Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 102, pp. 1–25. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu OPTIMAL TIME DECAY RATES FOR THE COMPRESSIBLE NAVIER-STOKES SYSTEM WITH AND WITHOUT YUKAWA-TYPE POTENTIAL QING CHEN, GUOCHUN WU, YINGHUI ZHANG, LAN ZOU Communicated by Hongjie Dong Abstract. We consider the time decay rates of smooth solutions to the Cauchy problem for the compressible Navier-Stokes system with and without a Yukawa- type potential. We prove the existence and uniqueness of global solutions by the standard energy method under small initial data assumptions. Further- more, if the initial data belong to L1(R3), we establish the optimal time decay rates of the solution as well as its higher-order spatial derivatives. In particu- lar, we obtain the optimal decay rates of the highest-order spatial derivatives of the velocity. Finally, we derive the lower bound time decay rates for the solution and its spacial derivatives. 1. Introduction We consider the Cauchy problem of the compressible Navier-Stokes system with and without the Yukawa-type potential term in the whole space R3, ρt + div(ρu) = 0, (ρu)t + div(ρu⊗ u) +∇P (ρ) + γρ∇ψ = µ∆u+ (µ+ ν)∇ div u, −∆ψ + ψ = ρ− 1, (ρ, u)|t=0 = (ρ0, u0)→ (1, 0) as |x| → ∞. (1.1) Here ρ = ρ(t, x), u = u(t, x), P = P (t, x) and ψ(t, x) represent the density, the velocity vector field of the fluid, the pressure and the potential force exerted in the fluid respectively, at time t ≥ 0 and position x ∈ R3. The Lamé coefficients µ and ν satisfy µ > 0 and 2 3µ+ν > 0. And the constant γ ∈ R may be arbitrary and it is es- sential on its sign. When γ = 0, (1.1) reduces to compressible Navier-Stokes system, which describes the motion of a barotropic viscous compressible flow. When γ 6= 0, (1.1) becomes the compressible Navier-Stokes equations with a Yukawa-potential, which is a simplified hydrodynamical model describing the nuclear matter [3, 8]. In this paper, we consider the compressible Navier-Stokes system with and without a Yukawa-potential. For technical consideration, we assume that the constant γ ≥ 0 and the pressure P is some smooth function depending only on ρ and P ′(ρ) > 0. 2010 Mathematics Subject Classification. 35Q30, 76N15. Key words and phrases. Compressible flow; energy method; optimal decay rates. c©2020 Texas State University. Submitted February 2, 2020. Published September 29, 2020. 1 2 Q. CHEN, G. C. WU, Y. H. ZHANG, L. ZOU EJDE-2020/102 There are many important investigations on large time behavior of the solu- tions to the compressible Navier-Stokes system in multi-dimensional space, see [5, 10, 11, 12, 18, 20, 21, 23, 28, 27] and references therein. Matsumura and Nishida [20, 21] first proved the existence of the small global solutions in H3(R3) to compressible Navier-Stokes equations, and particularly, for the initial perturbation small in L1(R3)∩H3(R3), by employing time-decay properties of the linear system, the authors in [20] obtained the decay rate of the solution in L2-norm: ‖(ρ− 1, u)‖L2(R3) ≤ C(1 + t)−3/4, which is the same as for the heat equation with initial data in L1(R3). Later, for small initial disturbance in H l(Rd) ∩ W l,1(Rd) with l ≥ 4, Ponce [23] gave the optimal Lp (p ≥ 2) decay rates of the solutions and their first and second order derivatives. For the initial perturbation in H3(R3)∩Lp(R3) with 1 ≤ p < 6 5 , Duan, Liu, Ukai and Yang [6] proved the optimal convergence rates of the solution in Lq-norm with 2 ≤ q ≤ 6 and its first order derivative in L2-norm. For the small data (ρ0 − 1,m0) with the momentum m = ρu in H l(R3) ∩ Ḃ−s1,∞(R3) with l ≥ 4 and 0 ≤ s ≤ 1, Li and Zhang [17] studied the Cauchy problem for compressible Navier-Stokes system (1.1) with γ = 0 and obtained the following decay rate ‖(ρ− 1,m)‖L2(R3) ≤ C(1 + t)−( 3 4+ s 2 ). Moreover, they established the lower bound of the time-decay rate for the global solution. For the small initial perturbation in ‖(ρ0 − 1, u0)‖Hl(R3) with l ≥ 3 and the data bounded in Ḣ−s(R3) with s ∈ [0, 32 ), instead of resorting to the decay properties of the linear system, Guo and Wang [10] developed a general energy method and obtained the optimal decay rates of the higher-order spatial derivatives of solutions ‖∇k(ρ− 1, u)‖L2(R3) ≤ C(1 + t)− k+s 2 , 1 ≤ k ≤ l − 1. Recently, Danchin and Xu [5] studied the Lp decay rates of the global solutions in the critical Lp framework and Xin and Xu [27] improved the result by removing some low frequencies conditions. Under the discontinuous initial data assumption, Hu and Wu [13] established the optimal convergence rates of the solutions with low regularity in Lp-norm with 2 ≤ p ≤ ∞ and of the first order derivative of the velocity in L2-norm. Compared with the compressible Navier-Stokes equations, there are few results on the system (1.1) with γ 6= 0. For instance, Chikami [3] studied the existence and uniqueness of the solution in the critical space, and they also developed a blow-up criterion of the solution. In this paper, we first establish the global existence of the smooth solutions for the system (1.1), and then we will continue to address the optimal decay rates for the solutions. In particular, we can derive the optimal decay rate on the highest-order derivatives of the velocity. The system (2.1) can be rewritten as Ut = DU +N , U |t=0 = U0, with the solution U = (ρ− 1, u) and the matrix-valued differential operator D has the form D = ( 0 −div −P ′(1)∇− γ∇(1−∆)−1 µ∆ + (µ+ ν)∇ div ) . EJDE-2020/102 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 3 Thus the solution can be expressed as U(t) = E(t)U0 + ∫ t 0 E(t− τ)N (U(τ))dτ, where E(t) = etD is the solution semigroup. Due to the estimates on the semigroup and the energy estimates on the solution to the nonlinear problem (cf. [6, 17, 20]), it is difficult to show that the optimal decay rate of the high-order derivatives of the solution since the nonlinear term involves the derivatives. To improve the known results, motivated by [2, 4], we introduce Hodge decomposition to analyze the system (1.1). Hence the second equation of (1.1) can be divided into two systems. One is a mere heat equation on the “incompressible part”, whose decay rate is exponential; another one is a mixed system, which seems more complicated because of the nonlinear term. Fortunately, we find that the solution semigroup of the mixed system also keep good properties (the detail can be seen in the proof of Proposition 3.3). As a result, we can eventually derive the optimal decay rate of the highest order derivatives of the velocity. Notation. In this paper, ∇` with an integer ` ≥ 0 stands for the usual any spatial derivatives of order `. We use Lp(R3) with 1 ≤ p ≤ ∞ to denote the usual Lp spaces with norm ‖·‖Lp , and Hs(R3) to denote the usual Sobolev spaces with norm ‖·‖Hs . Furthermore, we use Ḣs(R3) to denote the homogenous Sobolev spaces with norm ‖ · ‖Ḣs defined as ‖f‖Ḣs =: ‖Λsf‖L2 = ‖|ξ|sf̂‖L2 with s ∈ R and here Λs is a Riesz potential operator of order s. We will employ the notation a . b to mean that a ≤ Cb for a universal constant C > 0 that only depends on the parameters coming from the problem. And Ci(i = 1, 2, 3, 4) will also denote some positive constants depending only on the given problems. Our main results are stated in the following theorems. Theorem 1.1. Assume that ‖(ρ0 − 1, u0)‖Hl with an integer l ≥ 3 is sufficiently small. Then there exists a unique global solution (ρ(t, x), u(t, x), ψ(t, x)) to the initial value problem (1.1) such that ‖(ρ− 1, u)(t)‖2Hl + ‖ψ(t)‖2Hl+2 + ∫ t 0 (‖∇ρ(τ)‖2Hl−1 + ‖∇u(τ)‖2Hl + ‖∇ψ(τ)‖2Hl+1)dτ . ‖(ρ0 − 1, u0)‖2Hl . (1.2) If further ‖(ρ0 − 1, u0)‖L1 < +∞, then for k = 0, 1, . . . , l − 1, ‖∇k(ρ− 1)(t)‖L2 + ‖∇kψ(t)‖H2 . (1 + t)−( 3 4+ k 2 ), (1.3) and for k = 0, 1, . . . , l, ‖∇ku(t)‖L2 . (1 + t)−( 3 4+ k 2 ). (1.4) Remark 1.2. The results in Theorem 1.1 indicate that the optimal time decay rates are same for the solutions of the Navier-Stokes system and what with a Yukawa-type potential. It is worth noting that the optimal time decay rates of the highest-order spa- tial derivatives of the velocity are obtained. By comparison, the optimal time decay rates of the compressible Navier-Stokes equations and Navier-Stokes-Poisson 4 Q. CHEN, G. C. WU, Y. H. ZHANG, L. ZOU EJDE-2020/102 equations, see [6, 7, 10, 15, 16, 26] and the references therein for instance, can be obtained but except the highest-order one. This is because of the decomposition on the system. To obtain the optimal time decay rates of the higher-order derivatives of the solution, we can represent the spatial derivatives of the solutions to the equation Ut = BU + G with the initial data U |t=0 = U0 which follows from the Duhamel principle as (cf. [25]) ∇kU = ∇kS(t)U0 + ∫ t/2 0 ∇kS(t− τ)G(τ)dτ + ∫ t t/2 ∇rS(t− τ)∇k−rG(τ)dτ, (1.5) where S(t) := etB is the solution semigroup and 0 ≤ r ≤ k. Note that the decay rates for the solutions and their derivatives above are op- timal. Indeed, we shall establish the lower bound of the time decay rates for the global solution. Theorem 1.3. Besides the assumptions of Theorem 1.1, assume that the Fourier transform (F[ρ0 − 1],F[m0]) with m0 = ρ0u0 satisfies |F[ρ0 − 1](ξ)| > c0K0, and F[m0](ξ) = 0 for 0 ≤ |ξ| � 1 with a positive constant c0 and K0 = ‖(ρ0 − 1, u0)‖L1∩Hl . Then, the global solution (ρ, u, ψ) given by Theorem 1.1 satisfies for t ≥ t0 with t0 > 0 a sufficiently large time such that for k = 0, . . . , l − 1, c1K0(1 + t)−( 3 4+ k 2 ) ≤ min { ‖∇k(ρ− 1)(t)‖L2 , ‖∇ku(t)‖L2 , ‖∇kψ(t)‖H2 } ≤ C(1 + t)−( 3 4+ k 2 ), (1.6) and c1K0(1 + t)−( 3 4+ k 2 ) ≤ ‖∇lu(t)‖L2 ≤ C(1 + t)−( 3 4+ k 2 ), (1.7) where c1 is a positive constant independent of time. Remark 1.4. Compared to the lower bound of the time-decay rate for the solu- tion obtained in [17], we can also get the lower-time-decay-rate for the high-order derivatives of the solution as well as the highest-order derivatives of the velocity. The rest of this article is organized as follows. In Section 2, we reformulate the problem and state the equivalent theorem and propositions. In Section 3, we use the decomposition of the momentum to analyze the linearized system and establish the linear L2 decay estimates. In Section 4, we prove the global existence Proposition 2.2. In Section 5, we prove the optimal time decay rates Proposition 2.3 and the lower time decay rates Proposition 2.5 respectively. 2. Reformulated system Denoting % = ρ− 1, then we rewrite (1.1) in a perturbation form as %t + div u = N1, ut + P ′(1)∇%+ γ∇ψ − µ∆u− (µ+ ν)∇ div u = N2, −∆ψ + ψ = %, (%, u)|t=0 = (%0, u0) = (ρ0 − 1, u0), (2.1) where the nonlinear terms are N1 = −div(%u), (2.2) EJDE-2020/102 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 5 N2 = −u · ∇u− (P ′(ρ) ρ − P ′(1) ) ∇%− µ ρ %∆u− µ+ ν ρ %∇ div u. (2.3) For any T > 0, we define the solution space by X(0, T ) = { (%, u, ψ) : % ∈ C0(0, T ;H l(R3)) ∩ C1(0, T ;H l−1(R2)), u ∈ C0(0, T ;H l(R3)) ∩ C1(0, T ;H l−2(R2)), ψ ∈ C0(0, T ;H l+2(R3)) ∩ C1(0, T ;H l+1(R2)), ∇% ∈ L2(0, T ;H l−1(R3)),∇u ∈ L2(0, T ;H l(R3)), ∇ψ ∈ L2(0, T ;H l+1(R3)) } , (2.4) and the solution norm by χ2(T ) = sup 0≤t≤T { ‖(%, u)(t)‖2Hl + ‖ψ‖2Hl+2 + ∫ T 0 (‖∇%(t)‖2Hl−1 + ‖∇u(t)‖2Hl + ‖∇ψ(t)‖2Hl+1)dt } . (2.5) We now state a local existence theorem for the system (2.1), which can be es- tablished by a standard contraction mapping argument; we may refer to [15]. Proposition 2.1 (local existence). Let (%0, v0) ∈ H l(R3) for an integer l ≥ 3 and inf x∈R3 {%0 + 1} > 0. (2.6) Then there exists a positive constant T0 depending on χ(0) such that the problem (2.1) has a unique solution (%, u, ψ) ∈ X(0, T0) satisfying inf x∈R3,0≤t≤T0 {%(t, x) + 1} > 0 and χ(T0) ≤ 2χ(0). (2.7) It is easy to check that the global existence part of Theorem 1.1 is equivalent to the following proposition. Proposition 2.2 (Global existence). Assume ‖(%0, u0)‖l with an integer l ≥ 3 is sufficiently small. Then there exists a unique global solution (ρ(t, x), u(t, x), ψ(t, x)) to the initial value problem (2.1) such that ‖(%, u)(t)‖2Hl + ‖ψ(t)‖2Hl+2 + ∫ t 0 (‖∇%(τ)‖2Hl−1 + ‖∇u(τ)‖2Hl + ‖∇ψ(τ)‖2Hl+1)dτ ≤ C‖(%0, u0)‖2l . (2.8) To obtain the lower time decay rate for the system (1.1), we consider the following linearized system, which is equivalent to (1.1): %t + divm = 0, mt + P ′(1)∇%+ γ∇ψ − µ∆m− (µ+ ν)∇divm = N, −∆ψ + ψ = %, (%,m)|t=0 = (%0,m0) = (ρ0 − 1, ρ0u0), (2.9) 6 Q. CHEN, G. C. WU, Y. H. ZHANG, L. ZOU EJDE-2020/102 where N =: divF =: div ( (−P (1 + %) + P (1) + P ′(1)%)I3 + γ∇ψ ⊗∇ψ − γ 2 (|ψ|2 + |∇ψ|2)I3 − m⊗m 1 + % − µ∇ ( %m 1 + % ) − (µ+ ν) div ( %m 1 + % ) I3 ) . (2.10) For simplicity, we will also use the system (2.9) to analyze the upper decay rate for the solutions in the following proposition. Proposition 2.3 (Optimal time-decay-rate). Under the assumptions in Proposi- tion 2.2, and that ‖(%0, u0)‖L1 < +∞, for k = 0, . . . , l − 1, we have ‖∇k%(t)‖L2 + ‖∇kψ(t)‖H2 . (1 + t)−( 3 4+ k 2 ), (2.11) ‖∇l%(t)‖L2 + ‖∇l+2ψ(t)‖L2 . (1 + t)−( 3 4+ l−1 2 ), (2.12) and for k = 0, . . . , l, we have ‖∇km(t)‖L2 . (1 + t)−( 3 4+ k 2 ). (2.13) Remark 2.4. Regarding Proposition 2.3 we have the following observations. • ‖(%0, u0)‖L1 < +∞ and ‖ρ0‖L∞ < +∞ imply ‖m0‖L1 < +∞ (cf. [9]). • By (2.11) and (2.13), we can check that under the assumptions of Proposi- tion 2.3, (1.4) holds for k = 0, . . . , l. Proposition 2.5 (lower-time decay rate). Assume that the conditions in Proposi- tions 2.2 and 2.3 hold, |F[%0](ξ)| > c0K0, and F[m0](ξ) = 0 for 0 ≤ |ξ| � 1. Then for t ≥ t0 with t0 > 0 a sufficiently large time and for k = 0, . . . , l − 1, the global solution (ρ,m,ψ) given by Proposition 2.2 satisfies c2K0(1 + t)−( 3 4+ k 2 ) ≤ min { ‖∇k(%,m)(t)‖L2 , ‖ψ(t)‖H2 } ≤ C(1 + t)−( 3 4+ k 2 ), (2.14) c2K0(1 + t)−( 3 4+ l 2 ) ≤ ‖∇lm(t)‖L2 ≤ C(1 + t)−( 3 4+ l 2 ), (2.15) where c2 is a positive constant independent of time. By using the estimates on the upper decay rates of the solution, we can conclude from (2.14)–(2.15) that (1.7) holds for k = 0, . . . , l. 3. Spectral analysis and linear L2 estimates In this section, we focus on the decay rate of the solution to the linear system %t + divm = 0, mt + ( P ′(1)∇+ γ∇(1−∆)−1 ) %− µ∆m− (µ+ ν)∇divm = 0, (%,m)|t=0 = (%0,m0) = (ρ0 − 1, ρ0u0). (3.1) Motivated by [2], we decompose the momentum m to analyze the above system (3.1) similarly as Hodge decomposition of the vector field, the system (3.1) can be transformed into two systems. One is a mere heat equation on the “incompressible part”, and another one has distinct eigenvalues. Let n = Λ−1 divm and M = EJDE-2020/102 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 7 Λ−1 curlm (with curl z = (∂x2 z3 − ∂x3 z2, ∂x3 z1 − ∂x1 z3, ∂x1 z2 − ∂x2 z1)t), then we can rewrite (2.9) as follows %t + Λn = 0, nt − Λ(P ′(1) + γ(1−∆)−1)%− (2µ+ ν)∆n = 0, Mt − µ∆M = 0, (%, n,M)|t=0 = (%0, n0,M0) = (%0,Λ −1 divm0,Λ −1 curlm0). (3.2) Indeed, as the definition of n and M , and relation m = −Λ−1∇n− Λ−1 curlM (3.3) involve pseudo-differential operators of degree zero, the estimates in the space H l(R3) for the original function m can be derived from n and M . Now, we study the time decay rates of the solutions for the system (3.1). It follows immediately that the convergence rate of M is exponential in any norm. Hence, it suffices to consider the system %t = −Λn, nt = Λ ( P ′(1) + γ(1−∆)−1 ) %+ (2µ+ ν)∆n, (%, n)|t=0 = (%0, n0). (3.4) In terms of the semigroup theory, by denoting V = (%, n)t, we may express (3.4) as, Vt = BV, V |t=0 = V0 (3.5) with B = ( 0 −Λ Λ ( P ′(1) + γ(1−∆)−1 ) (2µ+ ν)∆ ) Applying the Fourier transform to the system (3.5), we have V̂t = A(ξ)V̂ , V̂ |t=0 = V̂0, (3.6) where V̂ (t, ξ) = FV (t, x), ξ = (ξ1, ξ2, ξ3)t and A(ξ) is defined by A(ξ) = ( 0 −|ξ|( P ′(1) + γ 1+|ξ|2 ) |ξ| −(2µ+ ν)|ξ|2 ) . (3.7) The eigenvalues of the matrix A(ξ) are computed from the determinant det(A(ξ)− λI) = λ2 + (2µ+ ν)|ξ|2λ+ ( P ′(1) + γ 1 + |ξ|2 ) |ξ|2 = 0, (3.8) which implies the eigenvalues of the matrix A can be expressed as λ±(|ξ|) = − ( µ+ ν 2 ) |ξ|2 ± √( µ+ ν 2 )2|ξ|4 − (P ′(1) + γ 1 + |ξ|2 ) |ξ|2. (3.9) The semigroup S(t) = etA can be decomposed into etA(ξ) = eλ+tP+(ξ) + eλ−tP−(ξ), (3.10) where the projector P±(ξ) is P±(ξ) = A(ξ)− λ∓I λ± − λ∓ . (3.11) 8 Q. CHEN, G. C. WU, Y. H. ZHANG, L. ZOU EJDE-2020/102 To estimate the semigroup etA in L2 frame, we analyze the asymptotical expan- sions of λ±, P± and etA(ξ) for both lower and higher frequencies. From [21], we have the following lemma by careful computation. Lemma 3.1. (a) For |ξ| � 1, the spectral has the Taylor series expansion λ± = − ( µ+ ν 2 ) |ξ|2 +O(|ξ|4)± i ( (P ′(1) + γ)1/2|ξ|+O(|ξ|3) ) . (3.12) (b) For |ξ| � 1, the spectral has the Laurent expansion λ+ = − P ′(1) 2µ+ ν +O(|ξ|−2), λ− = −(2µ+ ν)|ξ|2 + P ′(1) 2µ+ ν +O(|ξ|−2). (3.13) By (3.10)–(3.11) and Lemma 3.1, we obtain the following estimates for the so- lution V̂ (t, ξ) to the system (3.6). Lemma 3.2. (a) For |ξ| � 1, we have |%̂|, |n̂| . e−(µ+ ν 2 )|ξ| 2t(|%̂0|+ |n̂0|), (3.14) (b) For |ξ| � 1, we have |%̂| . e−Rt ( |%̂0|+ |ξ|−1|n̂0| ) , (3.15) |n̂| . |ξ|−1e−Rt|%̂0|+ ( e−(µ+ ν 2 )|ξ| 2t + |ξ|−2e−Rt ) |n̂0| (3.16) for some positive constant R. Proof. By the formula (3.10)–(3.11), we can calculate the semigroup S as follows. S(t, ξ) = (Sij(t, ξ))2×2 (3.17) = ( g1(λ+, λ−) −|ξ|g2(λ+, λ−) |ξ| ( P ′(1) + γ 1+|ξ|2 ) g2(λ+, λ−) g1(λ+, λ−)− (2µ+ ν)|ξ|2g2(λ+, λ−) ) , where g1(λ+, λ−) = λ+e λ−t − λ−eλ+t λ+ − λ− , g2(λ+, λ−) = eλ+t − eλ−t λ+ − λ− . From the expansions (3.12) and (3.13) of λ±, we can estimate gi(λ+, λ−) (i = 1, 2) as follows g1(λ+, λ−) =  e−(µ+ ν 2 )|ξ| 2 ( (µ+ ν 2 )|ξ| 2 (P ′(1)+γ)1/2|ξ|+O(|ξ|3) sin (( (P ′(1) + γ)1/2|ξ| +O(|ξ|3) ) t ) + cos (( (P ′(1) + γ)1/2|ξ|+O(|ξ|3) ) t )) , if |ξ| � 1, O(1)e(−(2µ+ν)|ξ|2+O(1))t+ ( (2µ+ν)|ξ|2+O(1) ) e (−P ′(1) 2µ+ν +O(|ξ|−2))t (2µ+ν)|ξ|2+O(1) , if |ξ| � 1, (3.18) EJDE-2020/102 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 9 and g2(λ+, λ−) =  sin (( (P ′(1)+γ)1/2|ξ|+O(|ξ|3) ) t ) (P ′(1)+γ)1/2|ξ|+O(|ξ|3) e−(µ+ ν 2 )|ξ| 2 , |ξ| � 1, e ( −P ′(1) 2µ+ν +O(|ξ|−2) ) t−e ( −(2µ+ν)|ξ|2+O(1) ) t (2µ+ν)|ξ|2+O(1) , |ξ| � 1. (3.19) Hence we conclude that |g1(λ+, λ−)| . { e−(µ+ ν 2 )|ξ| 2t, |ξ| � 1, e−Rt, |ξ| � 1 (3.20) for a positive constant R, and |g2(λ+, λ−)| . { |ξ|−1e−(µ+ ν 2 )|ξ| 2t, |ξ| � 1, |ξ|−2e−Rt, |ξ| � 1. (3.21) Moreover, by delicate calculations, we have the estimate |g1(λ+, λ−)− (2µ+ ν)|ξ|2g2(λ+, λ−)| . { e−(µ+ ν 2 )|ξ| 2t, |ξ| � 1, e−(µ+ ν 2 )|ξ| 2t + |ξ|−2e−Rt, |ξ| � 1. (3.22) Now we represent the solution of (3.6) as V̂ (t, ξ) = etAV̂0. (3.23) Therefore, by plugging (3.20)–(3.22) into the expression (3.17) of S(t), we obtain (3.14)–(3.16). � As in [21], we obtain the decay rates for the solution (%, n,M) of the linear system (3.2) as follows. Proposition 3.3. Assume that (%0,m0) ∈ H l ∩ L1. Let n = Λ−1 divm and M = Λ−1 curlm. Then the solution (%, n,M) of the linear system (3.2) satisfies (a) ‖%‖2L2 . (1 + t)−3/2‖(%0, n0)‖2L1 + e−2Rt‖(%0, n0)‖2L2 , (3.24) and for 1 ≤ k ≤ l, ‖∇k%‖2L2 . (1 + t)−( 3 2+k)‖(%0, n0)‖2L1 + e−2Rt ( ‖∇k%0‖2L2 + ‖∇k−1n0‖2L2 ) . (3.25) (b) For k = 0, 1, ‖∇kn‖2L2 . (1 + t)−( 3 2+k)‖(%0, n0)‖2L1 + e−2Rt‖(%0, n0)‖2L2 , (3.26) and for 2 ≤ k ≤ l, ‖∇kn‖2L2 . (1+t)−( 3 2+k)‖(%0, n0)‖2L1 +e−2Rt ( ‖∇k−1%0‖2L2 +‖∇k−2n0‖2L2 ) , (3.27) and for 0 ≤ k ≤ l, ‖∇kM‖2L2 . (1 + t)−( 3 2+k)‖M0‖2L1 . (3.28) 10 Q. CHEN, G. C. WU, Y. H. ZHANG, L. ZOU EJDE-2020/102 Proof. We only prove (3.27). By Lemma 3.2, Plancherel theorem and Hausdorff- Young’s inequality, from (3.14) and (3.16) we have that for each 2 ≤ k ≤ l and for some η > 0, ‖∇kn‖2L2 = ‖|ξ|kn̂‖2L2 . ∫ |ξ|<η |ξ|2ke−(2µ+ν)|ξ| 2t(|%̂0|2 + |n̂0|2)dξ + ∫ |ξ|≥η |ξ|2ke−(2µ+ν)|ξ| 2t|n̂0|2dξ + ∫ |ξ|≥η e−2Rt(|ξ|k−1|%̂0|+ |ξ|k−2|n̂0|)2dξ . ( ‖%̂0‖2L∞ + ‖n̂0‖2L∞ ) ∫ |ξ|<∞ |ξ|2ke−(2µ+ν)|ξ| 2tdξ + e−2Rt ∫ |ξ|≥η (|ξ|k−1|%̂0|+ |ξ|k−2|n̂0|)2dξ . (1 + t)−( 3 2+k)‖(%0, n0)‖2L1 + e−2Rt ( ‖∇k−1%0‖2L2 + ‖∇k−2n0‖2L2 ) . (3.29) � Finally, to obtain the lower decay rates for the solution, we also need the following decay rates for the solution (%, n,M) of the linear system (3.2). Proposition 3.4. Assume that %0 ∈ H l ∩ Ḣ−s and m0 ∈ L2 ∩ Ḣ−s. Let n = Λ−1 divm and M = Λ−1 curlm. Then the solution (%, n,M) of the linear system (3.2) satisfies (a) ‖%‖2L2 . (1 + t)−s‖(%0, n0)‖2 Ḣ−s + e−2Rt‖(%0, n0)‖2L2 , (3.30) ‖∇%‖2L2 . (1 + t)−(1+s)‖(%0, n0)‖2 Ḣ−s + e−2Rt ( ‖∇%0‖2L2 + ‖n0‖L2 ) . (3.31) (b) For k = 0, 1, ‖∇kn‖2L2 . (1 + t)−(k+s)‖(%0, n0)‖2 Ḣ−s + e−2Rt‖(%0, n0)‖2L2 , (3.32) ‖∇kM‖2L2 . (1 + t)−(k+s)‖M0‖2Ḣ−s . (3.33) See [1] for a proof of this proposition, or use an argument similar to the one in Proposition 3.3. 4. Energy estimates To prove Proposition 2.2, by the standard continuity argument, it suffices to derive the following a priori energy estimates. Proposition 4.1 (a priori estimate). Let (%0, u0) ∈ H l(R3) with an integer l ≥ 3. Suppose that (2.1) has a solution (%, u, ψ) ∈ X(0, T ), where T is a positive constant. Then there exists a small constant δ > 0, independent of T , such that if sup 0≤t≤T { ‖(%, u)(t)‖Hl + ‖ψ(t)‖2Hl+2 } ≤ δ, (4.1) then for any t ∈ [0, T ], ‖(%, u)(t)‖2Hl + ‖ψ(t)‖2Hl+2 + ∫ t 0 (‖∇%(τ)‖2Hl−1 + ‖∇u(τ)‖2Hl + ‖∇ψ(τ)‖2Hl+1)dτ . ‖(%0, u0)‖2Hl . (4.2) EJDE-2020/102 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 11 First from (2.1)3, we can easily deduce the following lemma and we omit the proof. Lemma 4.2. Under the assumption of Proposition 4.1, for k = 0, . . . , l, it holds ‖∇kψ‖H2 ≈ ‖∇k%‖L2 . (4.3) Next we derive an energy estimates for (%, u). Lemma 4.3. Under the assumption of Proposition 4.1, there exists a positive con- stant C1, such that d dt {P ′(1) 2 ‖%(t)‖2L2 + 1 2 ‖u‖2L2 + γ 2 ‖ψ‖2H1 + µ 4C1 〈∇%, u〉 } + µ 2 ‖∇u(t)‖2L2 + µP ′(1) 16C1 ‖∇%(t)‖2L2 + µγ 4C1 ‖∇ψ‖2H1 ≤ C‖∇lu(t)‖2L2 . (4.4) Proof. From (4.1) and the Sobolev inequality, we obtain upper and lower bounds 1 2 ≤ ρ ≤ 3 2 . (4.5) Multiplying (2.1)1 and(2.1)2 with P ′(1)% and u respectively, and then integrating the resulting equalities over R3, one has d dt {P ′(1) 2 ‖%(t)‖2L2 + 1 2 ‖u‖2L2 } + 〈γ∇ψ, u〉+ µ‖∇u‖2L2 + (µ+ ν)‖ div u‖2L2 = 〈N1, P ′(1)%〉+ 〈N2, u〉. (4.6) The terms in (4.6) can be estimated as follows. By using (2.1)1, (2.1)3, and inte- gration by parts, we have 〈γ∇ψ, u〉 = 〈γψ,−div u〉 = 〈γψ, %t + div(%u)〉 = 〈γψ, (−∆ψ + ψ)t〉+ 〈γψ, % div u+∇% · u〉 ≥ γ 2 d dt ‖ψ‖2H1 − C‖ψ‖L3 (‖%‖L6‖ div u‖L2 + ‖∇%‖L2‖u‖L6) ≥ γ 2 d dt ‖ψ‖2H1 − Cδ ( ‖∇%‖2L2 + ‖∇u‖2L2 ) , (4.7) where the a priori assumption (4.1), Hölder’s inequality, Cauchy’s inequality and the Sobolev embedding theorem are used. In addition, by (4.5), we have 〈N1, %〉 = 〈− div(%u), %〉 ≤ Cδ ( ‖∇%‖2L2 + ‖∇u‖2L2 ) , (4.8) 12 Q. CHEN, G. C. WU, Y. H. ZHANG, L. ZOU EJDE-2020/102 〈N2, u〉 = 〈−u · ∇u, u〉+ 〈 − (P ′(ρ) ρ − P ′(1) ) ∇%, u 〉 + 〈 µ∇ (% ρ ) · ∇u, u 〉 + 〈µ ρ %∇u,∇u 〉 + 〈 (µ+ ν)∇ (% ρ ) div u, u 〉 + 〈µ+ ν ρ %div u,div u 〉 ≤ C ( ‖u‖L3‖∇u‖L2‖u‖L6 + ‖%‖L3‖∇%‖L2‖u‖L6 + ‖u‖L∞‖∇%‖L2‖∇u‖L2 + ‖%‖L∞‖∇u‖2L2 ) ≤ Cδ(‖∇%‖2L2 + ‖∇u‖2L2). (4.9) Plugging (4.7)–(4.9) into (4.6) and using Cauchy’s inequality and the smallness of δ, we can deduce that d dt {P ′(1) 2 ‖%(t)‖2L2 + 1 2 ‖u‖2L2 + γ 2 ‖ψ‖2H1 } + 3µ 4 ‖∇u‖2L2 + (µ+ ν)‖ div u‖2L2 ≤ Cδ‖∇%‖2L2 . (4.10) Next we shall deal with the L2-norm of∇%. By taking 〈∇(2.1)1, u〉+〈(2.1)2,∇%〉, we have d dt 〈∇%, u〉+ P ′(1)‖∇%(t)‖2L2 + 〈γ∇ψ,∇%〉 = 〈∇(−div u+N1), u〉+ 〈µ∆u+ (µ+ ν)∇div u+N2,∇%〉. (4.11) By using (2.1)3, we obtain the following estimates: 〈γ∇ψ,∇%〉 = 〈γ∇ψ,∇(−∆ψ + ψ)〉 = γ‖∇ψ‖2H1 , (4.12) 〈∇(−div u+N1), u〉 = 〈div u+ div(%u),div u〉 ≤ ‖div u‖2L2 + ‖(%, u)‖L∞‖∇(%, u)‖L2‖∇u‖L2 ≤ C‖∇u‖2L2 + Cδ‖∇%‖2L2 , (4.13) and 〈µ∆u+ (µ+ ν)∇ div u+N2,∇%〉 ≤ P ′(1) 4 ‖∇ρ‖2L2 + C‖∇2u‖2L2 + C‖(%, u)‖L∞‖ ( ∇%,∇u,∇2u ) ‖L2‖∇%‖L2 ≤ P ′(1) 4 ‖∇ρ‖2L2 + C ( ‖∇u‖2L2 + ‖∇lu‖2L2 ) + Cδ ( ‖∇%‖2L2 + ‖∇u‖2L2 ) . (4.14) Thus plugging (4.12)–(4.14) into (4.11) yields d dt 〈∇%, u〉+ P ′(1) 2 ‖∇%(t)‖2L2 + γ‖∇ψ‖2H1 ≤ C1‖∇u‖2L2 + C‖∇lu‖2L2 , (4.15) where C1 is some positive number. Then the estimate (4.4) follows by taking the addition of (4.10) and µ/(4C1) times (4.15) and using the smallness of δ. � EJDE-2020/102 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 13 Lemma 4.4. Under the assumption of Proposition 4.1, there exist two positive constants C2 and C3 such that d dt {P ′(1) 2 ‖∇l−1%‖2H1 + 1 2 ‖∇l−1u‖2H1 + γ 2 ‖ ( ∇l−1ψ,∇lψ ) ‖2H1 + µ 4C2 〈∇l%,∇l−1u〉 } + µ 2 ‖∇lu(t)‖2H1 + µP ′(1) 16C2 ‖∇l%(t)‖2L2 + µγ 4C2 ‖∇lψ‖2H1 ≤ 0. (4.16) Proof. By taking derivatives with k = l − 1 or l, we obtain 1 2 d dt { P ′(1)‖∇k%‖2L2 + ‖∇ku‖2L2 } + µ‖∇k+1u(t)‖2L2 + (µ+ ν)‖∇k div u‖2L2 + 〈γ∇k∇ψ,∇ku〉 = 〈P ′(1)∇kN1,∇k%〉+ 〈∇kN2,∇ku〉. (4.17) By using (2.1)1, (2.1)3, and Lemmas 4.2, 6.1–Lemma 6.4, we can estimate the terms in (4.17) as follows. 〈γ∇k∇ψ,∇ku〉 = 〈γ∇kψ,∇k(−div u)〉 = 〈γ∇kψ,∇k(%t + div(%u))〉 = 〈γ∇kψ,∇k(−∆ψ + ψ)t〉+ 〈γ∇kψ,∇k div(%u)〉 = γ 2 d dt ‖∇kψ‖2H1 − 〈γ∇k+1ψ,∇k(%u)〉. (4.18) By ∣∣〈γ∇lψ,∇l−1(%u)〉 ∣∣ . ‖∇lψ‖L2 ( ‖∇l−1%‖L6‖u‖L3 + ‖%‖L3‖∇l−1u‖L6 ) . ‖(%, u)‖H1‖∇lψ‖L2‖∇l(%, u)‖L2 . δ‖∇l(%, u)‖2L2 , (4.19) and |〈γ∇l+1ψ,∇l(%u)〉| . ‖∇l+1ψ‖L2 ( ‖%‖L3‖∇lu‖L6 + ‖∇l%‖L2‖u‖L∞ ) . δ ( ‖∇l%‖2L2 + ‖∇l+1u‖2L2 ) , (4.20) we can conclude that for k = l − 1, l, 〈γ∇k∇ψ,∇ku〉 ≥ γ 2 d dt ‖∇kψ‖2H1 − Cδ ( ‖∇l%‖2L2 + ‖∇k+1u‖2L2 ) . (4.21) Similarly we have 〈∇kN1,∇k%〉 = 〈∇k(−∇% · u− %div u),∇k%〉 = ∫ R3 (div u) |∇k%|2 2 dx− 〈[∇k, u] · ∇ρ,∇kρ〉 + ‖∇k(%div u)‖L2‖∇k%‖L2 , (4.22) 14 Q. CHEN, G. C. WU, Y. H. ZHANG, L. ZOU EJDE-2020/102 where the commutator [∇k, f ]g is defined in (6.6). This together with ∫ R3 (div u) |∇l−1%|2 2 dx− 〈[∇l−1, u] · ∇ρ,∇l−1ρ〉 + ‖∇l−1(% div u)‖L2‖∇l−1%‖L2 . ( ‖∇(%, u)‖L3/2‖∇l−1(%, u)‖L6 + ‖%‖L3‖∇lu‖L2 ) ‖∇l−1%‖L6 . δ‖∇l(%, u)‖2L2 , (4.23) and ∫ R3 (div u) |∇l%|2 2 dx− 〈[∇l, u] · ∇ρ,∇lρ〉+ ‖∇l(%div u)‖L2‖∇l%‖L2 . ( ‖∇u‖L∞‖∇l%‖L2 + ‖∇%‖L3‖∇lu‖L6 + ‖%‖L∞‖∇l+1u‖L2 ) ‖∇l%‖L2 . δ ( ‖∇l%‖2L2 + ‖∇l+1u‖2L2 ) (4.24) implies that for k = l − 1, l, 〈∇kN1,∇k%〉 . δ ( ‖∇l%‖2L2 + ‖∇k+1u‖2L2 ) . (4.25) Furthermore, we have from (2.3) that for k = l − 1, 〈∇l−1N2,∇l−1u〉 = 〈∇l−1 ( − u · ∇u ) ,∇l−1u〉+ 〈 ∇l−1 ( − (P ′(ρ) ρ − P ′(1) ) ∇% ) ,∇l−1u 〉 + 〈 ∇l−1 (µ ρ %∇u ) ,∇lu 〉 + 〈 ∇l−1 ( ∇ (µ ρ % ) · ∇u ) ,∇l−1u 〉 + 〈 ∇l−1 (µ+ ν ρ %div u ) ,∇l−1 div u 〉 + 〈 ∇l−1 ( ∇ (µ+ ν ρ % ) div u ) ,∇l−1u 〉 . ( ‖(%, u)‖L3‖∇l(%, u)‖L2 + ‖∇l−1 (P ′(ρ) ρ − P ′(1), u ) ‖L6‖∇(%, u)‖L3/2 ) ‖∇l−1u‖L6 + ( ‖%‖L∞‖∇lu‖L2 + ‖∇l−1 (% ρ ) ‖L6‖∇u‖L3 ) ‖∇lu‖L2 + ( ‖∇%‖L3‖∇lu‖L2 + ‖∇l (% ρ ) ‖L2‖∇u‖L3 ) ‖∇l−1u‖L6 . δ ( ‖∇l%‖2L2 + ‖∇lu‖2L2 ) + δ ( ‖∇l (P ′(ρ) ρ − P ′(1) ) ‖L2 + ‖∇l ( % ρ ) ‖L2 ) ‖∇lu‖L2 . δ ( ‖∇l%‖2L2 + ‖∇lu‖2L2 ) , (4.26) EJDE-2020/102 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 15 where (6.9) is used, and for k = l, 〈∇lN2,∇lu〉 = 〈∇l(−u · ∇u),∇lu〉+ 〈 ∇l−1 ((P ′(ρ) ρ − P ′(1) ) ∇% ) ,∇l div u 〉 + 〈 ∇l−1 (µ ρ %∆u ) ,∇l div u 〉 + 〈 ∇l−1 (µ+ ν ρ %∇div u ) ,∇l div u 〉 . ( ‖u‖L3‖∇l+1u‖L2 + ‖∇lu‖L6‖∇u‖L3/2 ) ‖∇lu‖L6 + ( ‖%‖L∞‖∇l%‖L2 + ‖∇l−1 ( P ′(ρ) ρ − P ′(1) ) ‖L6‖∇%‖L3 + ‖%‖L∞‖∇l+1u‖L2 + ‖∇l−1 (% ρ ) ‖L6‖∇2u‖L3 ) ‖∇l+1u‖L2 . ‖(%, u)‖H3‖ ( ∇l%,∇l+1u ) ‖L2‖∇l+1u‖L2 . δ ( ‖∇l%‖2L2 + ‖∇l+1u‖2L2 ) . (4.27) Therefore, from (4.26) and (4.27) we conclude that for k = l − 1, l, 〈∇kN2,∇ku〉 . δ ( ‖∇l%‖2L2 + ‖∇k+1u‖2L2 ) . (4.28) Hence plugging (4.21), (4.25) and (4.28) into (4.17) yields that for k = l − 1, l, 1 2 d dt { P ′(1)‖∇k%‖2L2 + ‖∇ku‖2L2 + γ‖∇kψ‖2H1 } + 3µ 4 ‖∇k+1u(t)‖2L2 ≤ Cδ‖∇l%‖2L2 . (4.29) Here the smallness of δ is used again. Now we turn to estimate the L2-norm of ∇l%. By taking the derivative, we have d dt 〈∇l%,∇l−1u〉+ P ′(1)‖∇l%(t)‖2L2 + 〈γ∇lψ,∇l%〉 = 〈∇l(−div u+N1),∇l−1u〉+ 〈∇l−1 (µ∆u+ (µ+ ν)∇div u+N2) ,∇l%〉. (4.30) By using (2.1)3 and Lemmas 6.1–6.4, we obtain the following estimates. 〈γ∇lψ,∇l%〉 = 〈γ∇lψ,∇l(−∆ψ + ψ)〉 = γ‖∇lψ‖2H1 , (4.31) 〈∇l(−div u+N1),∇l−1u〉 = 〈∇l−1(div u+ div(%u)),∇l−1 div u〉 ≤ ‖∇l−1 div u‖2L2 + ‖(%, u)‖L∞‖∇l(%, u)‖L2‖∇lu‖L2 ≤ C‖∇lu‖2L2 + Cδ‖∇l%‖2L2 , (4.32) and as in the proof of (4.26) and (4.27), 〈∇l−1 (µ∆u+ (µ+ ν)∇div u+N2) ,∇l%〉 ≤ P ′(1) 4 ‖∇l%‖2L2 + C‖∇l+1u‖2L2 + C‖(%, u)‖H3‖ ( ∇l%,∇lu,∇l+1u ) ‖L2‖∇l%‖L2 ≤ 3P ′(1) 8 ‖∇l%‖2L2 + C‖∇lu‖2H1 . (4.33) 16 Q. CHEN, G. C. WU, Y. H. ZHANG, L. ZOU EJDE-2020/102 Thus plugging (4.31)–(4.33) into (4.30) yields d dt 〈∇l%,∇l−1u〉+ P ′(1) 2 ‖∇l%(t)‖2L2 + γ‖∇lψ‖2H1 ≤ C2‖∇lu‖2H1 , (4.34) where we take C2 > µ/(4 √ P ′(1)). Therefore, (4.16) can be deduced by adding (4.29) and µ/(4C2) times (4.34), and using the smallness of δ. � Now we are in a position to prove Proposition 4.1. Proof of Proposition 4.1. Choosing a sufficiently small positive number η1 and then taking η1 times (4.4) plus (4.16), we obtain d dt {η1P ′(1) 2 ‖%(t)‖2L2 + η1 2 ‖u‖2L2 + η1γ 2 ‖ψ‖2H1 + µη1 4C1 〈∇%, u〉 + P ′(1) 2 ‖∇l−1%‖2H1 + 1 2 ‖∇l−1u‖2H1 + γ 2 ‖(∇l−1ψ,∇lψ)‖2H1 + µ 4C2 〈∇l%,∇l−1u〉 } + η1µ 2 ‖∇u(t)‖2L2 + η1µP ′(1) 16C1 ‖∇%(t)‖2L2 + η1µγ 4C1 ‖∇ψ‖2H1 + µ 4 ‖∇lu(t)‖2H1 + µP ′(1) 16C2 ‖∇l%(t)‖2L2 + µγ 4C2 ‖∇lψ‖2H1 ≤ 0. (4.35) By integrating (4.35) with respect to t, and using Cauchy’s inequality, the smallness of δ and η1, the fact that C2 > µ 4 √ P ′(1) , and Sobolev interpolation inequality, we can finally deduce the a priori estimate (4.2). � 5. Decay estimates In this section, we shall prove Proposition 2.3 and Proposition 2.5, which imply the optimal decay rate for the solution. To this end, we define H(t) = sup 0≤τ≤t ∑ 0≤k≤l−1 (1 + τ) 3 2+k‖∇k(%,m)(τ)‖2L2 . (5.1) For technical considerations, we do not intend to estimate the optimal time de- cay rates of the highest-order derivatives of the density and the velocity at first. However, we will show that by energy estimates, ‖∇l(%, u)‖L2 can be controlled by ‖(∇l−1%,∇l−1u)‖L2 . First we shall introduce the following useful inequality. Lemma 5.1 ([6, 7]). Assume r1 > 1, r2 ∈ [0, r1], then we have∫ t 0 (1 + t− τ)−r1(1 + τ)−r2dτ ≤ C(r1, r2)(1 + t)−r2 . (5.2) Lemma 5.2. Under the assumption of Proposition 2.3, it holds that ‖∇l(%,m)‖2L2 . (1 + t)−( 1 2+l) { ‖(%0, u0)‖2Hl +H(t) } . (5.3) Proof. We define L(t) = P ′(1) 2 ‖∇l−1%‖2H1 + 1 2 ‖∇l−1u‖2H1 + γ 2 ‖ ( ∇l−1ψ,∇lψ ) ‖2H1 + µ 4C2 〈∇l%,∇l−1u〉. (5.4) EJDE-2020/102 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 17 Since by C2 > µ/(4P ′(1)) from (4.5), it is easy to verify that L(t) ≈ ‖∇l−1(%, u)‖2H1 ≈ ‖∇l−1(%,m)‖2H1 . (5.5) Then by taking (4.16) plus C3‖∇l−1(%, u)‖2L2 with some large number C3 > 0, one arrive at d dt L(t) + C4L((t) ≤ C3‖∇l−1(%, u)‖2L2 (5.6) for some constant C4 > 0. Hence by Gronwall’s inequality, the definition (5.1) of H(t) and Lemma 5.1, we have that L(t) ≤ e−C4tL(0) + C ∫ t 0 e−C4(t−τ)‖∇l−1(%, u)(τ)‖2L2dτ ≤ e−C4tL(0) + C ∫ t 0 e−C4(t−τ)(1 + τ)−( 1 2+l)H(t)dτ . (1 + t)−( 1 2+l){L(0) +H(t)}, (5.7) where the monotonicity of H(t) is used. Combining (5.7) with (5.5) yields ‖∇l(%,m)‖2L2 ≈ ‖∇l(%, u)‖2L2 . L(t) . (1 + t)−(l+ 1 2 ){L(0) +H(t)}, which completes the proof of Lemma 5.2. � Next we estimate the decay rate of the solution (%,m). Lemma 5.3. Under the assumption of Proposition 2.3, it holds that ‖(%,m)‖L2 . (1 + t)−3/4 ( ‖(%0, n0)‖L1∩L2 + δ √ H(t) ) . (5.8) Proof. By (3.24), (3.26), (3.28) and the Duhamel principle, we have ‖(%, n,M)‖L2 = ‖(%̂, n̂, M̂)‖L2 . (1 + t)−3/4 (‖(%0, n0,M0)‖L1 + ‖(%0, n0)‖L2) + ∫ t 0 (1 + t− τ)−3/4 ( ‖ ( Λ−1 divN,Λ−1 curlN ) ‖L1 + ‖Λ−1 divN‖L2 ) dτ. (5.9) We define F1 = ( − P (1 + %) + P (1) + P ′(1)% ) I3 + γ∇ψ ⊗∇ψ − γ 2 ( |ψ|2 + |∇ψ|2 ) I3 − m⊗m 1 + % , (5.10) F2 = %m 1 + % . (5.11) Then from the definition (2.10) of N , we have that |F | . |F1|+ |∇F2|. Moreover, F1 and F2 can be treated as the product of smooth functions depending on %, ψ, ∇ψ and/or m. Therefore, we can estimate the terms in (5.9) in the following: ‖ ( Λ−1 divN,Λ−1 curlN ) ‖L1 . ‖∇F‖L1 . ‖∇F1‖L1 + ‖∇2F2‖L1 . ‖(%,∇%, ψ,∇ψ,m)‖L2‖(∇%,∇ψ,∇m)‖H1 . δ(1 + t)−5/4 √ H(t), (5.12) 18 Q. CHEN, G. C. WU, Y. H. ZHANG, L. ZOU EJDE-2020/102 where (5.1) and Lemma 6.2 are used, and similarly, ‖Λ−1 divN‖L2 ≈ ‖divF‖L2 . ‖∇F1‖L2 + ‖∇2F2‖L2 . ‖(%,∇%, ψ,∇ψ,m)‖L∞‖(∇%,∇ψ,∇m)‖H1 . δ(1 + t)−5/4 √ H(t). (5.13) Plugging (5.12)–(5.13) into (5.9) yields ‖(%,m)‖L2 ≤ ‖(%̂, n̂, M̂)‖L2 . (1 + t)−3/4 (‖(%0, n0,M0)‖L1 + ‖(%0, n0)‖L2) + ∫ t 0 (1 + t− τ)−3/4δ(1 + τ)−5/4 √ H(t)dτ . (1 + t)−3/4 ( ‖(%0,m0)‖L1∩L2 + δ √ H(t) ) . (5.14) � Now we estimate the decay of ‖∇l−1(%,m)‖L2 . Lemma 5.4. Under the assumption of Proposition 2.3, it holds ‖∇l−1(%,m)‖L2 . (1 + t)−( 1 4+ l 2 ) ( ‖(%0,m0)‖L1∩Hl−1 + δ √ H(t) ) . (5.15) Proof. By using the Duhamel principal (1.5) with k = l − 1 and r = 1, we deduce from Proposition 3.3 that ‖∇l−1%‖L2 . (1 + t)−( 1 4+ l 2 ) (‖(%0, n0)‖L1 + ‖(%0, n0)‖Hl−1) + ∫ t/2 0 (1 + t− τ)−( 1 4+ l 2 )‖Λ−1 divN(τ)‖L1dτ + ∫ t t/2 (1 + t− τ)−5/4‖∇l−2Λ−1 divN(τ)‖L1dτ + ∫ t 0 e−R(t−τ)‖∇l−2Λ−1 divN(τ)‖L2dτ. (5.16) By Lemmaa 5.2, 6.2 and 6.4, we have ‖∇l−2Λ−1 divN‖L1 . ‖∇l−1F1‖L1 + ‖∇lF2‖L1 . ‖(%, ψ,∇ψ,m)‖L2‖∇l−1(%, ψ,m)‖H1 . δ(1 + t)−( 1 4+ l 2 ) ( ‖%0, u0)‖Hl + √ H(t) ) , (5.17) and ‖∇l−2Λ−1 divN‖L2 . ‖∇l−1F1‖L2 + ‖∇lF2‖L2 . ‖(%, ψ,∇ψ,m)‖L∞‖∇l−1(%, ψ,m)‖H1 . δ(1 + t)−( 1 4+ l 2 ) ( ‖%0, u0)‖Hl + √ H(t) ) . (5.18) EJDE-2020/102 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 19 Then, by substituting the estimates (5.12), (5.17) and (5.18) into (5.16), we obtain ‖∇l−1%‖L2 . (1 + t)−( 1 4+ l 2 )‖(%0, n0)‖L1∩Hl−1 + ∫ t/2 0 (1 + t− τ)−( 1 4+ l 2 )δ(1 + τ)−5/4 √ H(t)dτ + ∫ t t/2 (1 + t− τ)−5/4δ(1 + τ)−( 1 4+ l 2 ) ( ‖(%0, u0)‖Hl + √ H(t) ) dτ + ∫ t 0 e−R(t−τ)δ(1 + τ)−( 1 4+ l 2 ) ( ‖(%0, u0)‖Hl + √ H(t) ) dτ . (1 + t)−( 1 4+ l 2 ) ( ‖(%0,m0)‖L1∩Hl + δ √ H(t) ) . (5.19) Similarly as in (5.16), we have the estimate ‖∇l−1n‖L2 . (1 + t)−( 1 4+ l 2 ) (‖(%0, n0)‖L1 + ‖(%0, n0)‖Hl−1) + ∫ t/2 0 (1 + t− τ)−( 1 4+ l 2 )‖Λ−1 divN(τ)‖L1dτ + ∫ t t/2 (1 + t− τ)−5/4‖∇l−2Λ−1 divN(τ)‖L1dτ + ∫ t 0 e−R(t−τ)‖∇l−3Λ−1 divN(τ)‖L2dτ. (5.20) Here we only estimate the last term in (5.20) as follows. ‖∇l−3Λ−1 divN‖L2 . ‖∇l−2F1‖L2 + ‖∇l−1F2‖L2 . ‖(%, ψ,∇ψ,m)‖L3‖∇l−2(%, ψ,∇ψ,m)‖L6 + ‖∇l−3(P ′(1 + %)− P ′(1))‖L6‖∇%‖L3 + ‖(%,m)‖L∞‖∇l−1 ( % 1 + % ,m ) ‖L2 . ‖(%,m)‖H2‖∇l−1(%,m)‖L2 + ‖∇l−2%‖L2‖∇%‖L3 . ( ‖(%,m)‖H2 + ‖%‖ 1 l−1 L2 ‖%‖ l−2 l−1 L 6(l−2) 2l−5 ) ‖∇l−1(%,m)‖L2 . δ(1 + t)−( 1 4+ l 2 ) √ H(t). (5.21) Hence plugging (5.12), (5.17) and (5.21) into (5.20) leads to ‖∇l−1n‖L2 . (1 + t)−( 1 4+ l 2 ) ( ‖(%0,m0)‖L1∩Hl + δ √ H(t) ) . (5.22) Similarly we have ‖∇l−1M‖L2 . (1 + t)−( 1 4+ l 2 ) ( ‖(%0,m0)‖L1∩Hl + δ √ H(t) ) . (5.23) This combining with (5.19) and (5.22) yields (5.8). � By the definition (5.1) ofH(t) and using the smallness of δ, we have from Lemmas 5.3–5.4 and Sobolev interpolation inequality that H(t) ≤ CK2 0 . (5.24) 20 Q. CHEN, G. C. WU, Y. H. ZHANG, L. ZOU EJDE-2020/102 Moreover, by (5.24) and Lemma 5.2, we can obtain ‖∇l(%,m)‖L2 . (1 + t)−( 1 4+ l 2 ). (5.25) Finally, we only need to obtain the optimal estimate on the highest-order deriva- tives of m to complete the proof of Proposition 2.3. Lemma 5.5. Under the assumption of Proposition 2.3, it holds ‖∇lm‖L2 ≤ C(1 + t)−( 3 4+ l 2 ). (5.26) Proof. By using the Duhamel principal (1.5) with k = l and r = 2, we deduce from Proposition 3.3 that ‖∇ln‖L2 . (1 + t)−( 3 4+ l 2 ) {‖(%0, n0)‖L1 + ‖(%0, n0)‖Hl} + ∫ t/2 0 (1 + t− τ)−( 3 4+ l 2 )‖Λ−1 divN(τ)‖L1dτ + ∫ t t/2 (1 + t− τ)− 7 4 ‖∇l−2Λ−1 divN(τ)‖L1dτ + ∫ t 0 e−R(t−τ)‖∇l−2Λ−1 divN(τ)‖L2dτ. (5.27) Using (5.17) and (5.18), we have ‖∇l−2Λ−1 divN‖L1 . ‖(%,m)‖L2‖∇l−1(%,m)‖H1 . K2 0 (1 + t)−(1+ l 2 ), (5.28) ‖∇l−2Λ−1 divN‖L2 . ‖(%,m)‖H2‖∇l−1(%,m)‖H1 . K2 0 (1 + t)−(1+ l 2 ). (5.29) Plugging (5.12), (5.28) and (5.29) into (5.27), one arrives at ‖∇ln‖L2 . K0(1 + t)−( 3 4+ l 2 ) + ∫ t/2 0 (1 + t− τ)−( 3 4+ l 2 )δ(1 + τ)−5/4 √ H(t)dτ +K2 0 ∫ t t/2 (1 + t− τ)− 7 4 (1 + τ)−(1+ l 2 )dτ +K2 0 ∫ t 0 e−R(t−τ)(1 + τ)−(1+ l 2 )dτ . (K0 +K2 0 )(1 + t)−( 3 4+ l 2 ). (5.30) Similarly we deduce the same decay rate on ∇lM and obtain (5.26). � By combining the definition of H(t), the inequality (5.24), and Lemma 5.5, we complete the proof of Proposition 2.3. Now we estimate the lower boundedness of the decay rate for the solution to complete the proof of Proposition 2.5. First from (3.18) and (3.19), for |ξ| � 1 we have g1(λ+, λ−) ≥ cos (( (P ′(1) + γ)1/2|ξ|+O(|ξ|3) ) t ) e−(µ+ ν 2 )|ξ| 2t − C|ξ|e−(µ+ ν 2 )|ξ| 2t, (5.31) and g2(λ+, λ−) ∼ C|ξ|−1 sin ((√ P ′(1) + γ|ξ|+O(|ξ|3) ) t ) e−(µ+ ν 2 )|ξ| 2t, (5.32) EJDE-2020/102 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 21 Hence by using the Duhamel principal (1.5) and the condition that m̂0(ξ) = 0 for 0 ≤ |ξ| � 1, we deduce from (3.30) with s = 2 that ‖%‖2L2 = ‖%̂‖2L2 ≥ ∫ |ξ|<η |g1(λ+, λ−)|2|%̂0|2dξ − C ∫ t 0 ( (1 + t− τ)−2‖Λ−2Λ−1 divN(τ)‖2L2 + e−2R(t−τ)‖Λ−1 divN(τ)‖2L2 ) dτ. (5.33) In the spirit of [17], the first term in the right hand side of (5.33) can be estimated as∫ |ξ|<η |g1(λ+, λ−)|2|%̂0|2dξ ≥ ∫ |ξ|<η cos2 ((√ P ′(1) + γ|ξ|+O(|ξ|3) ) t ) e−(2µ+ν)|ξ| 2t|%̂0|2dξ − C ∫ |ξ|<η |ξ|2e−(2µ+ν)|ξ| 2t|%̂0|2dξ ≥ ∫ |ξ|<η cos2 ((√ P ′(1) + γ|ξ| ) t ) e−(2µ+ν)|ξ| 2t|%̂0|2dξ − C ∫ |ξ|<η ( |ξ|3t )2 e−(2µ+ν)|ξ| 2t|%̂0|2dξ − C ∫ |ξ|<η |ξ|2e−(2µ+ν)|ξ| 2t|%̂0|2dξ ≥ Cc20K2 0 t −3/2 ∫ r≤η √ t cos2 (√ P ′(1) + γr √ t ) e−(2µ+ν)r 2 r2dr − C(1 + t)−5/2‖%0‖2L1 ≥ Cc20K2 0 t −3/2 − CK2 0 (1 + t)−5/2, (5.34) where |%̂0| > c0K0 and t ≥ t0 for some sufficiently large t0 > 0. Furthermore, by using the estimates in (5.17), we have ‖Λ−2Λ−1 divN‖2L2 . ‖Λ−1F1‖2L2 + ‖F2‖2L2 . ‖F1‖2L6/5 + ‖F2‖2L2 . ‖(%,m)‖4H1 . (1 + t)−3, (5.35) where (6.10) is used, and obviously ‖Λ−1 divN‖2L2 ≤ C(1 + t)−3. (5.36) Plugging (5.34)–(5.36) into (5.33), and taking t0 large enough, one arrives at ‖%‖2L2 ≥ Cc20K2 0 t −3/2. (5.37) 22 Q. CHEN, G. C. WU, Y. H. ZHANG, L. ZOU EJDE-2020/102 As in (5.33), from (1.5) and (3.32) we have ‖n‖2L2 = ‖n̂‖2L2 ≥ ∫ |ξ|<η ∣∣|ξ|(P ′(1) + γ 1 + |ξ|2 ) g2(λ+, λ−) ∣∣2|%̂0|2dξ − C ∫ t 0 ( (1 + t− τ)−2‖Λ−2Λ−1 divN‖2L2 + e−2R(t−τ)‖Λ−1 divN‖2L2 ) dτ ≥ Cc20K2 0 t −3/2 − C(1 + t)−2, (5.38) which gives ‖n‖2L2 ≥ Cc20K2 0 t −3/2 (5.39) with t ≥ t0. Moreover, by using the condition m̂0(ξ) = 0 for 0 ≤ |ξ| � 1 again we have that ‖M‖2L2 = ‖M̂‖2L2 ≤ ∫ |ξ|≥η e−(2µ+ν)|ξ| 2t|M̂0|2dξ + C ∫ t 0 (1 + t− τ)−2‖Λ−2Λ−1 curlN‖2L2dτ ≤ ∫ |ξ|≥η e−(2µ+ν)η 2t|M̂0|2dξ + C(1 + t)−2 ≤ C(1 + t)−2, (5.40) Thus we conclude that for t ≥ t0, ‖m‖2L2 ≥ ‖n‖2L2 − ‖M‖2L2 ≥ CK2 0 t −3/2. (5.41) Next we shall estimate the lower boundedness on the first derivatives of the solution. As in (5.33), we have from (1.5) and (3.31) that ‖∇%‖2L2 ≥ ∫ |ξ|<η |g1(λ+, λ−)|2|ξ|2|%̂0|2dξ − C ∫ t 0 (1 + t− τ)−3 ( ‖Λ−2Λ−1 divN‖2L2 + ‖Λ−1 divN‖2L2 ) dτ ≥ CK2 0 t −5/2 − C(1 + t)−3 ≥ CK2 0 t −5/2 (5.42) with some sufficiently large positive constant t0. Similarly, we have ‖∇m‖2L2 ≥ CK2 0 t −5/2. (5.43) Moreover, by the Sobolev interpolation inequality, we can deduce that for 2 ≤ k ≤ l − 1, ‖∇%‖L2 . ‖%‖ k−1 k L2 ‖∇k%‖1/kL2 . (1 + t)− 3(k−1) 4k ‖∇k%‖1/kL2 , (5.44) which implies that for 2 ≤ k ≤ l − 1, ‖∇k%‖L2 ≥ (1 + t) 3(k−1) 4 ‖∇%‖kL2 ≥ t 3(k−1) 4 t− 5 4k = t−( 3 4+ k 2 ). (5.45) EJDE-2020/102 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 23 Similarly, for 2 ≤ k ≤ l, we have ‖∇lm‖L2 ≥ t−( 3 4+ k 2 ), (5.46) which complets the proof of Proposition 2.5. 6. Appendix: Analytic tools We will extensively use the Sobolev interpolation of the Gagliardo-Nirenberg inequality. Lemma 6.1. Let 0 ≤ i, j ≤ k, then we have ‖∇if‖Lp . ‖∇jf‖1−aLq ‖∇ kf‖aLr , (6.1) where a satisfies i 3 − 1 p = ( j 3 − 1 q ) (1− a) + (k 3 − 1 r ) a. (6.2) Especially, when p = q = r = 2, we have ‖∇if‖L2 . ‖∇jf‖ k−i k−j L2 ‖∇kf‖ i−j k−j L2 . (6.3) The above lemma is a special case of [22, theorem on p. 125]. To estimate the commutator and the product of two functions, we shall recall the following estimate. Lemma 6.2 ([2, 14]). For k ≥ 0, we have (i) ‖Λk(gh)‖Lp0 . ‖g‖Lp1 ‖Λkh‖Lp2 + ‖Λkg‖Lp3‖h‖Lp4 , (6.4) where p0, p2, p3 ∈ (1,∞) and 1 p0 = 1 p1 + 1 p2 = 1 p3 + 1 p4 . (ii) ‖∇k(gh)‖L1 . ‖g‖L2‖∇kh‖L2 + ‖∇kg‖L2‖h‖L2 , (6.5) Thus we can easily deduce from Lemma 6.2 the following commutator estimate, or one can refer to [19, pp. 98, Lemma 3.4]. Lemma 6.3 ([19]). Let f and g be smooth functions belonging to Hk ∩L∞ for any integer k ≥ 1 and define the commutator [∇k, f ]g = ∇k(fg)− f∇kg. (6.6) Then we have ‖[∇k, f ]g‖L2 . ‖∇f‖L∞‖∇k−1g‖L2 + ‖∇kf‖L2‖g‖L∞ . (6.7) Next, to estimate the L2-norm of the spatial derivatives of some smooth function F (f), we shall recall the following estimate. Lemma 6.4 ([1, 2]). Let F (f) be a smooth function of f with bounded derivatives of any order and f belong to Hk for any integer k ≥ 3, then we have ‖∇k(F (f))‖L2 . sup 0≤i≤k ‖F (i)(f)‖L∞ ( k∑ j=2 ‖f‖j−1− 3(j−1) 2k L2 ‖∇kf‖1+ 3(j−1) 2k L2 + ‖∇kf‖L2 ) . (6.8) 24 Q. CHEN, G. C. WU, Y. H. ZHANG, L. ZOU EJDE-2020/102 Moreover, if f has lower and upper bounds, and ‖f‖k ≤ 1, we have ‖∇k(F (f))‖L2 . ‖∇kf‖L2 . (6.9) Lemma 6.5 ([9, 24]). Let 0 < s < 3, 1 < p < q <∞, 1 q + s 3 = 1 p , then ‖Λ−sf‖Lq . ‖f‖Lp . (6.10) In particular, for −1 < s < 3/2, we have ‖Λ−sf‖L2 . ‖f‖ 2+2s 5 L1 ‖∇f‖ 3−2s 5 L2 . (6.11) Proof. (6.10) can be seen in [24, p. 119, Theorem 1], and [9, P. 10, Remark 1.2.2]. (6.11) can be proved using (6.10) and (6.1). � Acknowledgments. Qing Chen was supported by Natural Science Foundation of Fujian Province (No. 2018J01430). 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Qing Chen School of Applied Mathematics, Xiamen University of Technology, Xiamen, Fujian 361024, China Email address: chenqing@xmut.edu.cn Guochun Wu Fujian Province University Key Laboratory of Computational Science, School of Math- ematical Sciences, Huaqiao University, Quanzhou 362021, China Email address: guochunwu@126.com Yinghui Zhang (corresponding author) School of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi 541004, China Email address: yinghuizhang@mailbox.gxnu.edu.cn Lan Zou Fujian Province University Key Laboratory of Computational Science, School of Math- ematical Sciences, Huaqiao University, Quanzhou 362021, China Email address: zlyoung@163.com 1. Introduction Notation 2. Reformulated system 3. Spectral analysis and linear L2 estimates 4. Energy estimates 5. Decay estimates 6. Appendix: Analytic tools Acknowledgments References