)). In fact, allthese decay-in-time rates play an important role in obtaining the desired uniform bound (with respectto time) on the L (0 , T ; L ∞ )-norm of ∇ u (see (3.52)), which in particular implies L ∞ (0 , T ; L q )-normof the gradient of the viscosity µ ( ρ ) provided k∇ µ ( ρ ) k L q ≤ ε as stated in Theorem 1.1 (see (3.71)).Finally, the higher order estimates on solutions are obtained (see Lemma 3.7) by considering timeweighted type due to the lacking of the compatibility conditions.As a direct consequence of Theorem 1.1, we have the following global well-posedness and expo-nential decay of 2D nonhomogeneous Navier-Stokes equations with density-dependent viscosity. Theorem 1.2
For constant q ∈ (2 , ∞ ) , assume that the initial data ( ρ ≥ , u ) satisfies ρ ∈ W ,q , u ∈ H ,σ (Ω) , (1.12) then there exists a small positive constant ε depending only on Ω , µ, ¯ µ := sup [0 , k ρ k L ∞ ] µ ( ρ ) , ν, q, k ρ k L ∞ ,and k∇ u k L such that if k∇ µ ( ρ ) k L q ≤ ε , (1.13) the nonhomogeneous Navier-Stokes equations (1.1) – (1.3) with b = have a unique global strongsolution ( ρ ≥ , u ) , which satisfies (1.10) and (1.11) with b = and σ = µd k ρ k L ∞ . Remark 1.3
Compared with [12], on one hand, there is no need to impose the compatibility conditionon the initial data despite the presence of vacuum. On the other hand, exponential decay rates of thesolution is a new result.
Remark 1.4
We remark that the smallness condition (1.13) allows any given large initial data ( ρ , u ) satisfying (1.12) provided that the viscosity µ is a positive constant, which is in sharp contrastto [17] where the smallness assumption on k ρ k L ∞ is needed. emark 1.5 It is not hard to prove that the strong-weak uniqueness theorem [15, Theorem 2.7] stillholds for the initial data ( ρ , u ) satisfying (1.12) after modifying its proof slightly. Therefore, ourTheorem 1.2 can be regarded as the uniqueness and regularity theory of Lions’s weak solutions [15]in 2D case with ∇ µ ( ρ ) suitably small in the L q -norm. The rest of this paper is organized as follows. In Section 2, we collect some elementary facts andinequalities that will be used later. Section 3 is devoted to the a priori estimates. Finally, we givethe proof of Theorem 1.1 in Section 4.
In this section, we will recall some known facts and elementary inequalities that will be usedfrequently later.We begin with the local existence and uniqueness of strong solutions whose proof can be foundin [27].
Lemma 2.1
Assume that ( ρ , u , b ) satisfies (1.8) . Then there exist a small time T > and aunique strong solution ( ρ, u , b ) to the problem (1.1) – (1.3) in Ω × (0 , T ) . Next, the following Gagliardo-Nirenberg inequality (see [8, Theorem 10.1, p. 27]) will be usefulin the next section.
Lemma 2.2 (Gagliardo-Nirenberg)
Let Ω ⊂ R be a bounded smooth domain. Assume that ≤ q, r ≤ ∞ , and j, m are arbitrary integers satisfying ≤ j < m . If v ∈ W m,r (Ω) ∩ L q (Ω) , then wehave k D j v k L p ≤ C k v k − aL q k v k aW m,r , where − j + 2 p = (1 − a ) 2 q + a (cid:16) − m + 2 r (cid:17) , and a ∈ ( [ jm , , if m − j − r is a nonnegative integer , [ jm , , otherwise . The constant C depends only on m, j, q, r, a , and Ω . In particular, we have k v k L ≤ C k v k L k v k H , (2.1) which will be used frequently in the next section. Next, we give some regularity results for the following Stokes system with variable viscositycoefficient − div(2 µ ( ρ ) D ( u )) + ∇ P = F , x ∈ Ω , div u = 0 , x ∈ Ω , u = , x ∈ ∂ Ω , R P dx = 0 . (2.2) Lemma 2.3
Assume that ρ ∈ W ,q (Ω) with < q < ∞ , ≤ ρ ≤ ¯ ρ , µ ∈ C [0 , ∞ ) , and µ ≤ µ ( ρ ) ≤ ¯ µ .Let ( u , P ) ∈ H × L be the unique weak solution to the problem (2.2) , then there exists a positiveconstant C depending only on Ω , ¯ ρ, µ, ¯ µ such that the following regularity results hold true: If F ∈ L (Ω) , then ( u , P ) ∈ H × H and k u k H ≤ C k F k L (1 + k∇ µ ( ρ ) k L q ) qq − , k P k H ≤ C k F k L (1 + k∇ µ ( ρ ) k L q ) q − q − . • If F ∈ L r for some r ∈ (2 , q ) , then ( u , P ) ∈ W ,r × W ,r and k u k W ,r ≤ C k F k L r (1 + k∇ µ ( ρ ) k L q ) qr q − r ) , k P k W ,r ≤ C k F k L r (1 + k∇ µ ( ρ ) k L q ) qr q − r ) . Proof.
See [4, 12]. ✷ Finally, by zero extension of u outside Ω, we can derive the following lemma due to Desjardins(see [6, Lemma 1] or [7]), which plays a key role in the proof of Lemma 3.3 in the next section. Lemma 2.4
Let Ω ⊂ R be a bounded smooth domain. Suppose that ≤ ρ ≤ ¯ ρ and u ∈ H (Ω) ,then we have k√ ρ u k L ≤ C (¯ ρ, Ω)(1 + k√ ρ u k L ) k∇ u k L q log(2 + k∇ u k L ) . (2.3) In this section, we will establish some necessary a priori bounds for strong solutions ( ρ, u , b ) tothe problem (1.1)–(1.3) to extend the local strong solution. Thus, let T > ρ, u , b ) be the strong solution to (1.1)–(1.3) on Ω × (0 , T ] with initial data ( ρ , u , b ) satisfying(1.8). Before proceeding, we rewrite another equivalent form of the system (1.1) as the following ρ t + u · ∇ ρ = 0 ,ρ u t + ρ u · ∇ u − div(2 µ ( ρ ) D ( u )) + ∇ P = b · ∇ b , b t − ν ∆ b + u · ∇ b − b · ∇ b = , div u = div b = 0 . (3.1)In what follows, we denote by Z · dx = Z Ω · dx. We sometimes use C ( f ) to emphasize the dependence on f .First, since (3.1) is a transport equation, we have directly the following result. Lemma 3.1
For ( x, t ) ∈ Ω × [0 , T ] , it holds that ≤ ρ ( x, t ) ≤ k ρ k L ∞ . (3.2) Remark 3.1
Since µ ( ρ ) is a continuously differentiable function, we deduce from (3.2) and (1.4) that < µ ≤ µ ( ρ ) ≤ ¯ µ := sup [0 , k ρ k L ∞ ] µ ( ρ ) < ∞ , (3.3) and k µ ′ ( ρ ) k L ∞ (0 ,T ; L ∞ ) < ∞ . Next, the following lemma gives the basic energy estimates.6 emma 3.2
It holds that sup ≤ t ≤ T (cid:0) k√ ρ u k L + k b k L (cid:1) + Z T (cid:0) µ k∇ u k L + ν k∇ b k L (cid:1) dt ≤ k√ ρ u k L + k b k L , (3.4) and sup ≤ t ≤ T (cid:2) e σt (cid:0) k√ ρ u k L + k b k L (cid:1) (cid:3) + Z T e σt (cid:0) µ k∇ u k L + ν k∇ b k L (cid:1) dt ≤ k√ ρ u k L + k b k L , (3.5) where σ := min n µd k ρ k L ∞ , νd o with d the diameter of Ω .Proof.
1. Multiplying (3.1) by u , (3.1) by b , and integration (by parts) over Ω, we derive that12 ddt (cid:0) k√ ρ u k L + k b k L (cid:1) + 2 Z µ ( ρ ) D ( u ) · ∇ u dx + ν k∇ b k L = 0 . (3.6)Noting that 2 Z µ ( ρ ) D ( u ) · ∇ u dx = Z µ ( ρ )( ∂ i u j + ∂ j u i ) ∂ i u j dx = 12 Z µ ( ρ )( ∂ i u j + ∂ j u i )( ∂ i u j + ∂ j u i ) dx = 2 Z µ ( ρ ) | D ( u ) | dx, and 2 Z | D ( u ) | dx = 12 Z ( ∂ i u j + ∂ j u i )( ∂ i u j + ∂ j u i ) dx = Z |∇ u | dx + Z ∂ i u j ∂ j u i dx = Z |∇ u | dx, (3.7)we thus obtain from (3.6) and (1.4) that ddt (cid:0) k√ ρ u k L + k b k L (cid:1) + 2 (cid:0) µ k∇ u k L + ν k∇ b k L (cid:1) ≤ . (3.8)Integrating the above inequality over (0 , T ) gives the desired (3.4).2. It follows from Poincar´e’s inequality (see [24, (A.3), p. 266]) and (3.2) that k√ ρ u k L ≤ k ρ k L ∞ k u k L ≤ k ρ k L ∞ d k∇ u k L , (3.9)where d is the diameter of Ω. Hence, we get1 d k ρ k L ∞ k√ ρ u k L ≤ k∇ u k L , d k b k L ≤ k∇ b k L . (3.10)Consequently, letting σ := min n µd k ρ k L ∞ , νd o , then we derive from (3.8) and (3.10) that ddt (cid:2) e σt (cid:0) k√ ρ u k L + k b k L (cid:1) (cid:3) + e σt (cid:0) µ k∇ u k L + ν k∇ b k L (cid:1) ≤ . (3.11)Thus, integrating (3.11) with respect to t gives the desired (3.5). ✷ Lemma 3.3
Let q be as in Theorem 1.1 and ¯ µ be as in (3.3) , assume that sup ≤ t ≤ T k∇ µ ( ρ ) k L q ≤ , (3.12)7 hen there exists a positive constant C depending only on Ω , µ , ¯ µ , ν , q , k ρ k L ∞ , k∇ u k L , and k∇ b k L such that sup ≤ t ≤ T (cid:0) k∇ u k L + k∇ b k L (cid:1) + Z T (cid:0) k√ ρ u t k L + k ∆ b k L + k| b ||∇ b |k L (cid:1) dt ≤ C. (3.13) Moreover, for σ as in Lemma 3.2, one has sup ≤ t ≤ T (cid:2) e σt (cid:0) k∇ u k L + k∇ b k L (cid:1) (cid:3) + Z T e σt (cid:0) k√ ρ u t k L + k ∆ b k L + k| b ||∇ b |k L (cid:1) dt ≤ C. (3.14) Proof.
1. Since µ ( ρ ) is a continuously differentiable function, we obtain from (3.1) that[ µ ( ρ )] t + u · ∇ µ ( ρ ) = 0 . (3.15)Multiplying (3.1) by u t and integrating the resulting equation over Ω imply that2 Z µ ( ρ ) D ( u ) · ∇ u t dx + Z ρ | u t | dx = − Z ρ u · ∇ u · u t dx + Z b · ∇ b · u t dx. (3.16)Similarly to (3.11), we get2 Z µ ( ρ ) D ( u ) · ∇ u t dx = 2 Z µ ( ρ ) D ( u ) · D ( u ) t dx = ddt Z µ ( ρ ) | D ( u ) | dx − Z [ µ ( ρ )] t | D ( u ) | dx, which combined with (3.16) and (3.15) leads to ddt Z µ ( ρ ) | D ( u ) | dx + Z ρ | u t | dx = − Z ρ u ·∇ u · u t dx − Z u ·∇ µ ( ρ ) | D ( u ) | dx + Z b ·∇ b · u t dx. (3.17)By H¨older’s and Gagliardo-Nirenberg inequalities, we have (cid:12)(cid:12)(cid:12)(cid:12) − Z ρ u · ∇ u · u t dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k√ ρ u t k L + 2 k√ ρ u k L k∇ u k L ≤ k√ ρ u t k L + C (Ω) k√ ρ u k L k∇ u k L k u k H . (3.18)By Sobolev’s inequality, (3.12), and Gagliardo-Nirenberg inequality, we arrive at (cid:12)(cid:12)(cid:12)(cid:12) − Z u · ∇ µ ( ρ ) | D ( u ) | dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z | u ||∇ µ ( ρ ) ||∇ u | dx ≤ C k∇ µ ( ρ ) k L q k u k L qq − k∇ u k L ≤ C (Ω) k∇ u k L k∇ u k L k∇ u k H ≤ C k∇ u k L k u k H . (3.19)Integration by parts together with div b = 0 and b | ∂ Ω = , we infer from Sobolev’s inequality that Z b · ∇ b · u t dx = − ddt Z b · ∇ u · b dx + Z b t · ∇ u · b dx + Z b · ∇ u · b t dx = − ddt Z b · ∇ u · b dx + Z (∆ b − u · ∇ b + b · ∇ u ) · ∇ u · b dx + Z b · ∇ u · (∆ b − u · ∇ b + b · ∇ u ) dx ≤ − ddt Z b · ∇ u · b dx + ν k ∆ b k L + C k b k L + C k∇ u k L C k b k L k u k L ∞ k∇ b k L k∇ u k L ≤ − ddt Z b · ∇ u · b dx + ν k ∆ b k L + C k b k L k∇ b k L + C k∇ u k L k u k H + C k b k L k u k L k∇ u k L k∇ b k L k∇ u k L k∇ u k H ≤ − ddt Z b · ∇ u · b dx + ν k ∆ b k L + C k b k L k∇ b k L + C k∇ u k L k u k H + C k b k L k∇ b k L k∇ u k L k u k H , (3.20)where we have used the following Gagliardo-Nirenberg inequality k u k L ∞ ≤ C k u k L k∇ u k L , k∇ u k L ≤ C k∇ u k L k∇ u k H . Substituting (3.18)–(3.20) into (3.17), we derive ddt
Z (cid:0) µ ( ρ ) | D ( u ) | + b · ∇ u · b (cid:1) dx + 12 k√ ρ u t k L ≤ C (cid:0) k√ ρ u k L + k∇ u k L (cid:1) k∇ u k L k u k H + ν k ∆ b k L + C k∇ b k L + C k b k L k∇ b k L k∇ u k L k u k H . (3.21)2. Multiplying (3.1) by ∆ b and integrating the resulting equality over Ω, it follows from H¨older’sand Gagliardo-Nirenberg inequalities that ddt Z |∇ b | dx + ν Z | ∆ b | dx ≤ C Z |∇ u ||∇ b | dx + C Z |∇ u || b || ∆ b | dx ≤ C k∇ u k L k∇ b k L k ∆ b k L + C k∇ u k L k b k L k ∆ b k L ≤ C k∇ u k L k∇ u k L + C (1 + k b k L ) k∇ b k L + ν k ∆ b k L , (3.22)which together with (3.21) and (3.4) gives rise to B ′ ( t ) + 12 k√ ρ u t k L + ν k ∆ b k L ≤ C (cid:0) k√ ρ u k L + k∇ u k L (cid:1) k∇ u k L k u k H + C k∇ b k L + C k b k L k∇ b k L k∇ u k L k u k H , (3.23)where B ( t ) := Z (cid:0) µ ( ρ ) | D ( u ) | + |∇ b | + b · ∇ u · b (cid:1) dx (3.24)satisfies µ k∇ u k L + k∇ b k L − C k b k L ≤ B ( t ) ≤ C k∇ u k L + C k∇ b k L (3.25)owing to Gagliardo-Nirenberg inequality, (3.4), and the following estimate Z | b · ∇ u · b | dx ≤ µ k∇ u k L + C k b k L . (3.26)3. Multiplying(3.1) by | b | b and integrating the resulting equality by parts over Ω, we obtainfrom Gagliardo-Nirenberg inequality that14 ddt k b k L + k|∇ b || b |k L + 12 k∇| b | k L ≤ C k∇ u k L k| b | k L C k∇ u k L k| b | k L k∇| b | k L ≤ k∇| b | k L + C k∇ u k L k b k L , (3.27)which together with Gronwall’s inequality and (3.4) impliessup ≤ t ≤ T k b k L + Z T k| b ||∇ b |k L dt ≤ C. (3.28)This along with (3.23) yields B ′ ( t ) + 12 k√ ρ u t k L + ν k ∆ b k L ≤ C (cid:0) k√ ρ u k L + k∇ u k L (cid:1) k∇ u k L k u k H + C k∇ b k L + C k∇ b k L k∇ u k L k u k H . (3.29)4. Recall that ( u , P ) satisfies the following Stokes system with variable viscosity − div(2 µ ( ρ ) D ( u )) + ∇ P = − ρ u t − ρ u · ∇ u + b · ∇ b , x ∈ Ω , div u = 0 , x ∈ Ω , u = , x ∈ ∂ Ω . Applying Lemma 2.3 with F = − ρ u t − ρ u · ∇ u + b · ∇ b , we obtain from (3.2) and (3.12) that k u k H + k∇ P k L ≤ C ( k ρ u t k L + k ρ u · ∇ u k L + k b · ∇ b k L ) (1 + k∇ µ ( ρ ) k L q ) qq − ≤ C k√ ρ u t k L + C k√ ρ u k L k∇ u k L + C k| b ||∇ b |k L ≤ C k√ ρ u t k L + C k√ ρ u k L k∇ u k L k u k H + C k| b ||∇ b |k L ≤ C k√ ρ u t k L + C k√ ρ u k L k∇ u k L + 12 k u k H + C k| b ||∇ b |k L , and thus k u k H + k∇ P k L ≤ C k√ ρ u t k L + C k√ ρ u k L k∇ u k L + C k| b ||∇ b |k L . (3.30)Inserting (3.30) into (3.21) and applying Cauchy-Schwarz inequality, we deduce that B ′ ( t ) + 12 k√ ρ u t k L + ν k ∆ b k L ≤ k√ ρ u t k L + C k√ ρ u k L k∇ u k L + C k∇ u k L + C k∇ b k L + ε k| b ||∇ b |k L . (3.31)Noting that k∇ u k L k b k L ≤ C k∇ u k L k b k L k∇ b k L ≤ C k∇ u k L + C k∇ b k L due to (3.4) and Cauchy-Schwarz inequality. Thus, adding (3.27) multiplied by 4( C + 1) to (3.31)and choosing ε suitably small, we obtain after using (2.3) and (3.4) that ddt (cid:0) B ( t ) + ( C + 1) k b k L (cid:1) + k√ ρ u t k L + ν k ∆ b k L + k| b ||∇ b |k L ≤ C k∇ b k L + C k∇ u k L + C k√ ρ u k L k∇ u k L ≤ C k∇ b k L k∇ b k L + C k∇ u k L k∇ u k L + C k∇ u k L k∇ u k L log(2 + k∇ u k L ) . (3.32)Set f ( t ) := 2 + B ( t ) + ( C + 1) k b k L , g ( t ) := k∇ u k L + k∇ b k L , f ′ ( t ) ≤ Cg ( t ) f ( t ) + Cg ( t ) f ( t ) log f ( t ) , which yields (log f ( t )) ′ ≤ Cg ( t ) + Cg ( t ) log( f ( t )) . (3.33)We thus infer from (3.33), Gronwall’s inequality, (3.4), and (3.25) thatsup ≤ t ≤ T (cid:0) k∇ u k L + k∇ b k L + k b k L (cid:1) ≤ C. (3.34)Integrating (3.32) with respect to t together with (3.34) and (3.4) leads to Z T (cid:0) k√ ρ u t k L + k ∆ b k L + k| b ||∇ b |k L (cid:1) dt ≤ C. (3.35)This along with (3.34) gives the desired (3.13).5. Multiplying (3.32) by e σt and applying (3.34), we derive from (3.25) that ddt (cid:0) e σt B ( t ) + ( C + 1) e σt k b k L (cid:1) + e σt (cid:0) k√ ρ u t k L + ν k ∆ b k L + k| b ||∇ b |k L (cid:1) ≤ Ce σt (cid:0) k∇ b k L + k∇ u k L (cid:1) + σe σt B ( t ) + σ ( C + 1) e σt k b k L ≤ Ce σt (cid:0) k∇ b k L + k∇ u k L (cid:1) + Ce σt k b k L k∇ b k L ≤ Ce σt (cid:0) k∇ b k L + k∇ u k L (cid:1) . (3.36)Integrating (3.36) over (0 , T ) together with (3.25) leads to (3.14). ✷ Remark 3.2
Under the condition (3.12) , it follows from (2.3) , (3.2) , (3.4) , and (3.13) that sup ≤ t ≤ T k√ ρ u k L ≤ C. (3.37) Lemma 3.4
Let the condition (3.12) be in force, then there exists a positive constant C dependingonly on Ω , µ, ¯ µ, ν, q, k ρ k L ∞ , k∇ u k L , and k∇ b k L such that for i ∈ { , } , sup ≤ t ≤ T (cid:2) t i (cid:0) k√ ρ u t k L + k b t k L (cid:1)(cid:3) + Z T t i (cid:0) k∇ u t k L + k∇ b t k L (cid:1) dt ≤ C. (3.38) Moreover, for σ as that in Lemma 3.2, one has for < τ < T , sup τ ≤ t ≤ T (cid:2) e σt (cid:0) k√ ρ u t k L + k b t k L (cid:1) (cid:3) + Z Tτ e σt (cid:0) k∇ u t k L + k∇ b t k L (cid:1) dt ≤ C ( τ ) . (3.39) Proof.
1. Differentiating (3.1) with respect to t , we arrive at ρ u tt + ρ u · ∇ u t − div(2 µ ( ρ ) D ( u t ))= −∇ P t + ρ t ( u t + u · ∇ u ) − ρ u t · ∇ u + div(2 µ t D ( u )) + b t · ∇ b + b · ∇ b t . (3.40)Multiplying (3.40) by u t and integrating (by parts) over Ω and using (1.1) yield12 ddt Z ρ | u t | dx + 2 Z µ ( ρ ) D ( u t ) · ∇ u t dx = Z div( ρ u ) | u t | dx + Z div( ρ u ) u · ∇ u · u t dx − Z ρ u t · ∇ u · u t dx − Z µ t D ( u ) · ∇ u t dx + Z b t · ∇ b · u t dx + Z b · ∇ b t · u t dx =: X i =1 J i . (3.41)11y virtue of H¨older’s inequality, Sobolev’s inequality, (3.2), (3.13), and (3.15), we find that | J | = (cid:12)(cid:12)(cid:12)(cid:12) − Z ρ u · ∇| u t | dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ρ k L ∞ k u k L k√ ρ u t k L k∇ u t k L ≤ C k ρ k L ∞ k∇ u k L k√ ρ u t k L k√ ρ u t k L k∇ u t k L ≤ C k ρ k L ∞ k∇ u k L k√ ρ u t k L k∇ u t k L ≤ µ k∇ u t k L + C k√ ρ u t k L ; | J | = (cid:12)(cid:12)(cid:12)(cid:12) − Z ρ u · ∇ ( u · ∇ u · u t ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z (cid:0) ρ | u ||∇ u | | u t | + ρ | u | |∇ u || u t | + ρ | u | |∇ u ||∇ u t | (cid:1) dx ≤k ρ k L ∞ k u k L k∇ u k L k∇ u k L k u t k L + k ρ k L ∞ k u k L k∇ u k L k u t k L + k ρ k L ∞ k u k L k∇ u k L k∇ u t k L ≤ C k ρ k L ∞ k∇ u k L k u k H k∇ u t k L ≤ µ k∇ u t k L + C k u k H ; | J | ≤k∇ u k L k√ ρ u t k L ≤ C k∇ u k L k√ ρ u t k L k√ ρ u t k L ≤ C k ρ k L ∞ k∇ u k L k√ ρ u t k L k∇ u t k L ≤ µ k∇ u t k L + C k√ ρ u t k L ; | J | ≤ C Z | u ||∇ µ ( ρ ) ||∇ u ||∇ u t | dx ≤ C k u k L qq − k∇ µ ( ρ ) k L q k∇ u k L qq − k∇ u t k L ≤ C k∇ u k L k∇ u k H k∇ u t k L ≤ µ k∇ u t k L + C k u k H ; | J | = (cid:12)(cid:12)(cid:12)(cid:12) − Z b t · ∇ u t · b dx (cid:12)(cid:12)(cid:12)(cid:12) ≤k b t k L k∇ u t k L k b k L ≤ C k b t k L k∇ b t k L k∇ u t k L ≤ µ k∇ u t k L + C ( δ ) k b t k L + δ k∇ b t k L ; | J | = (cid:12)(cid:12)(cid:12)(cid:12) − Z b · ∇ u t · b t dx (cid:12)(cid:12)(cid:12)(cid:12) ≤k b k L k∇ u t k L k b t k L ≤ C k∇ u t k L k b t k L k∇ b t k L ≤ µ k∇ u t k L + C ( δ ) k b t k L + δ k∇ b t k L . Substituting the above estimates into (3.41) and noting that2 Z µ ( ρ ) D ( u t ) · ∇ u t dx ≥ µ k∇ u t k L , we derive that ddt k√ ρ u t k L + k∇ u t k L ≤ C k√ ρ u t k L + C k u k H + δ k∇ b t k L + C k b t k L . (3.42)12. It follows from (3.30) and (3.37) that k u k H + k∇ P k L ≤ C k√ ρ u t k L + C k∇ u k L + C k| b ||∇ b |k L . (3.43)This along with (3.1) , (3.13), Gagliardo-Nirenberg inequality, and Sobolev’s inequality leads to k b t k L ≤ C k ∆ b k L + C k u k L ∞ k∇ b k L + C k b k L k∇ u k L ≤ C k ∆ b k L + C k u k L k∇ u k L k∇ b k L + C k∇ b k L k∇ u k L k∇ u k H ≤ C k ∆ b k L + C k∇ u k L k∇ u k L k∇ u k H k∇ b k L + C k∇ b k L k∇ u k L k∇ u k H ≤ C k ∆ b k L + C k∇ u k H + C k∇ u k L + C k∇ b k L ≤ C (cid:0) k√ ρ u t k L + k ∆ b k L + k| b ||∇ b |k L (cid:1) + C (cid:0) k∇ u k L + k∇ b k L (cid:1) , (3.44)which combined with (3.42) and (3.43) gives ddt k√ ρ u t k L + k∇ u t k L ≤ C (cid:0) k√ ρ u t k L + k ∆ b k L + k| b ||∇ b |k L (cid:1) + C (cid:0) k∇ u k L + k∇ b k L (cid:1) + δ k∇ b t k L . (3.45)3. Differentiating (3.1) with respect to t and multiplying the resulting equations by b t , we obtainfrom integration by parts, (3.13), and Sobolev’s inequality that12 ddt k b t k L + ν k∇ b t k L ≤ C ( k| u t || b |k L + k| u || b t |k L ) k∇ b t k L ≤ C ( k u t k L k b k L + k u k L k b t k L ) k∇ b t k L ≤ C (cid:16) k∇ u t k L k∇ b k L + k∇ u k L k b t k L k∇ b t k L (cid:17) k∇ b t k L ≤ ν k∇ b t k L + C k∇ u t k L + C k b t k L , which together with (3.44) implies that ddt k b t k L + ν k∇ b t k L ≤ C k∇ u t k L + C (cid:0) k√ ρ u t k L + k ∆ b k L + k| b ||∇ b |k L (cid:1) + C (cid:0) k∇ u k L + k∇ b k L (cid:1) (3.46)for some positive constant C . Adding (3.45) multiplied by 2 C to (3.46) and then choosing δ = ν C ,we deduce that ddt (cid:16) C k√ ρ u t k L + k b t k L (cid:17) + C k∇ u t k L + ν k∇ b t k L ≤ C (cid:0) k√ ρ u t k L + k ∆ b k L + k| b ||∇ b |k L (cid:1) + C (cid:0) k∇ u k L + k∇ b k L (cid:1) . (3.47)4. Multiplying (3.47) by t i ( i ∈ { , } ) yields ddt (cid:16) C t i k√ ρ u t k L + t i k b t k L (cid:17) + C t i k∇ u t k L + ν t i k∇ b t k L ≤ Ct i (cid:0) k√ ρ u t k L + k ∆ b k L + k| b ||∇ b |k L (cid:1) + Ct i (cid:0) k∇ u k L + k∇ b k L (cid:1) + Ct i − (cid:0) k√ ρ u t k L + k b t k L (cid:1) ≤ Ct i (cid:0) k√ ρ u t k L + k ∆ b k L + k| b ||∇ b |k L (cid:1) + Ct i (cid:0) k∇ u k L + k∇ b k L (cid:1) + Ct i − (cid:0) k√ ρ u t k L + k ∆ b k L + k| b ||∇ b |k L (cid:1) + Ct i − (cid:0) k∇ u k L + k∇ b k L (cid:1) (3.48)due to (3.44). For σ as in Lemma 3.2 and any nonnegative integer k , we derive from (3.5) and (3.14)that Z T t k (cid:0) k∇ u k L + k∇ b k L (cid:1) dt ≤ sup ≤ t ≤ T (cid:0) t k e − σt (cid:1) Z T e σt (cid:0) k∇ u k L + k∇ b k L (cid:1) dt ≤ C, (3.49)13 T t k (cid:0) k√ ρ u t k L + k ∆ b k L + k| b ||∇ b |k L (cid:1) dt ≤ sup ≤ t ≤ T (cid:0) t k e − σt (cid:1) Z T e σt (cid:0) k√ ρ u t k L + k ∆ b k L + k| b ||∇ b |k L (cid:1) dt ≤ C. (3.50)Integrating (3.48) over (0 , T ) together with (3.49) and (3.50) leads to the desired (3.38).5. Multiplying (3.47) by e σt together with (3.44) gives ddt (cid:16) C e σt k√ ρ u t k L + e σt k b t k L (cid:17) + C e σt k∇ u t k L + ν e σt k∇ b t k L ≤ Ce σt (cid:0) k√ ρ u t k L + k ∆ b k L + k| b ||∇ b |k L (cid:1) + Ce σt (cid:0) k∇ u k L + k∇ b k L (cid:1) + Ce σt (cid:0) k√ ρ u t k L + k b t k L (cid:1) ≤ Ce σt (cid:0) k√ ρ u t k L + k ∆ b k L + k| b ||∇ b |k L (cid:1) + Ce σt (cid:0) k∇ u k L + k∇ b k L (cid:1) . (3.51)Integrating (3.51) over (0 , T ) together with (3.14) and (3.5) leads to the desired (3.39). ✷ Lemma 3.5
Let the condition (3.12) be in force, then there exists a positive constant C dependingonly on Ω , µ, ¯ µ, ν, q, k ρ k L ∞ , k∇ u k L , and k∇ b k L such that Z T k∇ u k L ∞ dt ≤ C. (3.52) Proof.
1. Choosing 2 < r < min { , q } , we infer from Sobolev’s inequality, Lemma 2.3, and (3.2) that k∇ u k L ∞ ≤ C k u k W ,r ≤ C ( k ρ u t k L r + k ρ u · ∇ u k L r + k b · ∇ b k L r ) (1 + k∇ µ ( ρ ) k L q ) qr q − r ) ≤ C k ρ u t k L + C k u k L ∞ k∇ u k L + C k b k L ∞ k∇ b k L ≤ C k ρ u t k L + C k u k H + C k b k L k∇ b k L ≤ C k ρ u t k L + C k u k H + C k∇ b k L k∇ b k L k∇ b k H ≤ C k ρ u t k L + C k u k H + C k∇ b k L + C k∇ b k L , (3.53)where we have used the following Gagliardo-Nirenberg inequality k b k L ∞ ≤ C k b k L k∇ b k L , k∇ b k L ≤ C k∇ b k L k∇ b k H . By H¨older’s inequality, Sobolev’s inequality, and (3.2), we have k ρ u t k L ≤ k ρ k L ∞ k√ ρ u t k L k√ ρ u t k L ≤ C k√ ρ u t k L k∇ u t k L . (3.54)As a consequence, if T ≤
1, we obtain from (3.54) and H¨older’s inequality that Z T k ρ u t k L dt ≤ C Z T k√ ρ u t k L k∇ u t k L dt ≤ C h Z T t − · t − t k√ ρ u t k L dt i × h Z T t k∇ u t k L · t k∇ u t k L dt i ≤ C sup ≤ t ≤ T (cid:16) t k√ ρ u t k L (cid:17) (cid:16) Z T t − dt (cid:17) (cid:16) Z T t k∇ u t k L dt (cid:17) × (cid:16) Z T t k∇ u t k L dt (cid:17) ≤ CT ≤ C. (3.55)14f T >
1, one deduces from (3.55) and (3.54) that Z T k ρ u t k L dt = Z k ρ u t k L dt + Z T k ρ u t k L dt ≤ C + C h Z T t − k√ ρ u t k L dt i × h Z T t k∇ u t k L · t k∇ u t k L dt i ≤ C + C (cid:16) sup ≤ t ≤ T t k√ ρ u t k L (cid:17) (cid:16) Z T t − · t − dt (cid:17) × (cid:16) Z T t k∇ u t k L dt (cid:17) (cid:16) Z T t k∇ u t k L dt (cid:17) ≤ C + C (cid:16) − T − (cid:17) ≤ C. (3.56)Hence, we derive the desired (3.52) from (3.53), (3.55), (3.56), (3.43), (3.4), and (3.13). ✷ With Lemma 3.5 at hand, we immediately have the following result. The detailed proof can befound in [30, Lemma 3.5] and we omit it for simplicity.
Lemma 3.6
Let the condition (3.12) be in force, then there exists a positive constant C dependingonly on Ω , µ, ¯ µ, ν, q, k ρ k L ∞ , k∇ u k L , and k∇ b k L such that for r ∈ [2 , q ) , sup ≤ t ≤ T (cid:0) k ρ k W ,q + k ρ t k L r (cid:1) ≤ C. (3.57) Lemma 3.7
Let the condition (3.12) be in force, then there exists a positive constant C dependingonly on Ω , µ, ¯ µ, ν, q, k ρ k L ∞ , k∇ u k L , and k∇ b k L such that for r ∈ (2 , q ) , sup ≤ t ≤ T (cid:2) t (cid:0) k u k H + k∇ P k L + k b k H (cid:1)(cid:3) + Z T t (cid:0) k∇ u k W ,r + k∇ P k L r + k b k H (cid:1) dt ≤ C. (3.58) Moreover, for σ as that in Lemma 3.2, one has for < τ < T , sup τ ≤ t ≤ T (cid:2) e σt (cid:0) k u k H + k∇ P k L + k b k H (cid:1)(cid:3) ≤ C ( τ ) . (3.59) Proof.
1. We obtain from (3.1) , (3.13), Sobolev’s inequality, and Gagliardo-Nirenberg inequalitythat k b k H ≤ C (cid:0) k b t k L + k u · ∇ b k L + k b · ∇ u k L + k b k H (cid:1) ≤ C k b t k L + C k u k L k∇ b k L + C k b k L ∞ k∇ u k L + C k∇ b k L ≤ C k b t k L + C k∇ u k L k∇ b k L k∇ b k H + C k b k L k b k H k∇ u k L + C k∇ b k L ≤ C k b t k L + 12 k b k H + C k∇ b k L , which gives k b k H ≤ C k b t k L + C k∇ b k L . (3.60)This combined with (3.38) and (3.14) leads tosup ≤ t ≤ T (cid:0) t k b k H (cid:1) ≤ C. (3.61)From (3.43), Sobolev’s inequality, and (3.13), we have k u k H + k∇ P k L ≤ C (cid:0) k√ ρ u t k L + k| b ||∇ b |k L + k∇ u k L (cid:1) C (cid:0) k√ ρ u t k L + k b k H k∇ b k L + k∇ u k L (cid:1) ≤ C k√ ρ u t k L + C k b k H + C k∇ u k L . (3.62)This along with (3.38), (3.61), and (3.14) yieldssup ≤ t ≤ T (cid:2) t (cid:0) k u k H + k∇ P k L (cid:1)(cid:3) ≤ C. (3.63)2. It follows from (3.60), (3.39), and (3.14) that for 0 < τ < T ,sup τ ≤ t ≤ T (cid:0) e σt k b k H (cid:1) ≤ C ( τ ) , (3.64)which together with (3.62) and (3.14) givessup τ ≤ t ≤ T (cid:2) e σt (cid:0) k u k H + k∇ P k L (cid:1)(cid:3) ≤ C ( τ ) . (3.65)3. For r ∈ (2 , q ), we infer from Lemma 2.3, (3.2), (3.12), (3.13), Sobolev’s inequality, (3.60),(3.62), and (3.44) that k∇ u k W ,r + k∇ P k L r ≤ C (cid:0) k ρ u t k L r + k ρ u · ∇ u k L r + k b · ∇ b k L r (cid:1) (1 + k∇ µ ( ρ ) k L q ) qr q − r ) ≤ C k ρ k ∞ k u t k L r + C k ρ k ∞ k u k L qrq − r k∇ u k L q + C k b k L qrq − r k∇ b k L q ≤ C k ρ k ∞ k∇ u t k L + C k ρ k ∞ k∇ u k L k∇ u k H + C k∇ b k L k∇ b k H ≤ C k∇ u t k L + C (cid:0) k√ ρ u t k L + k ∆ b k L + k| b ||∇ b |k L (cid:1) + C (cid:0) k∇ u k L + k∇ b k L (cid:1) , which together with (3.38), (3.14), and (3.5) yields Z T t (cid:0) k∇ u k W ,r + k∇ P k L r (cid:1) dt ≤ C. (3.66)Similarly, we can show that Z T t k b k H dt ≤ C. (3.67)This finishes the proof of Lemma 3.7. ✷ Lemma 3.8
Let the condition (3.12) be in force, then there exists a positive number ε dependingonly on Ω , µ, ¯ µ, ν, q, k ρ k L ∞ , k∇ u k L , and k∇ b k L such that sup ≤ t ≤ T k∇ µ ( ρ ) k L q ≤
12 (3.68) provided that k∇ µ ( ρ ) k L q ≤ ε . (3.69) Proof.
Taking spatial derivative ∇ on the transport equation (3.15) leads to( ∇ µ ( ρ )) t + u · ∇ µ ( ρ ) + ∇ u · ∇ µ ( ρ ) = . (3.70)Multiplying (3.70) by q |∇ µ ( ρ ) | q − ∇ µ ( ρ ) and integrating the resulting equation over Ω give rise to ddt Z |∇ µ ( ρ ) | q dx + q Z u · ∇ µ ( ρ ) · |∇ µ ( ρ ) | q − ∇ µ ( ρ ) dx = − q Z ∇ u · ∇ µ ( ρ ) · |∇ µ ( ρ ) | q − ∇ µ ( ρ ) dx. u = 0 yields q Z u · ∇ µ ( ρ ) · |∇ µ ( ρ ) | q − ∇ µ ( ρ ) dx = Z u · ∇ ( |∇ µ ( ρ ) | q ) dx = − Z |∇ µ ( ρ ) | q div u dx = 0 . Thus, we get ddt k∇ µ ( ρ ) k qL q ≤ q Z |∇ u ||∇ µ ( ρ ) | q dx ≤ q k∇ u k L ∞ k∇ µ ( ρ ) k qL q . This implies that ddt k∇ µ ( ρ ) k L q ≤ k∇ u k L ∞ k∇ µ ( ρ ) k L q . which combined with Gronwall’s inequality leads tosup ≤ t ≤ T k∇ µ ( ρ ) k L q ≤ k∇ µ ( ρ ) k L q e R T k∇ u k L ∞ dt . This along with (3.52) gives sup ≤ t ≤ T k∇ µ ( ρ ) k L q ≤ C k∇ µ ( ρ ) k L q (3.71)for some constant C depending only on Ω , µ, ¯ µ, ν, q, k ρ k L ∞ , k∇ u k L , and k∇ b k L . Hence, setting ε = C , we obtain the desired (3.68) provided the condition (3.69) holds true. ✷ Suppose that the initial data ( ρ , u , b ) satisfies (1.8), according to Lemma 2.1, there exists a T ∗ > ρ, u , b ) on Ω × (0 , T ∗ ].We plan to extend it to a global one. To this end, let ε be the constant stated in Lemma 3.8 and k∇ µ ( ρ ) k L q ≤ ε . (4.1)It follows from (3.57) that ρ ∈ C ([0 , T ∗ ]; W ,q ) . (4.2)Since µ ∈ C [0 , ∞ ), we have ∇ µ ( ρ ) = µ ′ ∇ ρ ∈ C ([0 , T ∗ ]; L q ) , (4.3)which combined with (4.1) yields that there is a T ∈ (0 , T ∗ ) such thatsup ≤ t ≤ T k∇ µ ( ρ ) k L q ≤ . Setting T ∗ := sup { T | ( ρ, u , b ) is a strong solution on Ω × (0 , T ] } , (4.4)and T ∗ := sup n T (cid:12)(cid:12) ( ρ, u , b ) is a strong solution on Ω × (0 , T ] and sup ≤ t ≤ T k∇ µ ( ρ ) k L q ≤ o . Then T ∗ ≥ T >
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