AMP Advances in Mathematical Physics 1687-9139 1687-9120 Hindawi 10.1155/2017/2708483 2708483 Research Article Decay of the 3D Quasilinear Hyperbolic Equations with Nonlinear Damping Qiu Hongjun 1 http://orcid.org/0000-0002-5237-5707 Zhang Yinghui 2 Mei Ming 1 College of Mathematics and Computer Science Hunan Normal University Changsha 410081 China hunnu.edu.cn 2 Department of Mathematics Hunan Institute of Science and Technology Yueyang 414006 China hnist.cn 2017 2352017 2017 13 02 2017 19 04 2017 2352017 2017 Copyright © 2017 Hongjun Qiu and Yinghui Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate the 3D quasilinear hyperbolic equations with nonlinear damping which describes the propagation of heat wave for rigid solids at very low temperature, below about 20 K. The global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in the sense of H3-norm. Furthermore, if, additionally, Lp-norm (1p<6/5) of the initial perturbation is finite, we also prove the optimal Lp-L2 decay rates for such a solution without the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton.

Hunan Provincial Natural Science Foundation of China 2017JJ2105 National Natural Science Foundation of China 11571280 11301172 11226170 National Scholarship Fund in Hunan Province Cooperation Projects
1. Introduction

In this paper, we consider the following 3D quasilinear hyperbolic system with nonlinear damping:(1)vt-divhvP=0,xR3,t>0,Pt+σv=fvP,xR3,t>0,with initial data(2)v,Px,0=v0,P0xv¯,0,as x,v¯>0,where σ(v)<0, h(v)>0, f(v)<0, and v>0. The above system is derived in [1, 2] and describes the propagation of heat wave for rigid solids at very low temperatures, below about 20 K. The first equation in (1) comes from the balance of energy, which takes the form(3)εϑt+divq=0,where ϑ>0 is the absolute temperature, ε is the internal energy, and q is the heat flux. The second equation in (1) is the evolution equation for an internal parameter P, which is introduced to account for memory effects of the heat flux. The effect of memory may be considered, for example, as a functional of a history of temperature gradient,(4)q=-αϑ-te-bt-sϑx,sds,αϑ>0,b>0.By defining(5)P=-te-bt-sϑx,sds,then (4) can be equivalently replaced with(6)q=-αϑP,(7)Pt=-bP+ϑ.Equation (7), related to (4) via (5), is however linear and does not fully describe the properties of heat propagation in solids (cf.  and references therein). To improve the model, one may generalize the history dependence of q by modifying (4) or, as was done in , by introducing a suitable nonlinear dependence in (7),(8)Pt=g1ϑϑ+g2ϑP.The functions α, g1, and g2 present in (6) and (8) are material functions. The second law of thermodynamics imposes the restrictions that α(ϑ)=ψ20ϑ2g1(ϑ) and g2(ϑ)<0, where the constant ψ20 comes from the Helmholtz free energy ψ which has the form ψ=ψ1(ϑ)+(1/2)ψ20ϑP2. We additionally make an assumption that g1(ϑ)>0 (cf. ). Combining (3) with (8) gives the following system:(9)εϑt-divαϑP=0,Pt+G1ϑ=g2ϑP,G1ϑ=-g1ϑ.

Finally, by employing the substitution ε(ϑ)=v with σ=G1ε-1, f=g2ε-1, and h=αε-1, system (9) is exactly system (1).

To go directly to the theme of this paper, we now only review some former results closely related. For the one-dimensional version of model (1),(10)vt-hvPx=0,xR,t>0,Pt+σvx=fvP,xR,t>0,the existence and asymptotic behavior of smooth solutions for the Cauchy problem and initial boundary value problem have been considered by [5, 6] and , respectively. In , they obtained the convergence rates of the smooth solutions for the Cauchy problem under the following technical assumptions for the nonlinear damping:(11)fv<0,3fv2-fvfv<0.The authors in  removed restriction (11). Recently, the authors in  proved Lp convergence rates for the half space problem. Moreover, the authors in  and  considered phase transition and the effect of damping respectively. When f(v)=-b with b being a given positive constant and h(v)=1, system (10) reduces to the well-known p-system with linear damping. The investigation of the case of p-system (h(v)=1) with linear damping (f(v)=-b) has been extensively studied. We refer reader to  and references therein.

To sum up, it is still unknown on the decay rates of solutions to the 3D quasilinear hyperbolic system with nonlinear damping (1)-(2). In this paper, we will give a positive answer to this question. More precisely, we prove global existence and uniqueness of strong solutions when the initial data is near its equilibrium in the sense of H3-norm. Moreover, if, additionally, Lp-norm (1p<6/5) of the initial perturbation is finite, we also show the optimal Lp-L2 decay rates for the solutions without the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton in .

Before stating the main results, we introduce some notations for the use throughout this paper. We use ·m and ·m,r to denote the usual Sobolev spaces with norm Hm(R3) and Wm,r(R3), respectively, for m0 and r1. In particular, for m=0, we will use · and ·Lr for simplicity. We use ·,· to denote the inner-product in L2(R3) and Λs to denote the pseudo differential operator:(12)Λsz=F-1ξsFz,for sR.We will use the notation ab to mean that aCb for a generic constant C>0 which may vary at different places.

Now, we are in a position to state our main results.

Theorem 1.

Assume that σC4,  σ<0,  hC4,  h(v)>0,  fC3,  f(v)<0,  v¯>0, and v0-v¯,P03 is small enough. Then the Cauchy problem (1)-(2) has a unique global solution (v,P) with v>0, which satisfies(13)v-v¯,PC00,;H3R3C10,;H2R3.Furthermore, for any t0, the following energy estimates hold:(14)v-v¯,Pt32+0tvs22+Ps32dsv0-v¯,P032.Finally, if further v0-v¯Lp+P0L3p/(3-p) is bounded for some 1p<6/5, then the following decay estimates of the solution (v,P) hold:(15)v-v¯tLq1+t-3/21/p-1/q,2q6,v-v¯tLq1+t-3/21/p-1/2-1/2,6q,PtLq1+t-3/21/p-1/2-1/2,2q,v,Pt21+t-3/21/p-1/2-1/2,tv,Pt1+t-3/21/p-1/2-1/2.

We now comment on the proof of Theorem 1. Roughly speaking, we follow the framework of [47, 48] on two-phase fluid model, and the proof consists of following three steps.

Firstly, we deduce the optimal Lp-L2 decay rates on the solutions (v,P) to the corresponding linearized system. By making delicate pointwise analysis on the linearized system, we can prove that the variables v and P have the L2 decay rates (1+t)-(3/2)(1/p-1/2) and (1+t)-(3/2)(1/p-1/2)-1/2, respectively, when v0-v¯Lp+P0L3p/(3-p) is finite with 1p2. Due to the fact that the Fourier transform of (30)1-(30)2 has multiple eigenvalue and the semigroup theory, to investigate spectral structure of (30)1-(30)2, we need to analyze the corresponding Jordan structure. Our main idea is to introduce Hodge decomposition to overcome this difficulty. By applying Hodge decomposition to system (30)1-(30)2, we can transform system (30)1-(30)2 into two systems. One only has one equation, and another has two different eigenvalues.

Secondly, we establish the uniform energy estimates to the original system (18)1-(18)2. Based on the system’s special dissipation structure and delicate analysis and interpolation technique on the nonlinear terms, the desired energy estimates can be achieved. Compared to the one-dimensional results in , the approach is new and quite different here. In , the antiderivative technique plays an essential role in proving their main results. However, the antiderivative technique does not work in our high dimensional problem since the antiderivative technique is basically limited to one-dimensional problem. The main difficulties in this step of the paper arise from the terms involving k+1v or kdivP which are not included in left hand side of (65) (see Lemma 6 for more details). We use the special dissipation structure of (18)1-(18)2, the technique on interpolation and energy estimates, and a technical lemma on estimating the spatial derivatives of nonlinear function to tackle these difficulties. It is worth mentioning that, in our proofs, we do not need the technical assumptions (11) as in .

Finally, we deduce the decay rates on the solutions. By virtue of the results obtained in the first two steps and by virtue of the uniform nonlinear energy estimates and the optimal Lp-L2 decay rates for the linearized system, we can prove the optimal Lp-L2 decay rates for the solutions.

The rest of the paper is organized as follows. In Section 2, we first reformulate the problem and then prove Theorem 1. In Sections 3 and 4, we derive the optimal Lp-L2 decay estimates for the linearized system and the uniform nonlinear estimates for the original equations, respectively. In Section 5, we prove Proposition 3 by combining the optimal Lp-L2 decay estimates obtained in Section 3 and the uniform nonlinear estimates obtained in Section 4.

2. Reformulation and the Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>

Denoting(16)λ=-σv¯hv¯,λ1=--σv¯hv¯and making change of variables by(17)v,Pv+v¯,λP,we can reformulate the Cauchy problem (1)-(2) as(18)vt+λ1divP=F,Pt+λ1v-fv¯P=G,v,Pt=0v0,P00,0,asx,where(19)Fv,P=hv+v¯-hv¯divP+P·hv,Gv,P=fv+v¯-fv¯P+1λσv¯-σv+v¯v.Here and in the sequel, for the notational simplicity, we will denote the reformulated variables by (v,P).

As usual, Theorem 1 will be proved by combining the local existence result together with uniform a priori energy estimates.

Proposition 2 (local existence).

Let (v0,P0)H3(R3) be such that(20)infxR3v0x+v¯>0.Then there exists a positive constant T0 depending on v0,P03 such that the unique solution of the Cauchy problem (18) exists on [0,T0] and satisfies(21)v,PC00,T0;H3R3C10,T0;H2R3.Furthermore, one has the following estimates:(22)v,P32v0,P03,infxR3,0tT0vx,t+v¯>0.

Proposition 3 (a priori estimate).

Let (v0,P0)H3(R3). Assume that the Cauchy problem (18) has a solution (v,P)(x,t) in the same function class as in Proposition 2 on R3×[0,T], where T>0 is a positive constant. Then there exists a small positive constant δ, which is independent of T, such that if(23)sup0tTv,Pt3δ,then, for any t[0,T], it holds that(24)v,Pt32+0tvs22+Ps32dsv0,P032.Furthermore, if additionally v0-v¯Lp+P0L3p/(3-p) is bounded for some 1p<6/5, then, for any t[0,T], the following decay estimates of the solution (v,P) hold:(25)vtLq1+t-3/21/p-1/q,2q6,(26)vtLq1+t-3/21/p-1/2-1/2,6q,(27)PtLq1+t-3/21/p-1/2-1/2,2q,(28)v,Pt21+t-3/21/p-1/2-1/2,(29)tv,Pt1+t-3/21/p-1/2-1/2.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.

Theorem 1 follows from Propositions 2 and 3 by the standard continuity argument. The proof of Proposition 2 is standard whose proof can be found in [49, 50]. Proposition 3 will be proved in Section 5.

3. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M125"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>P</mml:mi></mml:mrow></mml:msup><mml:mtext>-</mml:mtext><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> Decay Estimates for the Linearized Equations

The corresponding linearized system to (18) is(30)vt+λ1divP=0,Pt+λ1v-fv¯P=0,v,Pt=0v0,P00,0,asx.Due to the fact that the Fourier transform of (30)1-(30)2 has multiple eigenvalue and the semigroup theory, to analyze spectral structure of (30)1-(30)2, we need to consider corresponding Jordan structure. In spirit of [40, 47, 48], we will use Hodge decomposition technique to overcome this difficulty. By applying Hodge decomposition to system (30)1-(30)2, we can transform system (30)1-(30)2 into two systems. One only has one equation, and another has two different eigenvalues. To begin with, let(31)ϕ=Λ-1divPbe the “compressible part” of P, and let(32)ψ=Λ-1curlPbe the “incompressible part” of P (with (curlz)ij=jzi-izj), then we can rewrite system (30)1-(30)2 as follows:(33)vt+λ1Λϕ=0,ϕt-λ1Λv-fv¯ϕ=0,ψt-fv¯ψ=0.Noticing the definitions of ϕ and ψ and the relation(34)P=-Λ-1ϕ-Λ-1divψwhich involves pseudo differential operators with zero degree, one can easily see that the estimates in space H3(R3) for the velocity P are the same as for (ϕ,ψ).

In the rest of this section, we devote ourselves to show the following Lp-L2 decay estimates on the linearized system (33).

Proposition 4.

Assume that U0=(v0,P0)H3(R3) and v0Lp+P0L3p/(3-p) is bounded with 1p2. Denote U=(v,P) by the solution of (30)1-(30)2, and let ϕΛ-1divP and ψ=Λ-1curlP. Then, for 0k3, one has(35)kψtefv¯tkψ0,(36)kvt1+t-3/21/p-1/2-k/2v0Lp+P0L3p/3-p+kU0,(37)kϕ,Pt1+t-3/21/p-1/2-k/2-1/2v0Lp+P0L3p/3-p+kU0.

Proof.

Since the proof of (35) is trivial, we will focus on the proofs of (36) and (37). Firstly, in terms of semigroup theory, the solution (v,ϕ) of system (33)1-(33)2 has the following expression:(38)v,ϕtt=Gtv0,ϕ0t,t0,where G(t)G(x,t) is Green’s matrix of system (33)1-(33)2. Next, we derive the explicit expression for the Fourier transform G^(ξ,t) of Green’s matrix G(x,t). Taking the Fourier transform to (33)1-(33)2, we obtain(39)v^t+λ1ξϕ^=0,ϕ^t-λ1ξv^-fv¯ϕ^=0.From (39) and a simple calculation, we have(40)ϕ^tt-fv¯ϕ^t+λ12ξ2ϕ^=0,ϕ^ξ,0=ϕ^0,ϕ^tξ,0=λ1ξv^0+fv¯ϕ^0.By a straightforward computation, we can get the expression of the solution to the ODE (40)(41)ϕ^ξ,t=λ1ξeλ+t-eλ-tλ+-λ-v^0+λ+eλ+t-λ-eλ-tλ+-λ-ϕ^0,where λ±(ξ)=(fv¯/2)1±1-4λ12|ξ|2/(f(v¯))2 are the eigenvalues of the ODE (40). To compute v^, we substitute (41) into (39) to get(42)v^ξ,t=λ+eλ-t-λ-eλ+tλ+-λ-v^0-λ1ξeλ+t-eλ-tλ+-λ-ϕ^0.Thus, from (41)-(42), we obtain the explicit expression of the Fourier transformation G^(ξ,t) of Green’s matrix G(x,t) as(43)G^ξ,t=λ+eλ-t-λ-eλ+tλ+-λ--λ1ξeλ+t-eλ-tλ+-λ-λ1ξeλ+t-eλ-tλ+-λ-λ+eλ+t-λ-eλ-tλ+-λ-.

In order to deduce the long-time decay estimates of solutions in Lp-L2-framework, we need to verify the approximation of G^(ξ,t). By virtue of the definition λ±(ξ), for |ξ|1, it holds that(44)λ+eλ-t-λ-eλ+tλ+-λ-~fv¯eλ12ξ2/fv¯t-λ12ξ2/fv¯efv¯tfv¯,ξ1,(45)eλ+t-eλ-tλ+-λ-~efv¯t-eλ12ξ2/fv¯tfv¯,ξ1,(46)λ+eλ+t-λ-eλ-tλ+-λ-~fv¯efv¯t-λ12ξ2/fv¯eλ12ξ2/fv¯tfv¯,ξ1.Similarly, for high frequency, it also holds that(47)λ+eλ-t-λ-eλ+tλ+-λ-~cosηt-fv¯2sinηtηefv¯/2t,ξR,(48)eλ+t-eλ-tλ+-λ-~sinηtηefv¯/2t,ξR,(49)λ+eλ+t-λ-eλ-tλ+-λ-~cosηt+fv¯2sinηtηefv¯/2t,ξR,where R is some given positive constant and(50)η=-fv¯24λ12ξ2fv¯2-1~Oξ,ξR.By virtue of formula (43) and the asymptotical estimates on its elements, we are in a position to prove (36) and (37). By virtue of (43)–(45), (47)-(48), (50), Parsevel’s identity, Hausdorff-Young’s inequality, and Hölder inequality, we deduce that, for each 0k3,(51)kvt2=ξRξ2kv^ξ,t2dξ+ξRξ2kv^ξ,t2dξξRξkeλ12ξ2/fv¯t+ξ2efv¯tv^0+ξk+1eλ12ξ2/fv¯t+efv¯tϕ^02dξ+ξRefv¯tξ2kv^0+ϕ^02dξ1+t-3/22/p-1-kv^0Lp/p-12+ϕ^0L3p/4p-32+efv¯tξkv^0,ξkϕ^021+t-3/22/p-1-kv0Lp2+P0L3p/3-p2+kU02.This proves (36). Similarly, for each 0k3, we also have(52)kϕt2=ξRξ2kϕ^ξ,t2dξ+ξRξ2kϕ^ξ,t2dξξRξk+1eλ12ξ2/fv¯t+efv¯tϕ^0+ξkξ2eλ12ξ2/fv¯t+efv¯tv^02dξ+ξRefv¯tξ2kv^0+ϕ^02dξ1+t-3/22/p-1-k-1v^0Lp/p-12+ϕ^0L3p/4p-32+efv¯tξkv^0,ξkϕ^021+t-3/22/p-1-k-1v0Lp2+P0L3p/3-p2+kU02,which together with (34)-(35) yields (37).

Therefore, we have proved Proposition 4.

4. Uniform a Priori Estimates

In this section, we deduce the uniform a priori estimates stated in Proposition 3. Throughout this section, we assume that all the conditions in Proposition 3 are satisfied. Furthermore, we assume a priori that for sufficiently small δ>0(53)sup0tTv,Pt3δ.

We first derive the following lower order energy estimate of the solutions.

Lemma 5.

There exists a suitably large constant D1>0, which is independent of δ, such that(54)ddtD1v,Pt2+v,Pt+Cvt2+Pt2Pt2,for any 0tT.

Proof.

Multiplying (18)1-(18)2 by v, P, respectively, and then integrating them over R3, we have(55)12ddtv,P2-fv¯P2=v,F+P,G.We will estimate the two terms in the right-hand side of (55) as follows.

First, for the first term, by virtue of Hölder inequality, Cauchy inequality, and Lemmas A.1 and A.4, we get(56)v,Fv,hv+v¯-hv¯divP+v,P·hvvL6hv+v¯-hv¯L3P+vL3hvPL6vvH1P+vH1vPδv2+P2.Similarly, for the second term, it also holds that(57)P,GP,fv+v¯-fv¯P+P,σv¯-σv+v¯vfv+v¯-fv¯LP2+PL6vσv¯-σv+v¯L3vLP2+PvvH1δP2+P2+v2.Thus substituting (56) and (57) into (55) gives(58)ddtv,P2+CP2δv2+P2,since f(v¯)<0 and δ>0 is suitably small.

Next we shall estimate the term v2. Multiplying (18)2 by v and then integrating it over, we obtain(59)λ1v2=-Pt,v+fv¯P,v+G,v,where from (18)1, we can rewrite the first term in the right-hand side as(60)-Pt,v=-ddtv,P+vt,P=-ddtv,P-vt,divP=-ddtv,P+λ1divP-F,divP.By virtue of the definition of F, we obtain(61)-F,divPhv+v¯-hv¯LP2+PL6PhvL3vL+vH1P2δP2,where Lemma A.4 has been used. Then, by combining the relations (59)–(61), we have(62)λ1v2+ddtv,P=fv¯P,v+λ1divP-F,divP+G,vP2+λ14v2+λ1divP2+δP2+G,v.Similar to the proof of the estimate on P,G, it also holds that(63)G,vδv2.Putting (63) into (62) and noting that δ>0 is sufficiently small, we obtain(64)ddtv,P+λ12v2P12.Finally, multiplying (58) by a sufficiently large positive constant D1 and then adding it to (64), we can get (54) since δ>0 is small enough. Therefore, we have completed the proof of Lemma 5.

In the following lemma, we deduce the higher order energy estimate of the solutions.

Lemma 6.

There exists a suitably large constant D2>0, which is independent of δ, such that(65)ddtD2v,P22+1k3R3χλ1-χkv2+1λσv+v¯-σv¯kP2dx+1k2kv,kPt+C2vt12+Pt22δvt2,for any 0tT, where χ=h(v+v¯)-h(v¯).

Proof.

For 1k3, by applying k to (18)1-(18)2 and multiplying the resulting identities by kv, kP, respectively, summing up them, and then integrating over R3, we have(66)12ddtkv,P2-fv¯kP2=kv,kF+kP,kGI1+I2.Next, we will estimate the two terms in the right-hand side of (66). To do this, by the definition of F, we get(67)I1=kv,khv+v¯-hv¯divP+kv,kP·hvkv,kχvdivP+kv,kP·hv=kv,χvkdivP+kv,k,χvdivP+kv,P·khv+kv,k,PhvI11+I12+I13+I14.We will estimate the four terms I1i with 1i4. The main difficulties arise from terms involving k+1v or kdivP which are not included in left hand side of (65). The main observation is that we can tackle these difficulties by using the special dissipation structure of (18)1-(18)2, the technique on interpolation and energy estimates, and a technical lemma on estimating the spatial derivatives of nonlinear function to tackle these difficulties. It should be mentioned that, in our proofs, we do not need the technical assumptions (11) as in . Firstly, by virtue of (18)1, I11 can be rewritten as(68)I11=-χvkv,kvt-P·hvλ1-χv=-χvkv,kvtλ1-χv+χvkv,kP·hvλ1-χv=-χvλ1-χvkv,kvt+-χvkv,k,1λ1-χvvt+χvkvλ1-χv,P·khv+χvkv,k,Pλ1-χvhvI11,1+I11,2+I11,3+I11,4.From (18)1 and (53), we obtain(69)I11,1=12ddtR3-χλ1-χkv2dx+12χλ1-χt,kv212ddtR3-χλ1-χkv2dx+δR3vtkv2dx12ddtR3-χλ1-χkv2dx+δkv2.By virtue of (18)1, (53), Lemmas A.1A.4, Cauchy inequality, and Hölder inequality, we get(70)I11,2χLkvk-1vt1λ1-χL+k1λ1-χvtLvLkvk-1vtvL+kvvtLδkvkP+k-1P·hv+kvδkvkP+PLkhv+k-1PL6hvL3+kvδkv2+kP2.Next, we estimate the term I11,3. If k=1, from integration by parts, Lemmas A.1 and A.4, we obtain(71)I11,3=χvvλ1-χv,P·hv=12χvhvλ1-χvP,v2+χvhvP·vλ1-χv,v2=-12divχvhvλ1-χvP,v2+χvhvP·vλ1-χv,v2vLPL+PLvLv2δv2.If k=2, by integration by parts, Hölder inequality, and Lemmas A.1 and A.4, we have(72)I11,3=χv2vλ1-χv,P·2hv=12χvhvλ1-χvP,2v2+χv2vλ1-χv,P·3hvv·2v+hvv2v=-12divχvhvλ1-χvP,2v2+χv2vλ1-χv,P·3hvv·2v+hvv2vδ2v2+PLvL2v2+PL3vL6vL22vδ2v2.If k=3, by using similar arguments as the above, we obtain(73)I11,3=χv3vλ1-χv,P·3hv-12divχvhvλ1-χvP,3v2+R32v2+3vv+v43vPdxδ3v2+PL3v2vL32vL6+vL3v+vLvL3vδ3v2.Combining (71)–(73) gives that(74)I11,3δkv2.From (53), (56), Lemmas A.1A.4, Cauchy inequality, and Hölder inequality, we get(75)I11,4kvkhvPλ1-χL+kPλ1-χhvLδkvPLkv+vLkv,Pδkv2+kP2.Putting (69)-(70) and (74)-(75) into (68) leads to(76)I1112ddtR3-χλ1-χkv2dx+δkv2+kP2.Similarly, for the terms I12 and I14, we also have(77)I12kvk-1divPχvL+kχvdivPLδkvvLkv+PLkvδkv2+kP2,I14kvkPvL+kvPLδkvkP+kvδkv2+kP2.Using the similar arguments in obtaining (74), for the term I13, it also holds that(78)I13δkv2.Putting (76)–(78) into (67) leads to(79)I112ddtR3-χλ1-χkv2dx+δkv2+2P2.

Next, we estimate the term I2. By virtue of the definition of G, we separate I2 into three parts(80)I2=kP,kfv+v¯-fv¯P+1λσv¯-σv+v¯kP,kv+1λkP,k,σv¯-σv+v¯vI21+I22+I23.From (53), (56), Lemmas A.1A.4, Cauchy inequality, and Hölder inequality, we get(81)I21kPPLkv+vLkPδkv2+kP2,I23vLkvkPδkv2+kP2.Using similar arguments in obtaining (76), for I22, it also holds that(82)I23-12λddtR3σv+v¯-σv¯kP2dx+δkv2+kP2.Putting the estimates (81)-(82) into (80) gives(83)I2-12λddtR3σv+v¯-σv¯kP2dx+δkv2+kP2.Substituting (79) and (83) into (66) and then summing up the resultant equation for k=1 to 3, we obtain(84)ddtv,P22+1k3R3χλ1-χkv2+1λσv+v¯-σv¯kP2dx+P22δv22.

Finally, we derive the estimate on kv2 for 1k2. Applying kv to (18)2, multiplying the resulting identity by kv, and then integrating over R3, we obtain(85)λ1kv2=-kv,kPt+kv,fv¯kP+kv,kG,where from (18)1, we can rewrite the first term in the right-hand side of (85) as(86)-kv,kPt=-ddtkv,kP+kvt,kP=-ddtkv,kP-kvt,kdivP=-ddtkv,kP+λ1kdivP-kF,kdivP.By virtue of the definition of F, (53), (56), and Lemmas A.1A.4, we get(87)-kF,kdivPkdivPkdivPχL+kχdivPL+k+1vPL+kPL6vL3δk+1Pk+1P+k+1vδk+1v2+k+1P2.Then, combining relations (85)–(87) yields(88)λ1kv2+ddtkv,kP=fv¯kP,kv+λ1kdivP-kF,kdivP+kv,kGkP2+λ14kv2+kdivP2+δk+1v2+k+1P2+kv,kG.Similarly, for kP,kG, it also holds that(89)kP,kGδk+1v2+k+1P2.Putting (89) into (88) and summing up for k=1 to 2, we obtain(90)ddt1k2kv,2P+2v12P22.Therefore, multiplying (84) by a sufficiently large positive constant D2 and adding it to (90), we can obtain (65) since δ>0 is small enough. Therefore, we have completed the proof of Lemma 6.

5. Global Existence and Decay Rates

In this section, we devote ourselves to prove Proposition 3. To begin with, we give the following lemma which will be used later.

Lemma 7.

It holds that(91)v,PtK01+t-3/21/p-1/2-1/2+δ0t1+t-τ-3/21/p-1/2-1/2v,Pτ2dτ,for any 0tT, where K0=v0-v¯H3Lp+P0H3L3p/(3-p) is finite due to the assumptions of Proposition 3.

Proof.

Applying (36)-(37) with k=1 and using Duhamel’s principle, we have(92)v,PtK01+t-3/21/p-1/2-1/2+0t1+t-τ-3/21/p-1/2-1/2F,GτH1L1dτ.From (53), (56), Lemmas A.1 and A.4, and Hölder inequality, we can estimate the nonlinear source terms as follows:(93)F,GtL1δv,Pt1,F,Gt1δv,Pt2.Substituting these estimates into (92) gives (91) and thus proves Lemma 7.

Now, we are in a position to prove Proposition 3.

Proof of Proposition <xref ref-type="statement" rid="prop2.2">3</xref>.

The proof involves the following two steps.

Step  1. Since δ>0 is suitably small, from Lemmas 5 and 6, we can choose a suitably large positive constant D3 such that(94)ddtD3v,P32+1k3R3χλ1-χkv2+1λσv+v¯-σv¯kP2dx+1k2kv,kPt+Cvt22+Pt32δvt2,Due to (53), it is clear that the expression under d/dt in (94) is equivalent to (v,P)(t)32. Hence, by integrating (94) directly in time, we obtain (24).

Step  2. Define the temporal energy functional(95)Et=D2v,P22+1k3R3χλ1-χkv2+1λσv+v¯-σv¯kP2dx+1k2kv,kP,for any 0tT, where E(t) is equivalent to (v,P)22 since D2 can be large enough.

Due to Lemma 6, we have(96)ddtEt+C2v12+P22δvt2.Adding vt2 to both sides of (96) gives(97)ddtEt+D4Etvt2,where D4 is a positive constant independent of δ. Setting(98)Ht=sup0τt1+τ31/p-1/2+1Eτ,then(99)v,Pτ2Eτ1+τ-3/21/p-1/2-1/2Ht,0τtT.This together with (91) yields that(100)v,PtK01+t-3/21/p-1/2-1/2+δ0t1+t-τ-3/21/p-1/2-1/21+τ-3/21/p-1/2-1/2Htdτ1+t-3/21/p-1/2-1/2K0+δHt.Thus, from Gronwall’s inequality, (97), and (100), we have(101)EtE0e-D4t+0te-D4t-τvτ2dτE0e-D4t+K02+δ2Ht0te-D4t-τ1+τ-31/p-1/2-1dτ1+t-31/p-1/2-1E0+K02+δ2Ht.Since H is nondecreasing, from (101), we have(102)HtE0+K02+δ2Ht,for any 0tT, which together with the smallness of δ gives that(103)HtE0+K02K02.Therefore, (99) and (103) lead to(104)v,Pt2K01+t-3/21/p-1/2-1/2,0tTand this proves (28).

Next, we turn to the proofs of (25)–(27). Applying (36)-(37) with k=0, we have from Duhamel’s principle that(105)vtK01+t-3/21/p-1/2+0t1+t-τ-3/21/p-1/2F,GτL2L1dτK01+t-3/21/p-1/2+δ0t1+t-τ-3/21/p-1/2v,Pτ1dτK01+t-3/21/p-1/2+K0δ0t1+t-τ-3/21/p-1/21+τ-3/21/p-1/2-1/2dτK01+t-3/21/p-1/2,PtK01+t-3/21/p-1/2-1/2+0t1+t-τ-3/21/p-1/2-1/2F,GτL2L1dτK01+t-3/21/p-1/2-1/2+δ0t1+t-τ-3/21/p-1/2-1/2v,Pτ1dτK01+t-3/21/p-1/2-1/2+K0δ0t1+t-τ-3/21/p-1/2-1/21+τ-3/21/p-1/2-1/2dτK01+t-3/21/p-1/2-1/2,for any 0tT. On the other hand, from Lemma A.1 and (29), we also have(106)v,PtLqv,Pt1K01+t-3/21/p-1/2-1/2,6q.Thus, we can deduce (25)–(27) from the interpolation and (105)-(106) immediately.

Finally, we prove (29). By virtue of (18) and (28), we have(107)tv,PtP·v+divP+v+Pv+P1K01+t-3/21/p-1/2-1/2,for any 0tT. Therefore, (29) is proved and therefore we have completed the proof of Proposition 3.

Appendix Analytic Tools

We will extensively use the Sobolev interpolation of the Gagliardo-Nirenberg inequality.

Lemma A.1.

Let 0i,jk; then one has(A.1)ifLpjfLq1-θkfLrθ,where θ satisfies(A.2)i3-1p=j3-1q1-θ+k3-1rθ.

Proof.

This is a special case of [33, pp. 125, Theorem].

Next, we recall the following product estimate.

Lemma A.2.

Let k1 be an integer; then it holds that(A.3)kfgLpfLp1kgLp2+kfLp3gLp4,where p,p2,p3(1,+) and(A.4)1p=1p1+1p2=1p3+1p4.

Proof.

For p=p2=p3=2, it can be proved by using Lemma A.1. For the general cases, one may refer to [51, Lemma  3.1].

We also need the following commutator estimate.

Lemma A.3.

Let k1 be an integer and define the commutator(A.5)k,fg=kfg-fkg.Then one has(A.6)k,fgLpfLp1k-1gLp2+kfLp3gLp4,where p,p2,p3(1,+) and(A.7)1p=1p1+1p2=1p3+1p4.

Proof.

The proof can be found in [51, Lemma  3.1].

Finally, to estimate the Lp-norm of the spatial derivatives of f(v+v¯)-f(v¯),  h(v), and σ(v+v¯)-σ(v¯), we shall need the technique lemma.

Lemma A.4.

Assume that ϱL1. Let g(ϱ) be a smooth function of ϱ with bounded derivatives of any order, then, for any integer m1 and 1p, one has(A.8)mgϱLpmϱLp.

Proof.

Notice that, for m1,(A.9)mgϱ=a sum of products gγ1,,γnϱγ1ϱγnϱ,where the functions gγ1,,γn(ϱ) are some derivatives of g(ϱ) and 1γim,  i=1,,n, with γ1++γn=m. We then use the Sobolev interpolation of Lemma A.1 to bound(A.10)mgϱLPγ1ϱLpm/γ1γnϱLpm/γnϱL1-γ1/mmϱLpγ1/mϱL1-γn/mmϱLpγn/mϱLn-1mϱLp.Hence, we conclude our lemma since ϱL1.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This work was partially supported by the Hunan Provincial Natural Science Foundation of China no. 2017JJ2105, the National Natural Science Foundation of China no. 11571280, no. 11301172, and no. 11226170, and National Scholarship Fund in Hunan Province Cooperation Projects.

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