Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 35, pp. 1–33. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.35 EXISTENCE AND UNIQUENESS OF GLOBAL STRONG SOLUTIONS FOR 3D FRACTIONAL COMPRESSIBLE SYSTEMS MENGQIAN LIU, LEI NIU, ZHIGANG WU Abstract. In this article, we study the Cauchy problem for 3D fractional compressible isentropic generalized Navier-Stokes equations for viscous com- pressible fluid with one Levy diffusion process. We first obtain the existence and uniqueness of global strong solutions for small initial data by providing several commutators via the Littlewood-Paley theory. We then derive the L2-decay rate for the highest derivative of the strong solution without decay loss by using a cancellation of a low-medium-frequency quantity. Our results improve those provided recently in [36] . 1. Introduction In this article, we consider the 3D fractional compressible isentropic Navier- Stokes equations which describe the motion of viscous compressible fluid with one Levy diffusion process [36, 37] given by ρ̃t +∇ · (ρ̃u) = 0, (x, t) ∈ R3 × (0,∞), ρ̃ut + µ(−∆)αu+ ρ̃u · ∇u+∇P (ρ̃) = 0, (x, t) ∈ R3 × (0,∞), (1.1) where the initial data satisfy (ρ̃,u)(0, x) = (ρ̃0,u0)(x) → (ρ̃∞, 0), as |x| → ∞. (1.2) Here ρ̃ and u = (u1, u2, u3) are the unknown density and velocity respectively, the pressure P = P (ρ̃) is given by the power law P = Aρ̃γ with constants γ > 1 and A > 0, and µ > 0 denotes the coefficient of viscosity. The fractional Laplace operator (−∆)α is defined by the Fourier transform as ̂(−∆)αf(ξ) = |ξ|2αf̂(ξ), where α is a positive constant and f̂ is the Fourier transform of the function f . We write Λ = (−∆)1/2 for notational convenience. System (1.1) can be regarded as one direct extension of the classical compressible isentropic Navier-Stokes equations, which has been extensively studied over the past decades; see, for example, [13, 14, 15, 16, 18, 19, 20, 21, 25, 26, 28, 29, 30, 39, 40]. In particularly, Matsumura and Nishida [29, 30] first obtained the global existence 2020 Mathematics Subject Classification. 35A09, 35B40;,35Q35. Key words and phrases. Fractional Navier-Stokes equations; global well-posedness; uniqueness; optimal decay rate; Littlewood-Paley theory. ©2025. This work is licensed under a CC BY 4.0 license. Submitted June 30, 2024. Published April 6, 2025. 1 2 M. LIU, L. NIU, Z. WU EJDE-2025/35 of small solution in H3(R3), and further obtain the L2-decay rate of the solution as the heat equation under the additional assumption that the initial perturbation is small in L1. Duan et al. [13] obtained the Lp decay rate for the system with external force terms without requiring the initial perturbation to be small in L1- space. By replacing L1-space by Ḃ−s 1,∞ with s ∈ [0, 1], Li and Zhang [25] obtained a faster L2-decay rate of the solution. Further, Guo and Wang [16] developed a pure energy method to derive the L2-decay rate of the solution and its derivative in H l ∩ Ḣ−s with l ≥ 3 and s ∈ [0, 3 2 ). There are also a lot of results on the classical compressible Navier-Stokes equa- tions in Besov spaces. Based on scaling considerations, Danchin [10] established the global existence of strong solutions for the initial data in the vicinity of the equilibrium in (Ḃ d/2 2,1 ∩ Ḃ d 2−1 2,1 ) × Ḃ d 2−1 2,1 with d ≥ 2. Further, Charve and Danchin [4] and Chen et al. [7] extended Danchin’s result to the general Lp critical Besov spaces. Haspot [17] obtained the same results as in [4, 7] by using Hoff’s viscous effective flux. Moreover, Chen et al. [8] verified the ill-posedness, which means that the critical Besov space in [4, 7, 17] for the compressible Navier-Stokes equations can be regraded as the largest one in which the system is well-posed. Recently, Peng and Zhai [33] proved the global existence for d- dimensional compressible Navier-Stokes equations without heat conductivity for d ≥ 2 in L2-framework. We refer the readers to [9, 11, 12, 41, 42] for more results on critical spaces for the isen- tropic or non-isentropic compressible Navier-Stokes equations. Now, we go back to system (1.1). It is physically relevant by replacing the standard Laplacian opera- tors by the fractional diffusion operators when modelling the anomalous diffusion which has wide applications in physics, probability and finance; see, for example, [1, 22, 31] and the references therein. Some important results on fractional dissi- pation for many fluid models were developed in [3, 24]. In particular, Wang and Zhang in [36] obtained the existence and uniqueness of the global solution for sys- tem (1.1) in H4(R3), and the decay rate O(t− 3 4α ) for (ρ,u) under the assumptions that the initial data (ρ0,u0) is a small perturbation of the constant state (ρ̃∞, 0) when α ∈ ( 12 , 1]. In order to enclose the energy estimates, they actually need take advantage of the nonlocal operator Dm+α with m = 0, 1, 2, 3 and establish one elaborate spectral theory of one linearized nonlocal operator involved in the fractional dissipation viscosity Λ2α for the system, where the eigenvalues and the eigenvectors depend upon the fractional order derivative exponent α. The authors further proved that the results still hold in H3α+1(R3)-framework and obtained new commutator estimates in [37]. To be more precise, we state their main results in the following. Define κ = √ P ′(ρ̃∞) = √ Aγρ̃γ−1 ∞ , a = 2 γ − 1 , ρ̃ = ρ̃∞ κa (κ+ 1 a ρ)a. (1.3) Let ρ̃∞ = 1 and regard µ′ as κaµ. The initial problem for (1.1)-(1.2) is reformulated as follows ρt + u · ∇ρ+ (κ+ 1 a ρ)∇ · u = 0, ut + µ′ (κ+ 1 aρ) a Λ2αu+ u · ∇u+ (κ+ 1 a ρ)∇ρ = 0. (1.4) The associated initial condition (1.2) becomes (ρ,u)(0, x) = (ρ0,u0)(x), x ∈ R3 (1.5) EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 3 with ρ̃0 = 1 κa (κ+ 1 aρ0) a. Wang and Zhang [36, 37] obtained the following results. Lemma 1.1. ([36, 37]) Let α ∈ ( 12 , 1]. There exist constants C0 > 0 and ε0 > 0 such that if E0 = ∥(ρ0,u0)∥H3α+1 + ∥(ρ0 + u0)∥L1 < ε0, then the initial value problem (1.4)-(1.5) has a unique solution (ρ,u) globally in time, which satisfies ρ(t, x) ∈ C0(0,∞;H3α+1) ∩ C1(0,∞;H3α), u(t, x) ∈ C0(0,∞;H3α+1) ∩ C1(0,∞;Hα+1), and it has the decay rate ∥(ρ,u)(t)∥H2 ≤ C0E0(1 + t)− 3 4α . Because of the appearance of the fractional dissipation in (1.1), it is not easy to obtain the global existence in Hs(R3) with s > 3 2 as for the classical compressible isentropic Navier-Stokes equations (see [27] for example). The main reason is the fractional dissipation is weaker than the classical one, which forces us to use the L∞-estimate of the first derivative of the unknowns (Hs(R3) ↪→ L∞(R3) when s > 3 2 ). Therefore, how to weaken the regularity condition for the existence of global strong solutions is a very challenging problem. In addition, there are actually some difficulties to further reduce the regularity index to s + 1 − α, such as the nonlinear term∑ j≥0 22j(s+1−α)|⟨∆j(ρdivu), ρj⟩| ≲ ∥ρ divu∥Ḣs+1−α∥ρ∥Ḣs+1−α ≲ ∥ divu∥Hs+1−α∥ρ∥2Hs+1−α , where we need extra one order regularity of the velocity u. However, for system (1.1), we know that u can only achieve extra α ∈ (1/2, 1) order dissipation compared to initial data. Hence, it is not easy to close the energy argument in Hs+1−α space. The aim of this article is to further refine the global existence results and the decay rates in [36, 37] for the equivalent system (1.4)-(1.5) via the transform (1.3). Motivated by [27], we first establish an energy estimate by using the Littlewood- Paley decomposition theory in Sobolev spaces. In fact, the higher refinement of the low-high decomposition in Littlewood-Paley theorem is important to relax the requirement of the regularity. Then we apply the classical Friedrich’s regularization method to build global approximate solutions and prove the existence of a solution by compactness arguments for the small initial data. Also, we verify that the solution constructed is unique. As for the global existence, we do not need the initial condition in L1-space when deriving the a priori estimate. Moreover, we relax the requirement of the regularity H3α+1(R3) in [37] to Hs+1(R3) with s > 3 2 and develop a large number of complicated commutators which eventually help us to derive the priori estimate. We emphasize that these commutators on the fractional differential operator are general which can be applied to many other compressible fluid models with fractional dissipation. Finally, we shall deduce the optimal decay rates for all of the derivatives of the solution in Hs+1(R3)-framework by using some ideas in [38], where they got the optimal decay rates for all of the derivatives of the solution by virtue of Fourier theory and a new observation for cancellation of a low-medium-frequency quantity. Here, we generalize their results for classical compressible Navier-Stokes equations to the system (1.1). As a consequence, we 4 M. LIU, L. NIU, Z. WU EJDE-2025/35 improve the results on decay rates in [36, 37], which only addressed the optimal decay rate for the solution but not the optimal decay rates for the derivatives of the solution. Specifically, our main results are stated as follows. Theorem 1.2. Let α ∈ (1/2, 1). For ∥(ρ0, u0)∥Hs+1(R3) with s > 3/2, there exists a small constant η > 0 such that ∥ρ0∥Hs+1(R3) + ∥u0∥Hs+1(R3) ≤ η, (1.6) so that the Cauchy problem (1.4)-(1.5) has a unique global solution (ρ,u) satisfying (ρ,u) ∈ C(R+;Hs+1(R3)×Hs+1(R3)), (∇ρ,Λαu) ∈ L2(R+;Hs(R3)×Hs+1(R3)) (1.7) ∥(ρ,u)(t)∥2Hs+1(R3) + ∫ t 0 ∥∇ρ(τ)∥2Hs(R3) + ∥Λαu(τ)∥2Hs+1(R3)dτ ≲ ∥(ρ0,u0)∥2Hs+1(R3). (1.8) Theorem 1.3. Let α ∈ (1/2, 1). Assume that ∥(ρ0,u0)∥Hs+1(R3) is small and ∥(ρ0,u0)∥L1(R3) is bounded. Then the solution (ρ,u) for the Cauchy problem (1.4)- (1.5) satisfies the following optimal decay rate ∥Λσ(ρ,u)(t)∥ ≲ (1 + t)− 3 4α−σ 2 , 0 ≤ σ ≤ σ0 (1.9) with 5/2 < σ0 := s+ 1 < 3+4α 2 . Remark 1.4. The existence and uniqueness of a global solution to (1.4)-(1.5) in the two-dimensional case can be obtained in the same method and we can also get the corresponding decay results except the highest order derivative by similar arguments in the proof of Theorem 1.3. The rest of this article is organized as follows. In Section 2, we give some notations and several useful lemmas. In Section 3, we obtain the a priori estimate of the solution (ρ,u) to the system (1.4)-(1.5). In Section 4, we give the proof of the Theorem 1.2. The optimal decay rates of the solution are established in Section 5 based on the frequency decomposition given in the appendix. 2. Preliminaries We introduce notation that is used throughout this article. The norms in the Sobolev Spaces Hs(Rd) are denoted by ∥ · ∥Hs , where Rd is the d-dimensional Euclidean space. In particular, for s = 0, we will simply use ∥ ·∥ to denote L2-norm and ∥(f, g)∥2 = ∥f∥2 + ∥g∥2. We use ⟨f, g⟩ to denote the inner-product in L2(Rd). The symbol A ≲ B means that there exists a constant c > 0 independent of A and B such that A ≤ cB. The symbol A ≈ B represents A ≲ B and B ≲ A. We denote Di = ∂xi (i = 1, 2, · · · , d), Dk = ∂α1 x1 · · · ∂αd xd with α1 + · · ·+ αd = k. We first recall the Littlewood-Paley decomposition and the definition of Hilbert space. We refer the readers to [2] for more details. Let S(Rd) be the Schwartz class of rapidly decreasing functions. For given f ∈ S(Rd), the Fourier transform Ff = f̂ and its inverse Fourier transform F−1f = f̌ are defined, respectively, by f̂(ξ) ≜ ∫ Rd e−ix·ξf(x)dx and f̌(ξ) ≜ 1 (2π)d ∫ Rd eix·ξf(ξ)dξ. EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 5 Let φ ∈ S(Rd) be supported in the ring C̃ ≜ {ξ ∈ Rd : 3 4 ≤ |ξ| ≤ 8 3 }, and χ be a smooth function supported in the ball B̃ ≜ {ξ ∈ Rd : |ξ| ≤ 4 3 } such that ∑ j∈Z φ(2−jξ) = 1 for each ξ ∈ Rd \ {0}, χ(ξ) + ∑ j≥0 φ(2−jξ) = 1 for each ξ ∈ Rd. Then, for all u ∈ S′(Rd), we can define the nonhomogeneous dyadic blocks as follows ∆−1u ≜ χ(D)u = F−1(χFu), ∆ju ≜ φ(2−jD)u = F−1(φ(2−j ·)Fu), if j ≥ 0. The homogeneous dyadic blocks are defined by ∆̇ju ≜ φ(2−jD)u = F−1(φ(2−j ·)Fu), if j ∈ Z. Hence, u = ∑ j≥−1 ∆ju in S′(Rd) is called the nonhomogenous Littlewood-Paley decomposition of u. For s ∈ R, the nonhomogeneous Hilbert space Hs is given by Hs(Rd) ≜ {f ∈ S′(Rd) : ∥f∥Hs(Rd) ≜ ( ∑ j≥−1 22js∥∆jf∥2 )1/2 < ∞}, and the homogeneous Hilbert space Ḣs is defined as Ḣs(Rd) ≜ {f ∈ S′(Rd) : ∥f∥Ḣs(Rd) ≜ (∑ j∈Z 22js∥∆̇jf∥2 )1/2 < ∞}. One can deduce that there exist two positive constants c0 and C0 such that c0∥f∥Ḣs+1(Rd) ≤ ∥∇f∥Ḣs(Rd) ≤ C0∥f∥Ḣs+1(Rd), and ∥f∥Ḣs(Rd) ≲ ∥f∥Hs(Rd), if s > 0. We list some Bernstein-type inequalities for fractional derivatives which will be used below. Lemma 2.1 ([23]). Let α ≥ 0. Let 1 ≤ p ≤ q ≤ ∞. (i) If f satisfies supp f̂ ⊂ {ξ ∈ Rd : |ξ| ≤ K2j} for some integer j and a constant K > 0, then ∥(−∆)αf∥Lq(Rd) ≤ C12 2jα+jd( 1 p− 1 q )∥f∥Lp(Rd). (ii) If f satisfies supp f̂ ⊂ {ξ ∈ Rd : K12 j ≤ |ξ| ≤ K22 j} for some integer j and a constant 0 < K1 ≤ K2, then C12 2jα∥f∥Lq(Rd) ≤ ∥(−∆)αf∥Lq(Rd) ≤ C22 2jα+jd( 1 p− 1 q )∥f∥Lp(Rd), where C1 and C2 are constants depending on α, p and q only. 6 M. LIU, L. NIU, Z. WU EJDE-2025/35 We then recall some Sobolev inequalities which will be used below. Lemma 2.2 ([36, 37]). Let s > 0. Suppose g ∈ L∞∩Hs(R) and f ∈ C [s](Range(g)). Then f(g(x)) ∈ L∞ ∩Hs(R). Moreover, there exists a constant C > 0, depending on s and ∥g∥L∞ , such that ∥Ds xf(g(x))∥ ≤ C∥f∥C[s]∥Dsg∥. (2.1) In particular, for s ∈ (0, 1], one can directly apply the chain rule for fractional derivatives ∥Λs xf(g(x))∥ ≤ C∥Df∥L∞∥Λsg∥, (2.2) where C is a constant depending an s and ∥g∥L∞ . Lemma 2.3 ([30]). Assume that f(x) is a function on R3. (i) If f(x) ∈ Hs with s > 3 2 , then f ∈ L∞, and ∥f∥L∞ ≤ C∥f∥Hs , (2.3) where C is a positive constant. (i) If f(x) ∈ H1, then f ∈ Lp for any p ∈ [2, 6] and ∥f∥Lp ≤ C∥f∥H1 , (2.4) where C is a positive constant. Lemma 2.4 ([34, 35]). Let q > 1, 2 < p < ∞ with 1 p + α d = 1 q . There exists a constant C > 0 such that if for all f ∈ S ′ is such that f̂ is a function, then ∥f∥Lp(Rd) ≤ C∥Λαf∥Lq(Rd). (2.5) Lemma 2.5 ([2]). Let σ > 0 and σ1 ∈ R. Then we have, for all u, v ∈ Hσ(Rd) ∩ L∞(Rd), ∥uv∥Hσ(Rd) ≲ ∥u∥Hσ(Rd)∥v∥L∞(Rd) + ∥v∥Hσ(Rd)∥u∥L∞(Rd), ∥uv∥Ḣσ(Rd) ≲ ∥u∥Ḣσ(Rd)∥v∥L∞(Rd) + ∥v∥Ḣσ(Rd)∥u∥L∞(Rd). Moreover, if d ≥ 2, then we have, for u ∈ Hσ(Rd) ∩H d 2−1(Rd), v ∈ Hσ+1(Rd) ∩ L∞(R), ∥uv∥Ḣσ(Rd) ≲ ∥u∥Ḣσ(Rd)∥v∥L∞(Rd) + ∥v∥Ḣσ+1(Rd)∥u∥H d 2 −1(Rd) . If σ > d/2, then Hσ(Rd) embeds into L∞(Rd). Also, for all u, v ∈ Hσ(Rd), it holds that ∥uv∥Hσ(Rd) ≲ ∥u∥Hσ(Rd)∥v∥Hσ(Rd). Otherwise, if σ1 ≤ d 2 < σ and σ1 + σ > 0, then for all u ∈ Hσ(Rd), v ∈ Hσ1(Rd), it holds that ∥uv∥Hσ1 (Rd) ≲ ∥u∥Hσ(Rd)∥v∥Hσ1 (Rd). Lemma 2.6 ([2]). Let σ > 0 and f be a smooth function such that f(0) = 0. If u ∈ Hσ(Rd), then there exists a function C = C(σ, f, d) such that ∥f(u)∥Hσ(Rd) ≤ C(∥u∥L∞(Rd))∥u∥Hσ(Rd), ∥f(u)∥Ḣσ(Rd) ≤ C(∥u∥L∞(Rd))∥u∥Ḣσ(Rd). Lemma 2.7 ([2]). Let σ > d 2 and f be a smooth function such that f ′(0) = 0. If u, v ∈ Hσ(Rd), then there exists a function C = C(σ, f, d) such that ∥f(u)−f(v)∥Hσ(Rd) ≤ C(∥u∥L∞(Rd), ∥v∥L∞(Rd))∥u−v∥Hσ(Rd)(∥u∥Hσ(Rd)+∥v∥Hσ(Rd)). EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 7 Lemma 2.8 ([2]). Let σ > d 2 − 1. There exists a positive sequence ∥cj∥l2 = 1 satisfying {cj}j≥−1 such that ∥[u · ∇,∆j ]f∥L2(Rd) ≤ Ccj2 −j(σ+1)∥∇u∥Hσ+1(Rd)∥f∥Hσ+1(Rd). 3. A priori estimates To simplify notation, we will omit the R3 in the spaces and define the functional set (ρ,u) ∈ E(T ) for (ρ,u) ∈ C([0, T ];Hs+1 ×Hs+1), (∇ρ,Λαu) ∈ L2([0, T ];Hs ×Hs+1). The corresponding norms are defined by ∥(ρ,u)∥E(0) = ∥ρ0∥2Hs+1 + ∥u0∥2Hs+1 , ∥(ρ,u)∥E(T ) = ∥ρ∥2L∞ T (Hs+1) + ∥u∥2L∞ T (Hs+1) + ∥∇ρ∥2L2 T (Hs) + ∥Λαu∥2L2 T (Hs+1). Proposition 3.1. Let s > 3/2 and T > 0. Let (ρ,u) ∈ E(T ) be the solution of the Cauchy problem (1.4) with initial data (ρ0,u0). Suppose that ∥ρ(t, ·)∥L∞ ≤ 1/2. Then we have ∥(ρ,u)∥E(T ) ≲ ∥(ρ,u)∥E(0) + ∥(ρ,u)∥2E(T ). Proof. Multiplying ∆jρ∆j and ∆ju∆j on both sides of (1.4)1 and (1.4)2 respec- tively, summing and integrating over R3, we have 1 2 d dt (∥ρj∥2 + ∥uj∥2) + ⟨∆j( µ′ (κ+ 1 aρ) a Λ2αu),uj⟩ = −⟨∆j(u · ∇ρ), ρj⟩ − 1 a ⟨∆j(ρdivu), ρj⟩ − ⟨∆j(u · ∇u),uj⟩ − 1 a ⟨∆j(ρ∇ρ),uj⟩. Note that ⟨∆j( µ′ (κ+ 1 aρ) a Λ2αu),uj⟩ = µ⟨Λαuj ,Λ αuj⟩+ ⟨∆j ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ,uj⟩ = µ⟨Λαuj ,Λ αuj⟩+ ⟨[∆j , ( µ′ (κ+ 1 aρ) a − µ)]Λ2αu,uj⟩ + ⟨( µ′ (κ+ 1 aρ) a − µ)Λ2αuj ,uj⟩ = µ⟨Λαuj ,Λ αuj⟩+ ⟨[∆j , ( µ′ (κ+ 1 aρ) a − µ)]Λ2αu,uj⟩ + ⟨Λα ( ( µ′ (κ+ 1 aρ) a − µ)uj ) ,Λαuj⟩ = ⟨ µ′ (κ+ 1 aρ) a Λαuj ,Λ αuj⟩+ ⟨[∆j , ( µ′ (κ+ 1 aρ) a − µ)]Λ2αu,uj⟩ + ⟨[Λα, ( µ′ (κ+ 1 aρ) a − µ)]uj ,Λ αuj⟩, (3.1) 8 M. LIU, L. NIU, Z. WU EJDE-2025/35 so one has 1 2 d dt (∥ρj∥2 + ∥uj∥2) + ⟨ µ′ (κ+ 1 aρ) a Λαuj ,Λ αuj⟩ = −⟨∆j(u · ∇ρ), ρj⟩ − 1 a ⟨∆j(ρdivu), ρj⟩ − ⟨∆j(u · ∇u),uj⟩ − 1 a ⟨∆j(ρ∇ρ),uj⟩ − ⟨[∆j , ( µ′ (κ+ 1 aρ) a − µ)]Λ2αu,uj⟩ − ⟨[Λα, ( µ′ (κ+ 1 aρ) a − µ)]uj ,Λ αuj⟩. (3.2) Applying Di∆jρDi∆j and Di∆juDi∆j to (1.4)1 and (1.4)2 respectively and inte- grating over R3, we obtain 1 2 d dt (∥Diρj∥2 + ∥Diuj∥2) + ⟨Di∆j( µ′ (κ+ 1 aρ) a Λ2αu), Diuj⟩ = −⟨Di∆j(u · ∇ρ), Diρj⟩ − 1 a ⟨Di∆j(ρdivu), Diρj⟩ − ⟨Di∆j(u · ∇u), Diuj⟩ − 1 a ⟨Di∆j(ρ∇ρ), Diuj⟩. (3.3) Similar to (3.1), we have ⟨Di∆j( µ′ (κ+ 1 aρ) a Λ2αu), Diuj⟩ = ⟨ µ′ (κ+ 1 aρ) a DiΛ αuj , DiΛ αuj⟩+ ⟨∆j ( Di( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) , Diuj⟩ + ⟨[∆j , ( µ′ (κ+ 1 aρ) a − µ)]DiΛ 2αu, Diuj⟩ + ⟨[Λα, µ′ (κ+ 1 aρ) a − µ]Diuj , DiΛ αuj⟩, (3.4) and hence 1 2 d dt (∥Diρj∥2 + ∥Diuj∥2) + ⟨ µ′ (κ+ 1 aρ) a DiΛ αuj , DiΛ αuj⟩ = −⟨Di∆j(u · ∇ρ), Diρj⟩ − 1 a ⟨Di∆j(ρ divu), Diρj⟩ − ⟨Di∆j(u · ∇u), Diuj⟩ − 1 a ⟨Di∆j(ρ∇ρ), Diuj⟩ − ⟨∆j ( Di( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) , Diuj⟩ − ⟨[∆j , ( µ′ (κ+ 1 aρ) a − µ)]DiΛ 2αu, Diuj⟩ − ⟨[Λα, µ′ (κ+ 1 aρ) a − µ]Diuj , DiΛ αuj⟩. (3.5) EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 9 Summing ∑ 1≤i≤3 (3.5), we obtain 1 2 d dt ( 3∑ i=1 ∥Diρj∥2 + 3∑ i=1 ∥Diuj∥2) + 3∑ i=1 ⟨ µ′ (κ+ 1 aρ) a DiΛ αuj , DiΛ αuj⟩ = − 3∑ i=1 ⟨Di∆j(u · ∇ρ), Diρj⟩ − 3∑ i=1 1 a ⟨Di∆j(ρ divu), Diρj⟩ − 3∑ i=1 ⟨Di∆j(u · ∇u), Diuj⟩ − 3∑ i=1 1 a ⟨Di∆j(ρ∇ρ), Diuj⟩ − 3∑ i=1 ⟨∆j ( Di( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) , Diuj⟩ − 3∑ i=1 ⟨[∆j , ( µ′ (κ+ 1 aρ) a − µ)]DiΛ 2αu, Diuj⟩ − 3∑ i=1 ⟨[Λα, µ′ (κ+ 1 aρ) a − µ]Diuj , DiΛ αuj⟩. (3.6) Multiplying Λ2∆j (1.4)1 and Λ2∆j (1.4)2 by Λ2∆jρ and Λ2∆ju, respectively, and integrating with respect to x over R3, we have 1 2 d dt (∥Λ2ρj∥2 + ∥Λ2uj∥2) + ⟨Λ2∆j( µ′ (κ+ 1 aρ) a Λ2αu),Λ2uj⟩ = −⟨Λ2∆j(u · ∇ρ),Λ2ρj⟩ − 1 a ⟨Λ2∆j(ρdivu),Λ 2ρj⟩ − ⟨Λ2∆j(u · ∇u),Λ2uj⟩ − 1 a ⟨Λ2∆j(ρ∇ρ),Λ2uj⟩. (3.7) Note that Λ2 = −∆, so ⟨Λ2∆j( µ′ (κ+ 1 aρ) a Λ2αu),Λ2uj⟩ = µ⟨Λ2+αuj ,Λ 2+αuj⟩ − ⟨div∆j ( ∇( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ,Λ2uj⟩ − ⟨div∆j ( ( µ′ (κ+ 1 aρ) a − µ)∇Λ2αu ) ,Λ2uj⟩ = µ⟨Λ2+αuj ,Λ 2+αuj⟩ − ⟨div∆j ( ∇( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ,Λ2uj⟩ + ⟨[∆j , ( µ′ (κ+ 1 aρ) a − µ)]∇Λ2αu,∇Λ2uj⟩ + ⟨( µ′ (κ+ 1 aρ) a − µ)∇Λ2αuj ,∇Λ2uj⟩ = µ∥Λ2+αuj∥2 + ⟨( µ′ (κ+ 1 aρ) a − µ)∇Λ1+αuj ,∇Λ1+αuj⟩ − ⟨div∆j ( ∇( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ,Λ2uj⟩ 10 M. LIU, L. NIU, Z. WU EJDE-2025/35 + ⟨[∆j , ( µ′ (κ+ 1 aρ) a − µ)]∇Λ2αu,∇Λ2uj⟩ + ⟨[Λ1−α, µ′ (κ+ 1 aρ) a − µ]∇Λ2αuj ,∇Λ1+αuj⟩. (3.8) Similarly, − ⟨Λ2∆j(u · ∇ρ),Λ2ρj⟩ = ⟨div∆j(∇u · ∇ρ),Λ2ρj⟩ − ⟨[∆j ,u · ∇]∇ρ,∇Λ2ρj⟩ − ⟨u · ∇∇ρj ,∇Λ2ρj⟩ = ⟨div∆j(∇u · ∇ρ),Λ2ρj⟩ − ⟨[∆j ,u · ∇]∇ρ,∇Λ2ρj⟩+ ⟨∇u∇2ρj ,Λ 2ρj⟩ + 1 2 ⟨divu, |Λ2ρj |2⟩, (3.9) − 1 a ⟨Λ2∆j(ρdivu),Λ 2ρj⟩ − 1 a ⟨Λ2∆j(ρ∇ρ),Λ2uj⟩ = 1 a ⟨div∆j(∇ρdivu),Λ2ρj⟩ − 1 a ⟨[∆j , ρ]∇ divu,∇Λ2ρj⟩ − 1 a ⟨ρ∇ divuj ,∇Λ2ρj⟩+ 1 a ⟨div∆j(∇ρ∇ρ),Λ2uj⟩ − 1 a ⟨[∆j , ρ]∇2ρ,∇Λ2uj⟩ − 1 a ⟨ρ∇2ρj ,∇Λ2uj⟩ = 1 a ⟨div∆j(∇ρdivu),Λ2ρj⟩ − 1 a ⟨[∆j , ρ]∇ divu,∇Λ2ρj⟩ + 1 a ⟨div∆j(∇ρ∇ρ),Λ2uj⟩ − 1 a ⟨[∆j , ρ]∇2ρ,∇Λ2uj⟩ + 1 a ⟨∇ρ∇2ρj ,Λ 2uj⟩+ 1 a ⟨∇ρΛ2ρj ,Λ 2uj⟩ + 1 a ⟨∇ρ∇divuj ,Λ 2ρj⟩, (3.10) and − ⟨Λ2∆j(u · ∇u),Λ2uj⟩ = ⟨div∆j(∇u · ∇u),Λ2uj⟩ − ⟨[∆j ,u · ∇]∇u,∇Λ2uj⟩ − ⟨u · ∇∇uj ,∇Λ2uj⟩ = ⟨div∆j(∇u · ∇u),Λ2uj⟩ − ⟨[∆j ,u · ∇]∇u,∇Λ2uj⟩+ ⟨∇u∇2uj ,Λ 2uj⟩ + 1 2 ⟨divu, |Λ2uj |2⟩. (3.11) Inserting (3.8)-(3.11) into (3.7), we obtain 1 2 d dt (∥Λ2ρj∥2 + ∥Λ2uj∥2) + µ∥Λ2+αuj∥2 + ⟨( µ′ (κ+ 1 aρ) a − µ)∇Λ1+αuj ,∇Λ1+αuj⟩ = ⟨div∆j(∇u · ∇ρ),Λ2ρj⟩ − ⟨[∆j ,u · ∇]∇ρ,∇Λ2ρj⟩+ ⟨∇u∇2ρj ,Λ 2ρj⟩ + 1 2 ⟨divu, |Λ2ρj |2⟩+ ⟨div∆j(∇u · ∇u),Λ2uj⟩ − ⟨[∆j ,u · ∇]∇u,∇Λ2uj⟩+ ⟨∇u∇2uj ,Λ 2uj⟩ + 1 2 ⟨divu, |Λ2uj |2⟩+ 1 a ⟨div∆j(∇ρ divu),Λ2ρj⟩ EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 11 − 1 a ⟨[∆j , ρ]∇ divu,∇Λ2ρj⟩+ 1 a ⟨div∆j(∇ρ∇ρ),Λ2uj⟩ − 1 a ⟨[∆j , ρ]∇2ρ,∇Λ2uj⟩+ 1 a ⟨∇ρ∇2ρj ,Λ 2uj⟩ + 1 a ⟨∇ρΛ2ρj ,Λ 2uj⟩+ 1 a ⟨∇ρ∇ divuj ,Λ 2ρj⟩ + ⟨div∆j(∇( µ′ (κ+ 1 aρ) a − µ)Λ2αu),Λ2uj⟩ − ⟨[∆j , ( µ′ (κ+ 1 aρ) a − µ)]∇Λ2αu,∇Λ2uj⟩ − ⟨[Λ1−α, µ′ (κ+ 1 aρ) a − µ]∇Λ2αuj ,∇Λ1+αuj⟩. (3.12) It follows from the equations (1.4)1-(1.4)2 and integration by parts that d dt ⟨∇ρj ,uj⟩+ κ∥∇ρj∥2 = κ∥ divu∥2 − ⟨∇∆j(u · ∇ρ),uj⟩ − 1 a ⟨∇∆j(ρdivu),uj⟩ − 1 a ⟨∆j(ρ∇ρ),∇ρj⟩ − ⟨∆j(u · ∇u),∇ρj⟩ − ⟨∆j ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ,∇ρj⟩ − µ⟨∇ρj ,Λ 2αuj⟩ (3.13) and d dt ⟨Di∇ρj , Diuj⟩+ κ∥Di∇ρj∥2 = κ∥Di divu∥2 − ⟨Di∇∆j(u · ∇ρ), Diuj⟩ − 1 a ⟨Di∇∆j(ρdivu), Diuj⟩ − 1 a ⟨Di∆j(ρ∇ρ), Di∇ρj⟩ − ⟨∆jDi(u · ∇u), Di∇ρj⟩ − ⟨∆jDi ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) , Di∇ρj⟩ − µ⟨Di∇ρj , DiΛ 2αuj⟩. (3.14) Summing for 1 ≤ i ≤ 3 in (3.14), we have d dt 3∑ i=1 ⟨Di∇ρj , Diuj⟩+ κ 3∑ i=1 ∥Di∇ρj∥2 = κ 3∑ i=1 ∥Di divu∥2 + 3∑ i=1 ⟨Di∆j(u · ∇ρ),divDiuj⟩ + 1 a 3∑ i=1 ⟨Di∆j(ρdivu),divDiuj⟩ − 3∑ i=1 1 a ⟨Di∆j(ρ∇ρ), Di∇ρj⟩ − 3∑ i=1 ⟨∆jDi(u · ∇u), Di∇ρj⟩ − 3∑ i=1 ⟨∆jDi ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) , Di∇ρj⟩ 12 M. LIU, L. NIU, Z. WU EJDE-2025/35 − µ 3∑ i=1 ⟨Di∇ρj , DiΛ 2αuj⟩. (3.15) Next, summing (3.2), (3.6), (3.12), β1 × (3.13) and β2 × (3.15), one obtains 1 2 d dt (∥ρj∥2 + ∥uj∥2 + 3∑ i=1 ∥Diρj∥2 + 3∑ i=1 ∥Diuj∥2 + ∥Λ2ρj∥2 + ∥Λ2uj∥2 + 2β1⟨∇ρj ,uj⟩+ 2β2 3∑ i=1 ⟨Di∇ρj , Diuj⟩) + ⟨ µ′ (κ+ 1 aρ) a Λαuj ,Λ αuj⟩ + 3∑ i=1 ⟨ µ′ (κ+ 1 aρ) a DiΛ αuj , DiΛ αuj⟩+ ⟨( µ′ (κ+ 1 aρ) a − µ)∇Λ1+αuj ,∇Λ1+αuj⟩ + µ∥Λ2+αuj∥2 + β1κ∥∇ρj∥2 + β2κ 3∑ i=1 ∥Di∇ρj∥2 = β1κ∥divu∥2 + β2κ 3∑ i=1 ∥Di divu∥2 − ⟨∆j(u · ∇ρ), ρj⟩ − 1 a ⟨∆j(ρdivu), ρj⟩ − ⟨∆j(u · ∇u),uj⟩ − 1 a ⟨∆j(ρ∇ρ),uj⟩ − 3∑ i=1 ⟨Di∆j(u · ∇ρ), Diρj⟩ − 3∑ i=1 1 a ⟨Di∆j(ρdivu), Diρj⟩ − 3∑ i=1 (⟨Di∆j(u · ∇u), Diuj⟩ + 1 a ⟨Di∆j(ρ∇ρ), Diuj⟩ − ⟨∆j ( Di( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) , Diuj⟩) + ⟨div∆j(∇u · ∇ρ),Λ2ρj⟩+ 1 a ⟨div∆j(∇ρdivu),Λ2ρj⟩ + ⟨div∆j(∇u · ∇u),Λ2uj⟩+ 1 a ⟨div∆j(∇ρ∇ρ),Λ2uj⟩ + ⟨∇u∇2uj ,Λ 2uj⟩+ 1 2 ⟨divu, |Λ2uj |2⟩+ ⟨∇u∇2ρj ,Λ 2ρj⟩ + 1 2 ⟨divu, |Λ2ρj |2⟩+ 1 a ⟨∇ρ∇ divuj ,Λ 2ρj⟩+ 1 a ⟨∇ρ∇2ρj ,Λ 2uj⟩ + 1 a ⟨∇ρΛ2ρj ,Λ 2uj⟩+ ⟨div∆j(∇( µ′ (κ+ 1 aρ) a − µ)Λ2αu),Λ2uj⟩ − β1⟨∇∆j(u · ∇ρ),uj⟩ − β1 a ⟨∇∆j(ρdivu),uj⟩ − β1 a ⟨∆j(ρ∇ρ),∇ρj⟩ − β1⟨∆j(u · ∇u),∇ρj⟩ − β1⟨∆j ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ,∇ρj⟩ − β1µ⟨∇ρj ,Λ 2αuj⟩+ β2 3∑ i=1 ⟨Di∆j(u · ∇ρ),divDiuj⟩ + β2 a 3∑ i=1 ⟨Di∆j(ρdivu),divDiuj⟩ − β2 3∑ i=1 1 a ⟨Di∆j(ρ∇ρ), Di∇ρj⟩ EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 13 − β2 3∑ i=1 ⟨∆jDi(u · ∇u), Di∇ρj⟩ − β2 3∑ i=1 ⟨∆jDi ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) , Di∇ρj⟩ − β2µ 3∑ i=1 ⟨Di∇ρj , DiΛ 2αuj⟩ − ⟨([Λα, µ′ (κ+ 1 aρ) a − µ)]uj ,Λ αuj⟩ − 3∑ i=1 ⟨[Λα, µ′ (κ+ 1 aρ) a − µ]Diuj , DiΛ αuj⟩ − ⟨[Λ1−α, µ′ (κ+ 1 aρ) a − µ]∇Λ2αuj ,∇Λ1+αuj⟩ − ⟨[∆j , ( µ′ (κ+ 1 aρ) a − µ)]Λ2αu,uj⟩ − 3∑ i=1 ⟨[∆j , µ′ (κ+ 1 aρ) a − µ]DiΛ 2αu, Diuj⟩ − ⟨[∆j ,u · ∇]∇ρ,∇Λ2ρj⟩ − ⟨[∆j ,u · ∇]∇u,∇Λ2uj⟩ − 1 a ⟨[∆j , ρ]∇ divu,∇Λ2ρj⟩ − 1 a ⟨[∆j , ρ]∇2ρ,∇Λ2uj⟩ − ⟨[∆j , µ′ (κ+ 1 aρ) a − µ]∇Λ2αu,∇Λ2uj⟩ := 27∑ i=1 Ii. (3.16) Now, we estimate the terms on the right hand side of (3.16). By Lemma 2.1, Hölder inequality and Young’s inequality, we obtain |I1| = | − ⟨∆j(u · ∇ρ), ρj⟩ − 1 a ⟨∆j(ρ divu), ρj⟩| ≲ 2−j∥∆j(u · ∇ρ)∥∥∇ρj∥2 + 2−j∥∆j(ρ divu)∥∥∇ρj∥ ≲ 2−2jCϵ1∥∆j(u · ∇ρ)∥2 + 2−2jCϵ1∥∆j(ρ divu)∥2 + 2ϵ1∥∇ρj∥2, |I2| = | − ⟨∆j(u · ∇u),uj⟩ − 1 a ⟨∆j(ρ∇ρ),uj⟩| ≲ 2−jα∥∆j(u · ∇u)∥∥Λαuj∥+ 2−jα∥∆j(ρ∇ρ)∥∥Λαuj∥ ≲ 2−2jαCϵ1∥∆j(u · ∇u)∥2 + 2−2jαCϵ1∥∆j(ρ∇ρ)∥2 + 2ϵ1∥Λαuj∥2. Similarly, |I3| = | − 3∑ i=1 ⟨Di∆j(u · ∇ρ), Diρj⟩ − 3∑ i=1 1 a ⟨Di∆j(ρdivu), Diρj⟩| ≲ Cϵ1 3∑ i=1 ∥Di∆j(u · ∇ρ)∥2 + Cϵ1 3∑ i=1 ∥Di∆j(ρ divu)∥2 + 2ϵ1∥∇ρj∥2, 14 M. LIU, L. NIU, Z. WU EJDE-2025/35 |I4| = | − 3∑ i=1 ⟨Di∆j(u · ∇u), Diuj⟩ − 3∑ i=1 1 a ⟨Di∆j(ρ∇ρ), Diuj⟩| ≲ 2−2jαCϵ1 3∑ i=1 ( ∥Di∆j(u · ∇u)∥2 + ∥Di∆j(ρ∇ρ)∥2 ) + 2ϵ1 3∑ i=1 ∥ΛαDiuj∥2, |I5| = | − 3∑ i=1 ⟨∆j ( Di( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) , Diuj⟩| ≲ 2−2jαCϵ1 3∑ i=1 ∥∆j ( Di( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ∥2 + ϵ1 3∑ i=1 ∥ΛαDiuj∥2, |I6| = |⟨div∆j(∇u · ∇ρ),Λ2ρj⟩+ 1 a ⟨div∆j(∇ρdivu),Λ2ρj⟩| ≲ Cϵ1∥ div∆j(∇u · ∇ρ)∥2 + Cϵ1∥div∆j(∇ρdivu)∥2 + 2ϵ1∥Λ2ρj∥2, |I7| = |⟨div∆j(∇u · ∇u),Λ2uj⟩+ 1 a ⟨div∆j(∇ρ∇ρ),Λ2uj⟩| ≲ 2−2jαCϵ1∥ div∆j(∇u · ∇u)∥2 + 2−2jαCϵ1∥ div∆j(∇ρ∇ρ)∥2 + 2ϵ1∥ΛαΛ2uj∥2, |I8| = |⟨∇u∇2uj ,Λ 2uj⟩+ 1 2 ⟨divu, |Λ2uj |2⟩| ≲ 2−2jαCϵ1∥∇u∥2L∞∥∇2uj∥2 + 2−2jαCϵ1∥ divu∥2L∞∥Λ2uj∥2 + 2ϵ1∥ΛαΛ2uj∥2, |I9| = |⟨∇u∇2ρj ,Λ 2ρj⟩+ 1 2 ⟨divu, |Λ2ρj |2⟩+ 1 a ⟨∇ρ∇ divuj ,Λ 2ρj⟩| ≲ Cϵ1 ( ∥∇u∥2L∞∥∇2ρj∥2 + ∥ divu∥2L∞∥Λ2ρj∥2 + ∥∇ρ∥2L∞∥∇ divuj∥2 ) + 3ϵ1∥Λ2ρj∥2, |I10| = |1 a ⟨∇2ρj∇ρ,Λ2uj⟩+ 1 a ⟨∇ρΛ2uj ,Λ 2ρj⟩| ≲ 2−2jαCϵ1∥∇ρ∥2L∞∥∇2ρj∥2 + 2−2jαCϵ1∥∇ρ∥2L∞∥Λ2ρj∥2 + 2ϵ1∥ΛαΛ2uj∥2, |I11| = |⟨div∆j(∇( µ′ (κ+ 1 aρ) a − µ)Λ2αu),Λ2uj⟩| ≲ 2−2jαCϵ1∥ div∆j(∇( µ′ (κ+ 1 aρ) a − µ)Λ2αu∥2 + ϵ1∥ΛαΛ2uj∥2, |I12| = | − β1⟨∇∆j(u · ∇ρ),uj⟩ − β1 1 a ⟨∇∆j(ρdivu),uj⟩| ≲ 2−2jαCϵ1β 2 1∥∇∆j(u · ∇ρ)∥2 + 2−2jαCϵ1β 2 1∥∇∆j(ρdivu)∥2 + 2ϵ1∥Λαuj∥2, |I13| = | − β1 1 a ⟨∆j(ρ∇ρ),∇ρj⟩ − β1⟨∆j(u · ∇u),∇ρj⟩ − β1⟨∆j ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ,∇ρj⟩| ≲ Cϵ1β 2 1 ( ∥∆j(ρ∇ρ)∥2 + ∥∆j(u · ∇u)∥2 + ∥∆j ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ∥2 ) + 3ϵ1∥∇ρj∥2, EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 15 |I14| = | − β1µ⟨∇ρj ,Λ 2αuj⟩| ≲ Cϵ1β 2 1∥Λ2αuj∥2 + ϵ1∥∇ρj∥2, |I15| = |β2 3∑ i=1 ⟨Di∆j(u · ∇ρ),divDiuj⟩+ β2 1 a 3∑ i=1 ⟨Di∆j(ρdivu),divDiuj⟩| ≲ 2−2jαCϵ1β 2 2 ( 3∑ i=1 ∥Di∆j(u · ∇ρ)∥2 + 3∑ i=1 ∥Di∆j(ρdivu)∥2 ) + 2ϵ1 3∑ i=1 ∥Λα divDiuj∥2, |I16| = | − β2 3∑ i=1 1 a ⟨Di∆j(ρ∇ρ), Di∇ρj⟩ − β2 3∑ i=1 ⟨∆jDi(u · ∇u), Di∇ρj⟩, − β2 3∑ i=1 ⟨∆jDi ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) , Di∇ρj⟩| ≲ Cϵ1β 2 2 3∑ i=1 ∥Di∆j(ρ∇ρ)∥2 + Cϵ1β 2 2 3∑ i=1 ∥Di∆j(u · ∇u)∥2 + Cϵ1β 2 2∥ 3∑ i=1 Di∆j ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ∥2 + 3ϵ1 3∑ i=1 ∥Di∇ρj∥2, |I17| = | − β2µ 3∑ i=1 ⟨Di∇ρj , DiΛ 2αuj⟩| ≲ Cϵ1β 2 2 3∑ i=1 ∥DiΛ 2αuj∥2 + ϵ1 3∑ i=1 ∥Di∇ρj∥2. By using Hölder inequality, Young’s inequality, Lemma 2.6 and Commutator Esti- mates [37, Lemma 2.1], we obtain |I18| = | − ⟨[Λα, µ′ (κ+ 1 aρ) a − µ]uj ,Λ αuj⟩| ≲ ∥Λα( µ′ (κ+ 1 aρ) a − µ)∥L∞∥uj∥∥Λαuj∥ ≲ ∥( µ′ (κ+ 1 aρ) a − µ)∥Hs+α∥uj∥∥Λαuj∥ ≲ 2−2jαCϵ1∥ρ∥2Hs+α∥Λαuj∥2 + ϵ1∥Λαuj∥2, |I19| = | − 3∑ i=1 ⟨[Λα, µ′ (κ+ 1 aρ) a − µ]Diuj , DiΛ αuj⟩| ≲ Cϵ1∥ρ∥2Hs+α∥∇uj∥2 + ϵ1 3∑ i=1 ∥DiΛ αuj∥2, |I20| = | − ⟨[Λ1−α, µ′ (κ+ 1 aρ) a − µ]∇Λ2αuj ,∇Λ1+αuj⟩| ≲ Cϵ1∥ρ∥2Hs+1−α∥∇Λ2αuj∥2 + ϵ1∥∇Λ1+αuj∥2. According to Lemma 2.6 and Lemma 2.8, one has |I21| = | − ⟨[∆j , ( µ′ (κ+ 1 aρ) a − µ)]Λ2αu,uj⟩| 16 M. LIU, L. NIU, Z. WU EJDE-2025/35 ≲ cj2 −js∥ρ∥Hs+1∥∇Λ2α−2u∥Hs∥Λαuj∥. Then, I22 − I27 can be estimated in the same way as I21. Specifically, |I22| = | − 3∑ i=1 ⟨[∆j , µ′ (κ+ 1 aρ) a − µ]DiΛ 2αu, Diuj⟩ | ≲ 3∑ i=1 cj2 −js∥ρ∥Hs+1∥Di∇Λ2α−2u∥Hs∥Diuj∥, |I23| = | − ⟨[∆j ,u · ∇]∇ρ,∇Λ2ρj⟩| ≲ cj2 −js∥∇u∥Hs∥∇ρ∥Hs∥∇Λ2ρj∥, |I24| = | − ⟨[∆j ,u · ∇]∇u,∇Λ2uj⟩| ≲ cj2 −js∥∇u∥Hs∥∇u∥Hs∥∇Λ2uj∥, |I25| = | − 1 a ⟨[∆j , ρ]∇divu,∇Λ2ρj⟩| ≲ cj2 −js∥∇ρ∥Hs∥ divu∥Hs∥∇Λ2ρj∥, |I26| = | − 1 a ⟨[∆j , ρ]∇2ρ,∇Λ2uj⟩| ≲ cj2 −js∥∇ρ∥Hs∥∇ρ∥Hs∥∇Λ2uj∥, |I27| = |⟨[∆j , ( µ′ (κ+ 1 aρ) a − µ)]∇Λ2αu,∇Λ2uj⟩| ≲ cj2 −js∥ρ∥Hs+1∥Λ2αu∥Hs∥∇Λ2uj∥. Inserting the above inequalities about I1 − I27 into (3.16), we have 1 2 d dt (∥ρj∥2 + ∥uj∥2 + 3∑ i=1 ∥Diρj∥2 + 3∑ i=1 ∥Diuj∥2 + ∥Λ2ρj∥2 + ∥Λ2uj∥2 + 2β1⟨∇ρj ,uj⟩+ 2β2 3∑ i=1 ⟨Di∇ρj , Diuj⟩) + ⟨ µ′ (κ+ 1 aρ) a Λαuj ,Λ αuj⟩ + 3∑ i=1 ⟨ µ′ (κ+ 1 aρ) a DiΛ αuj , DiΛ αuj⟩ + ⟨( µ′ (κ+ 1 aρ) a − µ)∇Λ1+αuj ,∇Λ1+αuj⟩ + µ∥Λ2+αuj∥2 + β1κ∥∇ρj∥2 + β2κ 3∑ i=1 ∥Di∇ρj∥2 ≲ β1κ∥divu∥2 + β2κ 3∑ i=1 ∥Di divu∥2 + 8ϵ1∥∇ρj∥2 + 5ϵ1∥Λ2ρj∥2 + 4ϵ1 3∑ i=1 ∥Di∇ρj∥2 + 5ϵ1∥Λαuj∥2 (3.17) + 4ϵ1 3∑ i=1 ∥ΛαDiuj∥2 + 7ϵ1∥ΛαΛ2uj∥2 + 2ϵ1 3∑ i=1 ∥Λα divDiuj∥2 + ϵ1∥∇Λ1+αuj∥2 + Cϵ1β 2 1∥Λ2αuj∥2 + Cϵ1β 2 2 3∑ i=1 ∥DiΛ 2αuj∥2 + 2−2jCϵ1(∥∆j(u · ∇ρ)∥2 + ∥∆j(ρdivu)∥2) + 2−2jαCϵ1(∥∆j(u · ∇u)∥2 + ∥∆j(ρ∇ρ)∥2) EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 17 + Cϵ1 3∑ i=1 ( ∥Di∆j(u · ∇ρ)∥2 + ∥Di∆j(ρdivu)∥2 ) + 2−2jαCϵ1 3∑ i=1 ( ∥Di∆j(u · ∇u)∥2 + ∥Di∆j ( ρ∇ρ)∥2 + ∥∥∆j(Di( µ′ (κ+ 1 aρ) a − µ)Λ2αu) ∥∥2) + Cϵ1 ( ∥ div∆j(∇u · ∇ρ)∥2 + ∥ div∆j(∇ρdivu)∥2 ) + 2−2jαCϵ1∥∇u∥2L∞∥∇2uj∥2 + 2−2jαCϵ1∥ div u∥2L∞∥Λ2uj∥2 + 2−2jαCϵ1∥div∆j(∇u · ∇u)∥2 + 2−2jαCϵ1∥div∆j(∇ρ∇ρ)∥2 + Cϵ1∥∇u∥2L∞∥∇2ρj∥2 + Cϵ1∥ divu∥2L∞∥Λ2ρj∥2 + Cϵ1∥∇ρ∥2L∞∥∇ divuj∥2 + 2−2jαCϵ1∥∇ρ∥2L∞(∥∇2ρj∥2 + ∥Λ2ρj∥2) + 2−2jαCϵ1∥div∆j(∇( µ′ (κ+ 1 aρ) a − µ)Λ2αu)∥2 + 2−2jαCϵ1β 2 1(∥∇∆j(u · ∇ρ)∥2 + ∥∇∆j(ρdivu)∥2) + Cϵ1β 2 1 ( ∥∆j(ρ∇ρ)∥2 + ∥∆j(u · ∇u)∥2 + ∥∆j ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ∥2 ) + 2−2jαCϵ1β 2 2 ( 3∑ i=1 ∥Di∆j(u · ∇ρ)∥2 + 3∑ i=1 ∥Di∆j(ρ divu)∥2 ) + Cϵ1β 2 2 3∑ i=1 ( ∥Di∆j(ρ∇ρ)∥2 + ∥Di∆j(u · ∇u)∥2 + ∥Di∆j ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ∥2 ) + 2−2jαCϵ1∥ρ∥2Hs+α∥Λαuj∥2 + Cϵ1∥ρ∥2Hs+α∥∇uj∥2 + Cϵ1∥ρ∥2Hs+1−α∥∇Λ2αuj∥2 + cj2 −js∥ρ∥Hs+1∥∇Λ2α−2u∥Hs∥Λαuj∥ + 3∑ i=1 cj2 −js∥ρ∥Hs+1∥Di∇Λ2α−2u∥Hs∥Diuj∥ + cj2 −js∥∇u∥Hs∥∇ρ∥Hs∥∇Λ2ρj∥+ cj2 −js∥∇u∥Hs∥∇u∥Hs∥∇Λ2uj∥ + cj2 −js∥∇ρ∥Hs∥divu∥Hs∥∇Λ2ρj∥+ cj2 −js∥∇ρ∥Hs∥∇ρ∥Hs∥∇Λ2uj∥ + cj2 −js∥ρ∥Hs+1∥Λ2αu∥Hs∥∇Λ2uj∥. (3.18) Since ∥Λαuj∥2 ≲ ∥uj∥2 + ∥∇uj∥2, it holds that ∥Λ2αuj∥2 ≲ ∥Λαuj∥2 + ∥∇Λαuj∥2, ∥DiΛ 2αuj∥2 ≲ ∥DiΛ αuj∥2 + ∥Di∇Λαuj∥2. (3.19) 18 M. LIU, L. NIU, Z. WU EJDE-2025/35 Choosing β1, β2 and ϵ1 small enough, and combining (3.19) with the following facts: ∥(ρj ,u)∥2 + 3∑ i=1 ∥Di(ρj ,uj)∥2 + ∥Λ2(ρj ,uj)∥2 + 2β1⟨∇ρj ,uj⟩+ 2β2 3∑ i=1 ⟨Di∇ρj , Diuj⟩ ≈ ∥ρj∥2H2 + ∥uj∥2H2 ,〈 µ′ (κ+ 1 aρ) a Λαuj ,Λ αuj 〉 + 3∑ i=1 〈 µ′ (κ+ 1 aρ) a DiΛ αuj , DiΛ αuj 〉 + 〈 ( µ′ (κ+ 1 aρ) a − µ)∇Λ1+αuj ,∇Λ1+αuj 〉 + µ∥Λ2+αuj∥2 ≈ ∥Λαuj∥2H2 , ∥∇ρj∥2 + 3∑ i=1 ∥Di∇ρj∥2 ≈ ∥∇ρj∥2H1 , for each j ≥ 0, we can obtain from (3.18) that ∥ρj∥2H2 + ∥uj∥2H2 + ∫ t 0 ∥∇ρj∥2H1 + ∥Λαuj∥2H2dτ ≲ ∥∆jρ0∥2H2 + ∥∆ju0∥2H2 + ∫ t 0 { ∥∆j(u · ∇ρ)∥2 + ∥∆j(ρdivu)∥2 + ∥∆j(u · ∇u)∥2 + ∥∆j(ρ∇ρ)∥2 + ∥∇∆j(u · ∇ρ)∥2 + ∥∇∆j(ρdivu)∥2 + ∥∇∆j(u · ∇u)∥2 + ∥∇∆j(ρ∇ρ)∥2 + 3∑ i=1 ∥∆j ( Di( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ∥2 + ∥div∆j(∇u · ∇ρ)∥2 + ∥ div∆j(∇ρ divu)∥2 + ∥∇u∥2L∞∥∇2uj∥2 + ∥ divu∥2L∞∥Λ2uj∥2 + ∥ div∆j(∇u · ∇u)∥2 + ∥ div∆j(∇ρ∇ρ)∥2 + ∥∇u∥2L∞∥∇2ρj∥2 + ∥ divu∥2L∞∥Λ2ρj∥2 + ∥∇ρ∥2L∞∥∇ divuj∥2 + ∥∇ρ∥2L∞∥∇2ρj∥2 + ∥∇ρ∥2L∞∥Λ2ρj∥2 + ∥ div∆j(∇( µ′ (κ+ 1 aρ) a − µ)Λ2αu∥2 + ∥∆j ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ∥2 + 3∑ i=1 ∥Di∆j ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ∥2 + ∥ρ∥2Hs+α∥Λαuj∥2 + ∥ρ∥2Hs+α∥∇uj∥2 + ∥ρ∥2Hs+1−α∥∇Λ2αuj∥2 + cj2 −js∥ρ∥Hs+1∥∇Λ2α−2u∥Hs∥Λαuj∥ + 3∑ i=1 cj2 −js∥ρ∥Hs+1∥Di∇Λ2α−2u∥Hs∥Diuj∥ + cj2 −js∥∇u∥Hs∥∇ρ∥Hs∥∇Λ2ρj∥+ cj2 −js∥∇u∥Hs∥∇u∥Hs∥∇Λ2uj∥ + cj2 −js∥∇ρ∥Hs∥divu∥Hs∥∇Λ2ρj∥+ cj2 −js∥∇ρ∥Hs∥∇ρ∥Hs∥∇Λ2uj∥ EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 19 + cj2 −js∥ρ∥Hs+1∥Λ2αu∥Hs∥∇Λ2uj∥ } dτ. (3.20) By the same argument as in (3.16), we have 1 2 d dt (∥ρ∥2 + ∥u∥2 + 3∑ i=1 ∥Diρ∥2 + 3∑ i=1 ∥Diu∥2 + 2β1⟨∇ρ,u⟩) + µ⟨Λαu,Λαu⟩ + µ 3∑ i=1 ⟨DiΛ αu, DiΛ αu⟩+ β1κ∥∇ρ∥2 = −⟨u · ∇ρ, ρ⟩+ 1 a ⟨ρ∇ρ,u⟩ − ⟨u · ∇u,u⟩ − ⟨( µ′ (κ+ 1 aρ) a − µ)Λ2αu,u⟩ − 3∑ i=1 ⟨Di(u · ∇ρ), Diρ⟩ − 3∑ i=1 1 a ⟨Diρ divu, Diρ⟩ − 3∑ i=1 ⟨Di(u · ∇u), Diu⟩ − 3∑ i=1 ⟨Di( µ′ (κ+ 1 aρ) a − µ)Λ2αu, Diu⟩ − 3∑ i=1 ⟨( µ′ (κ+ 1 aρ) a − µ)DiΛ 2αu, Diu⟩+ β1κ∥ divu∥2 − β1⟨∇(u · ∇ρ),u⟩ − β1 1 a ⟨∇(ρdivu),u⟩ − β1 1 a ⟨ρ∇ρ,∇ρ⟩ − β1⟨u · ∇u,∇ρ⟩ − β1⟨( µ′ (κ+ 1 aρ) a − µ)Λ2αu,∇ρ⟩ − β1µ⟨∇ρ,Λ2αu⟩ := 11∑ i=1 Si. (3.21) From Lemma 2.3, Lemma 2.4 and Lemma 2.6, we bound the terms S1–S3 as follows: |S1| = | − ⟨u · ∇ρ, ρ⟩+ 1 a ⟨ρ∇ρ,u⟩| ≲ ∥u∥L3∥∇ρ∥L2∥ρ∥L6 ≲ ∥∇u∥2∥ρ∥2H1 + ϵ1∥∇ρ∥2, |S2| = | − ⟨u · ∇u,u⟩| ≲ ∥u∥L3/α∥∇u∥L2∥u∥ L 6 3−2α ≲ ∥∇u∥2∥u∥2H1 + ϵ1∥Λαu∥2, |S3| = | − ⟨( µ′ (κ+ 1 aρ) a − µ)Λ2αu,u⟩| ≲ ∥( µ′ (κ+ 1 aρ) a − µ)∥L3/α∥Λ2αu∥L2∥u∥ L 6 3−2α ≲ ∥ρ∥2H1∥Λ2αu∥2 + ϵ1∥Λαu∥2, where we used Hölder inequality and Young’s inequality. Similarly, |S4| = | − 3∑ i=1 ⟨Di(u · ∇ρ), Diρ⟩| ≲ ∥Λ2ρ∥2∥u∥2Hs + ϵ1∥∇ρ∥2, |S5| = | − 3∑ i=1 1 a ⟨Diρ divu, Diρ⟩| ≲ ∥∇ρ∥2∥divu∥2Hs + ϵ1∥∇ρ∥2, |S6| = | − 3∑ i=1 ⟨Di(u · ∇u), Diu⟩| ≲ ∥∇u∥2H1∥Λ2u∥2 + ϵ1∥Λαu∥2, 20 M. LIU, L. NIU, Z. WU EJDE-2025/35 |S7| = ∣∣∣− 3∑ i=1 ⟨Di( µ′ (κ+ 1 aρ) a − µ)Λ2αu, Diu⟩ ∣∣∣ ≲ 3∑ i=1 ∥Diρ∥2H1∥Λ2αu∥2 + ϵ1 3∑ i=1 ∥DiΛ αu∥2, |S8| = | − 3∑ i=1 ⟨( µ′ (κ+ 1 aρ) a − µ)DiΛ 2αu, Diu⟩| ≲ 3∑ i=1 ∥ρ∥2H1∥Λ2αDiu∥2 + ϵ1 3∑ i=1 ∥DiΛ αu∥2, |S9| = | − β1⟨∇(u · ∇ρ),u⟩ − β1 1 a ⟨∇(ρdivu),u⟩| ≲ β2 1∥∇u∥2∥ divu∥2H1 + β2 1∥ divu∥2H1∥ divu∥2 + 2ϵ1∥∇ρ∥2, |S10| = | − β1 1 a ⟨ρ∇ρ,∇ρ⟩ − β1⟨u · ∇u,∇ρ⟩| ≲ β2 1∥ρ∥2Hs∥∇ρ∥2 + β2 1∥u∥2Hs∥∇u∥2 + 2ϵ1∥∇ρ∥2, |S11| = | − β1⟨( µ′ (κ+ 1 aρ) a − µ)Λ2αu,∇ρ⟩ − β1µ⟨∇ρ,Λ2αu⟩| ≲ β2 1∥ρ∥2L∞∥Λ2αu∥2 + β2 1µ 2∥Λ2αu∥2 + ϵ1∥∇ρ∥2. A similar computation as in (3.20) yields that ∥ρ∥2H1 + ∥u∥2H1 + ∫ t 0 (∥∇ρ∥2 + ∥Λαu∥2H1)dτ ≲ ∥ρ0∥2H1 + ∥u0∥2H1 + ∫ t 0 {∥∇u∥2∥ρ∥2H1 + ∥∇u∥2∥u∥2H1 + ∥ρ∥2H1∥Λ2αu∥2 + ∥Λ2ρ∥2∥u∥2Hs + ∥∇ρ∥2∥ divu∥2Hs + ∥∇u∥2H1∥Λ2u∥2 + 3∑ i=1 ∥Diρ∥2H1∥Λ2αu∥2 + 3∑ i=1 ∥ρ∥2H1∥Λ2αDiu∥2 + β2 1∥∇u∥2∥ divu∥2H1 + β2 1∥divu∥2H1∥ divu∥2 + β2 1∥ρ∥2Hs∥∇ρ∥2 + β2 1∥u∥2Hs∥∇u∥2 + β2 1∥ρ∥2Hs∥Λ2αu∥2}dτ ≲ ∥ρ0∥2H1 + ∥u0∥2H1 + ∫ t 0 (∥ρ∥2Hs+1 + ∥u∥2Hs+1)(∥∇ρ∥2Hs + ∥Λαu∥2Hs+1)dτ. (3.22) Note that ∥∆−1ρ∥2Hs + ∥∆−1u∥2Hs + ∫ t 0 (∥∆−1∇ρ∥2 + ∥∆−1Λ αu∥2H1)dτ ≲ ∥ρ∥2H1 + ∥u∥2H1 + ∫ t 0 (∥∇ρ∥2 + ∥Λαu∥2H1)dτ. (3.23) EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 21 Now, multiplying (3.20) by 22j(s−1) and then summing over j ≥ 0, and using (3.22) and (3.23), we obtain ∥ρ∥2Hs+1 + ∥u∥2Hs+1 + ∫ t 0 ∥∇ρ∥2Hs + ∥Λαu∥2Hs+1dτ ≲ ∥ρ0∥2Hs+1 + ∥u0∥2Hs+1 + ∫ t 0 { ∥u · ∇ρ∥2 Ḣs−1 + ∥ρ divu∥2 Ḣs−1 + ∥u · ∇u∥2 Ḣs−1 + ∥ρ∇ρ∥2 Ḣs−1 + ∥∇(u · ∇ρ)∥2 Ḣs−1 + ∥∇(ρ divu)∥2 Ḣs−1 + ∥∇(u · ∇u)∥2 Ḣs−1 + ∥∇(ρ∇ρ)∥2 Ḣs−1 + 3∑ i=1 ∥Di( µ′ (κ+ 1 aρ) a − µ)Λ2αu∥2 Ḣs−1 + ∥div(∇u · ∇ρ)∥2 Ḣs−1 + ∥ div(∇ρdivu)∥2 Ḣs−1 + ∥∇u∥2L∞∥∇2u∥2 Ḣs−1 + ∥ divu∥2L∞∥Λ2u∥2 Ḣs−1 + ∥ div(∇u · ∇u)∥2 Ḣs−1 + ∥ div(∇ρ∇ρ)∥2 Ḣs−1 + ∥∇u∥2L∞∥∇2ρ∥2 Ḣs−1 + ∥ divu∥2L∞∥Λ2ρ∥2 Ḣs−1 + ∥∇ρ∥2L∞∥∇ divu∥2 Ḣs−1 + ∥∇ρ∥2L∞∥∇2ρ∥2 Ḣs−1 + ∥∇ρ∥2L∞∥Λ2ρ∥2 Ḣs−1 + ∥ div(∇( µ′ (κ+ 1 aρ) a − µ)Λ2αu)∥2 Ḣs−1 + ∥( µ′ (κ+ 1 aρ) a − µ)Λ2αu∥2 Ḣs−1 + 3∑ i=1 ∥Di ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ∥2 Ḣs−1 + ∥ρ∥2Hs+α∥Λαu∥2 Ḣs−1 + ∥ρ∥2Hs+α∥∇u∥2 Ḣs−1 + ∥ρ∥2Hs+1−α∥∇Λ2αu∥2 Ḣs−1 + ∥ρ∥Hs+1∥∇Λ2α−2u∥Hs∥Λαu∥Ḣs−2 + 3∑ i=1 ∥ρ∥Hs+1∥Di∇Λ2α−2u∥Hs∥Diu∥Ḣs−2 + ∥∇u∥Hs∥∇ρ∥Hs∥∇Λ2ρ∥Ḣs−2 + ∥∇u∥Hs∥∇u∥Hs∥∇Λ2u∥Ḣs−2 + ∥∇ρ∥Hs∥ divu∥Hs∥∇Λ2ρ∥Ḣs−2 + ∥∇ρ∥Hs∥∇ρ∥Hs∥∇Λ2u∥Ḣs−2 + ∥ρ∥Hs+1∥Λ2αu∥Hs∥∇Λ2u∥Ḣs−2 } dτ + ∫ t 0 (∥ρ∥2Hs+1 + ∥u∥2Hs+1)(∥∇ρ∥2Hs + ∥Λαu∥2Hs+1)dτ. (3.24) By Lemma 2.5, we deduce that ∥u · ∇ρ∥2 Ḣs−1 + ∥ρdivu∥2 Ḣs−1 + ∥u · ∇u∥2 Ḣs−1 + ∥ρ∇ρ∥2 Ḣs−1 + ∥∇(u · ∇ρ)∥2 Ḣs−1 + ∥∇(ρdivu)∥2 Ḣs−1 + ∥∇(u · ∇u)∥2 Ḣs−1 + ∥∇(ρ∇ρ)∥2 Ḣs−1 ≲ ∥u · ∇ρ∥2Hs + ∥ρdivu∥2Hs + ∥u · ∇u∥2Hs + ∥ρ∇ρ∥2Hs ≲ ∥u∥2Hs∥∇ρ∥2Hs + ∥ρ∥2Hs∥ divu∥2Hs + ∥u∥2Hs∥∇u∥2Hs + ∥ρ∥2Hs∥∇ρ∥2Hs ≲ (∥u∥2Hs + ∥ρ∥2Hs)(∥∇ρ∥2Hs + ∥divu∥2Hs + ∥∇u∥2Hs). (3.25) 22 M. LIU, L. NIU, Z. WU EJDE-2025/35 Similar to (3.25), we have 3∑ i=1 ∥Di( µ′ (κ+ 1 aρ) a − µ)Λ2αu∥2 Ḣs−1 + ∥ div(∇( µ′ (κ+ 1 aρ) a − µ)Λ2αu)∥2 Ḣs−1 ≲ ∥ρ∥2Hs+1∥Λ2αu∥2Hs , ∥ div(∇u · ∇ρ)∥2 Ḣs−1 + ∥ div(∇ρdivu)∥2 Ḣs−1 ≲ ∥∇u∥2Hs∥∇ρ∥2Hs + ∥∇ρ∥2Hs∥ divu∥2Hs , ∥ div(∇u · ∇u)∥2 Ḣs−1 + ∥ div(∇ρ∇ρ)∥2 Ḣs−1 ≲ ∥∇u∥2Hs∥∇u∥2Hs + ∥∇ρ∥2Hs∥∇ρ∥2Hs , ∥( µ′ (κ+ 1 aρ) a − µ)Λ2αu∥2 Ḣs−1 + 3∑ i=1 ∥Di ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ∥2 Ḣs−1 ≲ ∥ρ∥2Hs∥Λ2αu∥2Hs . (3.26) Inserting inequalities (3.25)-(3.26) into (3.24), one has ∥ρ∥2Hs+1 + ∥u∥2Hs+1 + ∫ t 0 ∥∇ρ∥2Hs + ∥Λαu∥2Hs+1dτ ≲ ∥ρ0∥2Hs+1 + ∥u0∥2Hs+1 + ∫ t 0 (∥ρ∥2Hs+1 + ∥u∥2Hs+1)(∥∇ρ∥2Hs + ∥Λαu∥2Hs+1)dτ, (3.27) where we used Hr ↪→ L∞(r > 3 2 ). This completes the proof. □ 4. Proof of Theorem 1.1 In this section, we shall use four steps to prove the existence and uniqueness of the solution to (1.1) with the small initial data. Step 1: Construction of the approximate solutions. We first construct the approximate solutions based on the classical Friedrich’s method as in [27]. Defining the smoothing operator Jεf = F−1(10≤|ξ|≤ 1 ε Ff), we consider the approximate system of (1.4), ∂Uε ∂t = Fε(Uε), Uε = (ρε,uε), (4.1) where Fε(Uε) = (F (1) ε (Uε), F (2) ε (Uε)) are defined by F (1) ε (Uε) = −κdiv(Jεuε)− Jε(Jεuε · ∇Jερε)− 1 a Jε(Jερε divJεuε), F (2) ε (Uε) = −κ∇Jερε − µΛ2αJεuε − Jε(( µ′ (κ+ 1 aJερε)a − µ)Λ2αJεuε) − Jε(Jεuε · ∇Jεuε)− 1 a Jε(Jερε∇Jερε). (4.2) Using that ∥Jεf∥Hk ≤ C(1 + 1 ε2 ) k 2 ∥f∥L2 , it is easy to see that ∥Fε(Uε)∥L2 ≤ Cεf(∥Uε∥L2), ∥Fε(Uε)− Fε(Ũε)∥L2 ≤ Cεg(∥Uε∥L2 , ∥Ũε∥L2)∥Uε − Ũε∥L2 , EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 23 where f and g are polynomials with positive coefficients. Therefore, the approxi- mate system can be viewed as an ODE system on L2. By using Cauchy-Lipschitz theorem, we know that there exists a maximal time Tε > 0 and a unique (ρε,uε) which is continuous in time with a value in L2. J 2 ε = Jε ensures that (Jερε,Jεuε) is also a solution of (4.1). Thus, (ρε,uε) = (Jερε,Jεuε). Then, (ρε,uε) satisfies ∂tρε + κdiv(uε) = −Jε(uε · ∇ρε)− 1 a Jε(ρε divuε), ∂tuε + κ∇ρε + µΛ2αuε = −Jε(( µ′ (κ+ 1 aρε) a − µ)Λ2αuε)− Jε(uε · ∇uε)− 1 a Jε(ρε∇ρε). (4.3) Moreover, we can also conclude that (ρε,uε) ∈ E(Tϵ). Step 2: Uniform energy estimates. Now we will prove that ∥(ρε,uε)∥E(T ) is uniformly bounded independently of ε by losing energy estimate. Suppose that η is small such that ∥ρ0∥L∞ ≤ 1/4. Since the solution depends continuously on the time variable, then there exist 0 < T0 < Tε and a positive constant C such that the solution (ρε,uε) satisfies ∥ρε(t, ·)∥L∞ ≤ 1 2 for all t ∈ [0, T0], ∥(ρε,uε)∥E(T0) ≤ 2C∥(ρ,u)∥E(0). We suppose that T0 is a maximal time so that the above inequalities hold without loss of generality. Next, we will get a refined estimate on [0, T0] for the solution. According to Proposition 3.1, for all 0 < T ≤ T0, one has ∥(ρε,uε)∥E(T ) ≤ C∥(ρ,u)∥E(0) + C∥(ρε,uε)∥2E(T ) ≤ C∥(ρ,u)∥E(0)(1 + 4C2η). (4.4) Let η < 1 8C2 . Then ∥(ρε,uε)∥E(T0) ≤ 3 2 C∥(ρ,u)∥E(0) < 2Cη. (4.5) A standard bootstrap argument yields for all 0 < T < ∞ that ∥(ρε,uε)∥E(T ) ≤ 2C∥(ρ,u)∥E(0). Step 3: Existence of the solution. (ρ,u) ∈ E(T ) of (1.4) can be deduced by a standard compactness argument to the approximation sequence (ρε, uε); for convenience, we omit here. Moreover, it holds for all 0 < T < ∞ that ∥(ρ,u)∥E(T ) ≤ 2C∥(ρ,u)∥E(0). Step 4: Uniqueness of the solution. The uniqueness of the solution can be guaranteed as Step 4 in [27], so we omit the proof here. 5. Optimal decay rates In this section, we will provide a detailed proof of Theorem 1.3. We shall see that the optimal decay rates for all of the derivatives of the solutions may be established by virtue of Fourier theory and a new observation for cancellation of a low-medium-frequency quantity. 24 M. LIU, L. NIU, Z. WU EJDE-2025/35 5.1. Energy estimates on the highest-order derivative. In this subsection, based on the energy method together with low-high frequency decompositon, we obtain the estimates on the σ0-order (σ0 > 5 2 ) derivative of solution (ρ,u) to the Cauchy problem (1.4)-(1.5). Lemma 5.1. There exist sufficiently small constants β3 > 0 and ϵ1 > 0 such that 1 2 d dt (∥Λσ0ρ∥2 + ∥Λσ0u∥2 + 2β3⟨∇Λσ0−1ρ,Λσ0−1u⟩) + ( µ′ (κ+ 1 aρ) a − 2ϵ1)∥Λσ0Λαu∥2 + β3(κ− ϵ1)∥∇Λσ0−1ρ∥2 ≲ ( η + η2 )( ∥Λσ0ρ∥2 + ∥Λσ0u∥2 + ∥Λ 3+4α 2 u∥2 ) + β3(η + Cϵ1)∥Λσ0−1Λ2αu∥2. (5.1) Here η > 0 is defined in (1.6) and sufficiently small. Proof. Multiplying Λσ0(1.4)1−Λσ0(1.4)2 by Λσ0ρ and Λσ0u, respectively, integrat- ing the resultant inequality with respect to x over R3 and using Young’s inequality, we obtain 1 2 d dt (∥Λσ0ρ∥2 + ∥Λσ0u∥2) + ⟨ µ′ (κ+ 1 aρ) a Λσ0Λαu,Λσ0Λαu⟩ = −⟨[Λσ0 ,u]∇ρ,Λσ0ρ⟩+ 1 2 ⟨divuΛσ0ρ,Λσ0ρ⟩ − 1 a ⟨[Λσ0 , ρ] divu,Λσ0ρ⟩ − 1 a ⟨[Λσ0 , ρ]∇ρ,Λσ0u⟩+ 1 a ⟨∇ρΛσ0u,Λσ0ρ⟩ − ⟨[Λσ0 , µ′ (κ+ 1 aρ) a − µ]Λ2αu,Λσ0u⟩ − ⟨[Λα, µ′ (κ+ 1 aρ) a − µ]Λσ0u,Λσ0Λαu⟩ − ⟨[Λσ0 ,u]∇u,Λσ0u⟩ + 1 2 ⟨divuΛσ0u,Λσ0u⟩ ≲ ∥[Λσ0 ,u]∇ρ∥∥Λσ0ρ∥+ 1 2 ∥ divu∥L∞∥Λσ0ρ∥2 + 1 a ∥[Λσ0 , ρ] divu∥∥Λσ0ρ∥ + 1 a ∥[Λσ0 , ρ]∇ρ∥∥Λσ0u∥+ 1 a ∥∇ρ∥L∞∥Λσ0u∥∥Λσ0ρ∥ + ∥[Λσ0 , µ′ (κ+ 1 aρ) a − µ]Λ2αu∥ L 6 6−2σ0+4α ∥Λσ0u∥ L 6 2σ0−4α + ∥[Λα, µ′ (κ+ 1 aρ) a − µ]Λσ0u∥∥Λσ0Λαu∥+ ∥[Λσ0 ,u]∇u∥∥Λσ0u∥ + 1 2 ∥ divu∥L∞∥Λσ0u∥2. (5.2) EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 25 By using Lemma 2.4, Young’s inequality and Commutator Estimates [37, Lemma 2.1], one obtains ∥[Λσ0 , µ′ (κ+ 1 aρ) a − µ]Λ2αu∥ L 6 6−2σ0+4α ∥Λσ0u∥ L 6 2σ0−4α ≲ ( ∥Λσ0 ( µ′ (κ+ 1 aρ) a − µ ) ∥∥Λ2αu∥ L 6 3−2σ0+4α + ∥D ( µ′ (κ+ 1 aρ) a − µ ) ∥ L 6 5−2σ0+2α ∥Λσ0−1Λ2αu∥ L 6 1+2α ) ∥Λ 3+4α 2 u∥ ≲ ( ∥ρ∥Hσ0∥Λσ0u∥+ ∥Λσ0−α−1Dρ∥∥Λσ0+αu∥ ) ∥Λ 3+4α 2 u∥ ≲ ∥ρ∥Hσ0 ( ∥Λσ0u∥2 + ∥Λ 3+4α 2 u∥2 ) + Cϵ1∥Λσ0−α−1Dρ∥2∥Λ 3+4α 2 u∥2 + ϵ1∥Λσ0+αu∥2 ≲ (η + η2) ( ∥Λσ0u∥2 + ∥Λ 3+4α 2 u∥2 ) + ϵ1∥Λσ0+αu∥2, (5.3) and ∥[Λσ0 ,u]∇ρ∥∥Λσ0ρ∥ ≲ ( ∥Λσ0u∥∥∇ρ∥L∞ + ∥Du∥L∞∥Λσ0−1∇ρ∥ ) ∥Λσ0ρ∥ ≲ η(∥Λσ0u∥2 + ∥Λσ0−1∇ρ∥2 + ∥Λσ0ρ∥2). (5.4) Similarly, we obtain ∥[Λσ0 , ρ] divu∥∥Λσ0ρ∥ ≲ η(∥Λσ0ρ∥2 + ∥Λσ0−1 divu∥2), ∥[Λσ0 , ρ]∇ρ∥∥Λσ0u∥ ≲ η(∥Λσ0u∥2 + ∥Λσ0−1∇ρ∥2 + ∥Λσ0ρ∥2), ∥[Λσ0 ,u]∇u∥∥Λσ0u∥ ≲ η(∥Λσ0u∥2 + ∥Λσ0−1∇u∥2), ∥[Λα, µ′ (κ+ 1 aρ) a − µ]Λσ0u∥∥Λσ0Λαu∥ ≲ η2∥Λσ0u∥2 + ϵ1∥Λσ0Λαu∥2. (5.5) Inserting (5.3), (5.4) and (5.5) into (5.2), we deduce that 1 2 d dt (∥Λσ0ρ∥2 + ∥Λσ0u∥2) + ⟨ µ′ (κ+ 1 aρ) a Λσ0Λαu,Λσ0Λαu⟩ ≲ ( η + η2 )( ∥Λσ0u∥2 + ∥Λσ0−1∇ρ∥2 + ∥Λσ0ρ∥2 + ∥Λσ0−1 divu∥2 + ∥Λ 3+4α 2 u∥2 + ∥Λσ0−1∇u∥2 ) + 2ϵ1∥Λσ0Λαu∥2 ≲ ( η + η2 )( ∥Λσ0ρ∥2 + ∥Λσ0u∥2 + ∥Λ 3+4α 2 u∥2 ) + 2ϵ1∥Λσ0Λαu∥2, (5.6) where we used Hr(Rd) ↪→ L∞(Rd) with r > d 2 and Young’s inequality. 26 M. LIU, L. NIU, Z. WU EJDE-2025/35 Multiplying Λσ0−1(1.4)2 by ∇Λσ0−1ρ and using (1.4)1, we have d dt ⟨∇Λσ0−1ρ,Λσ0−1u⟩+ κ∥∇Λσ0−1ρ∥2 = κ∥div Λσ0−1u∥2 + ⟨Λσ0−1(u · ∇ρ),divσ0−1 u⟩ + 1 a ⟨Λσ0−1(ρdivu),div Λσ0−1u⟩ − ⟨∇Λσ0−1ρ,Λσ0−1 ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ⟩ − µ⟨∇Λσ0−1ρ,Λσ0−1Λ2αu⟩ − ⟨∇Λσ0−1ρ,Λσ0−1 ( u · ∇u ) ⟩ − 1 a ⟨∇Λσ0−1ρ,Λσ0−1 ( ρ∇ρ ) ⟩. (5.7) By (2.5) and Young’s inequality, it holds that |⟨∇Λσ0−1ρ,Λσ0−1 ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ⟩| ≲ ∥∇Λσ0−1ρ∥∥Λσ0−1 ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ∥ ≲ ( ∥Λσ0−1( µ′ (κ+ 1 aρ) a − µ)∥ L 6 2s−4α ∥Λ2αu∥ L 6 3−2σ0+4α + ∥Λσ0−1Λ2αu∥( µ′ (κ+ 1 aρ) a − µ)∥L∞ ) ∥∇Λσ0−1ρ∥ ≲ ∥∇Λσ0−1ρ∥ ( ∥Λ 1+4α 2 ρ∥∥Λσ0u∥+ ∥Λσ0−1Λ2αu∥∥ρ∥L∞ ) ≲ η ( ∥∇Λσ0−1ρ∥2 + ∥Λσ0u∥2 + ∥Λσ0−1Λ2αu∥2 ) , (5.8) |⟨Λσ0−1(u · ∇ρ),div Λσ0−1u⟩| ≲ ∥Λσ0−1(u · ∇ρ)∥∥div Λσ0−1u∥ ≲ ( ∥Λσ0−1u∥L6∥∇ρ∥L3 + ∥Λσ0−1∇ρ∥∥u∥L∞ ) ∥ div Λσ0−1u∥ ≲ η ( ∥∇Λσ0−1u∥2 + ∥Λσ0−1∇ρ∥2 + ∥ div Λσ0−1u∥2 ) . (5.9) In the same way, we have |⟨Λσ0−1(ρ divu),div Λσ0−1u⟩| ≲ η ( ∥∇Λσ0−1ρ∥2 + ∥ div Λσ0−1u∥2 ) , |⟨∇Λσ0−1ρ,Λσ0−1(u · ∇u)⟩| ≲ η ( ∥∇Λσ0−1ρ∥2 + ∥∇Λσ0−1u∥2 ) , ⟨∇Λσ0−1ρ,Λσ0−1 ( ρ∇ρ ) ⟩| ≲ η∥∇Λσ0−1ρ∥2. (5.10) Combining (5.7)-(5.10), one has d dt ⟨∇Λσ0−1ρ,Λσ0−1u⟩+ κ∥∇Λσ0−1ρ∥2 ≲ η ( ∥∇Λσ0−1u∥2 + ∥Λσ0−1∇ρ∥2 + ∥ div Λσ0−1u∥2 + ∥Λσ0u∥2 + ∥Λσ0−1Λ2αu∥2 ) + ϵ1∥∇Λσ0−1ρ∥2 + Cϵ1∥Λσ0−1Λ2αu∥2 + κ∥div Λσ0−1u∥2. (5.11) EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 27 Summing (5.6) and β3 × (5.11) with sufficiently small β3 > 0, we obtain 1 2 d dt (∥Λσ0ρ∥2 + ∥Λσ0u∥2 + β3⟨∇Λσ0−1ρ,Λσ0−1u⟩) + ⟨ µ′ (κ+ 1 aρ) a Λσ0Λαu,Λσ0Λαu⟩ − 2ϵ1∥Λσ0Λαu∥2 + β3(κ− ϵ1)∥∇Λσ0−1ρ∥2 ≲ ( η + η2 )( ∥Λσ0ρ∥2 + ∥Λσ0u∥2 + ∥Λ 3+4α 2 ρ∥2 ) + β3η ( ∥∇Λσ0−1u∥2 + ∥Λσ0−1∇ρ∥2 + ∥ div Λσ0−1u∥2 + ∥Λσ0u∥2 + ∥Λσ0−1Λ2αu∥2 ) + β3Cϵ1∥Λσ0−1Λ2αu∥2 + β3κ∥div Λσ0−1u∥2 ≲ ( η + η2 )( ∥Λσ0ρ∥2 + ∥Λσ0u∥2 + ∥Λ 3+4α 2 u∥2 ) + β3(η + Cϵ1)∥Λσ0−1Λ2αu∥2 + β3κ∥div Λσ0−1u∥2. (5.12) This completes the proof. □ 5.2. Cancellation of a low-frequency part. Lemma 5.2. It holds that ∥(Λσ0ρ,Λσ0u)∥2 ≲ e−C2t∥(Λσ0ρ0,Λ σ0u0)∥2 + ∫ t 0 e−C2(t−τ)∥(Λσ0ρL,Λσ0uL)∥2dτ, (5.13) where the positive constant C2 is independent of η. Proof. Multiplying Λσ0−1(1.4)2 by ∇Λσ0−1ρL and using (1.4)1, we have d dt ⟨∇Λσ0−1ρL,Λσ0−1u⟩ = κ⟨div Λσ0−1uL,div Λσ0−1u⟩+ ⟨Λσ0−1(u · ∇ρ)L,div Λσ0−1u⟩ + 1 a ⟨Λσ0−1(ρdivu)L,div Λσ0−1u⟩ − ⟨∇Λσ0−1ρL,Λσ0−1 ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ⟩ − µ⟨∇Λσ0−1ρL,Λσ0−1Λ2αu⟩ − ⟨∇Λσ0−1ρL,Λσ0−1(u · ∇u)⟩ 1 a ⟨∇Λσ0−1ρL,Λσ0−1(ρ∇ρ)⟩ − κ⟨∇Λσ0−1ρL,∇Λσ0−1ρ⟩. (5.14) Similar to (5.8)-(5.10), we obtain |⟨∇Λσ0−1ρL,Λσ0−1 ( ( µ′ (κ+ 1 aρ) a − µ)Λ2αu ) ⟩| ≲ η ( ∥∇Λσ0−1ρL∥2 + ∥Λσ0u∥2 + ∥Λσ0−1Λ2αu∥2 ) , |⟨Λσ0−1(u · ∇ρ)L,div Λσ0−1u⟩| ≲ η∥∇Λσ0−1u∥2 + ϵ1∥Λσ0−1∇ρ∥2 + η2Cϵ1∥ div Λσ0−1u∥2, |⟨Λσ0−1(ρ divu)L,div Λσ0−1u⟩| ≲ ϵ1∥Λσ0−1∇ρ∥2 + η2Cϵ1∥ div Λσ0−1u∥2 + η∥ div Λσ0−1u∥2, |⟨∇Λσ0−1ρL,Λσ0−1(u · ∇u)⟩| ≲ η ( ∥∇Λσ0−1ρL∥2 + ∥∇Λσ0−1u∥2 ) , 28 M. LIU, L. NIU, Z. WU EJDE-2025/35 |⟨∇Λσ0−1ρL,Λσ0−1(ρ∇ρ)⟩| ≲ Cϵ1η 2∥∇Λσ0−1ρL∥2 + 2ϵ1∥∇Λσ0−1ρ∥2. Combining these inequalities with (5.14), we have − d dt ⟨∇Λσ0−1ρL,Λσ0−1u⟩ ≲ κ 2 ∥div Λσ0−1uL∥2 + ( µ 2 + η)∥Λσ0−1Λ2αu∥2 + ( µ 2 + Cϵ1κ 2 + η + Cϵ1η 2)∥Λσ0−1∇ρL∥2 + η ( ∥∇Λσ0−1u∥2 + ∥Λσ0−1∇ρ∥2 + ∥Λσ0u∥2 ) + ( κ 2 + η + Cϵ1η 2)∥div Λσ0−1u∥2 + 5ϵ1∥∇Λσ0−1ρ∥2. (5.15) Adding (5.12) and β3 × (5.15) with sufficiently small β3 > 0, we obtain 1 2 d dt (∥Λσ0ρ∥2 + ∥Λσ0u∥2 + β3⟨∇Λσ0−1ρH ,Λσ0−1u⟩) + ⟨ µ′ (κ+ 1 aρ) a Λσ0Λαu,Λσ0Λαu⟩ − 2ϵ1∥Λσ0Λαu∥2 + β3(κ− 6ϵ1)∥∇Λσ0−1ρ∥2 ≲ β3 κ 2 ∥div Λσ0−1uL∥2 + β3( µ 2 + Cϵ1κ 2 + η + Cϵ1η 2)∥Λσ0−1∇ρL∥2 + β3( 3κ 2 + η + Cϵ1η 2)∥div Λσ0−1u∥2 + β3( µ 2 + η + Cϵ1)∥Λσ0−1Λ2αu∥2 + ( η + η2 )( ∥Λσ0ρ∥2 + ∥Λσ0u∥2 + ∥Λ 3+4α 2 u∥2 ) . (5.16) Then, there exists a C2 > 0 such that d dt (∥Λσ0ρ∥2 + ∥Λσ0u∥2) + C2(∥Λσ0ρ∥2 + ∥Λσ0u∥2) ≲ ∥Λσ0ρL∥2 + ∥Λσ0uL∥2 + ∥Λ 3+4α 2 uL∥2 + ∥Λσ0−1Λ2αuL∥2 ≲ ∥Λσ0ρL∥2 + ∥Λσ0uL∥2. (5.17) Multiplying (5.17) by eC2t and integrating with respect to t over [0, t], we have ∥Λσ0ρ∥2 + ∥Λσ0u∥2 ≲ e−C2t(∥Λσ0ρ0∥2 + ∥Λσ0u0∥2) + ∫ t 0 e−C2(t−τ)(∥Λσ0ρL∥2 + ∥Λσ0uL∥2)dτ. (5.18) This completes the proof. □ 5.3. Decay rates for the nonlinear system. Based on spectral analysis in [36], we have the following lemma. Lemma 5.3. Suppose that U = (ρ,u) is the solution of the Cauchy problem of the nonlinear problem ρt + κ∇ · u = −u · ∇ρ− 1 a ρ∇ · u := F1, ut + µΛ2αu+ κ∇ρ = −( µ′ (κ+ 1 aρ) a − µ)Λ2αu− u · ∇u − 1 a ρ∇ρ := F2, (5.19) EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 29 with the initial data U0 = (ρ0,u0). Then for each integer j ≥ 0, we have that ∥∂j x(ρ L,uL)(t)∥L2(R3) ≲ (1 + t)− 3 4α− j 2 ∥(ρ0,u0)∥L1(R3) + ∫ t 2 0 (1 + t− τ)− 3 4α− j 2 ∥F (U)∥L1(R3)(τ)dτ + ∫ t t 2 (1 + t− τ)− j 2 ∥F (U)∥L2(R3)(τ)dτ, (5.20) where F (U) = (F1(U), F2(U)). In this subsection, by combining Lemma 5.1 with Lemma 5.2 and Lemma 5.3, we obtain the time-decay rates of the solution to the nonlinear Cauchy problem of (1.4). Lemma 5.4. Under the assumptions of Theorem 1.3, it holds that ∥Λσ(ρ,u)(t)∥L2(R3) ≲ (1 + t)− 3 4α−σ 2 , 0 ≤ σ ≤ σ0. (5.21) Proof. Let M(t) := sup 0≤τ≤t σ0∑ m=0 (1 + τ) 3 4α+m 2 ∥Λm(ρ,u)∥. Notice that M(t) is non-decreasing, so for 0 ≤ m ≤ σ0, ∥Λm(ρ,u)(τ)∥ ≤ C3(1 + τ)− 3 4α−m 2 M(t), 0 ≤ τ ≤ t holds for some positive constant C3 independent of η. By using Hölder’s inequality, we have ∥F (U)(τ)∥L1 ≲ ∥u∥(∥∇ρ∥+ ∥∇u∥) + ∥ρ∥(∥ divu∥+ ∥∇ρ∥) + ∥ρ∥∥Λ2αu∥ ≲ ηM(t)(1 + τ)− 3 4α− 1 2 , (5.22) and ∥F (U)(τ)∥ ≲ ∥u∥L3(∥∇ρ∥L6 + ∥∇u∥L6) + ∥ρ∥L3(∥ divu∥L6 + ∥∇ρ∥L6) + ∥ρ∥ L 6 2σ0−4α ∥Λ2αu∥ L 6 3−2σ0+4α ≲ ∥u∥H1(∥∇∇ρ∥+ ∥∇∇u∥) + ∥ρ∥H1(∥∇ divu∥+ ∥∇∇ρ∥) + ∥Λ 3+4α−2σ0 2 ρ∥∥Λσ0u∥ ≲ η1−ϵ2M(t)1+ϵ2(t)(1 + τ)− 3 4α−1− 3 4α ϵ2 , (5.23) where ϵ2 ∈ (0, 1/2). By [15, Lemma 2.5], Lemma 5.3, (5.22) and (5.23), for 0 ≤ σ ≤ σ0, we have ∥(ΛσρL,ΛσuL)(t)∥ ≲ (1 + t)− 3 4α−σ 2 ∥(ρ0,u0)∥L1(R3) + ηM(t) ∫ t 2 0 (1 + t− τ)− 3 4α−σ 2 (1 + τ)− 3 4α− 1 2 dτ + η1−ϵ2M(t)1+ϵ2(t) ∫ t t 2 (1 + t− τ)− σ 2 (1 + τ)− 3 4α−1− 3 4α ϵ2dτ ≲ ( ∥(ρ0,u0)∥L1(R3) + ηM(t) + η1−ϵ2M(t)1+ϵ2(t) ) (1 + t)− 3 4α−σ 2 . (5.24) 30 M. LIU, L. NIU, Z. WU EJDE-2025/35 It follows from (5.18) that ∥(Λσ0ρ,Λσ0u)(t)∥2 ≲ e−C2t(∥(Λσ0ρ0,Λ σ0u0)∥2) + ∫ t 0 e−C2(t−τ) ( ∥(ρ0,u0)∥2L1(R3) + η2M2(t) + η2−2ϵ2M(t)2+2ϵ2(t) ) (1 + τ)− 3 2α−σ0dτ ≲ ( ∥(ρ0,u0)∥Hσ0∩L1 + η2M2(t) + η2−2ϵ2M(t)2+2ϵ2(t) ) (1 + τ)− 3 2α−σ0 . (5.25) Moreover, the decomposition (6.1) for 0 ≤ σ ≤ σ0 yields ∥(Λσρ,Λσu)(t)∥2 ≲ ∥(Λσρ,Λσu)L(t)∥2 + ∥(Λσρ,Λσu)H(t)∥2 ≲ ∥(Λσρ,Λσu)L(t)∥2 + ∥(Λσ0ρ,Λσ0u)(t)∥2. (5.26) From (5.24), (5.25) and (5.26), for 0 ≤ σ ≤ σ0, we have ∥(Λσρ,Λσu)(t)∥2 ≲ ( ∥(ρ0,u0)∥Hσ0∩L1 + η2M2(t) + η2−2ϵ1M(t)2+2ϵ2(t) ) (1 + τ)− 3 2α−σ. (5.27) By noting the definition of M(t) and using the smallness of η, from (5.27), there exists a positive constant C4 independent of η such that M2(t) ≤ C4 ( ∥(ρ0,u0)∥Hσ0∩L1 + η2M2(t) + η2−2ϵ2M(t)2+2ϵ2(t) ) . (5.28) By using Young’s inequality, we obtain C4η 2−2ϵ2M(t)2+2ϵ2(t) ≤ 1− ϵ2 2 C 2 1−ϵ2 4 + 1 + ϵ2 2 η 4(1−ϵ2) 1+ϵ2 M4(t). (5.29) For simplicity, we denote K0 := C4∥(ρ0,u0)∥Hσ0∩L1 + 1− ϵ2 2 C 2 1−ϵ2 4 , (5.30) Cη := 1 + ϵ2 2 η 4(1−ϵ2) 1+ϵ2 . (5.31) From (5.28) and the smallness of η, we have M2(t) ≤ K0 + C4M 4(t). (5.32) Notice that M(t) is non-decreasing and continuous, we have M(t) ≤ C for any t ∈ [0,+∞). This implies ∥Λσ(ρ,u)(t)∥L2(R3) ≲ (1 + t)− 3 4α−σ 2 , 0 ≤ σ ≤ σ0, and completes the proof. □ 6. Appendix To obtain the optimal decay rates of the solution, we need the following results. First, we need the frequency decomposition of the solution f l(x) = χ0(∂x)f(x), fh(x) = χ1(∂x)f(x), fm(x) = (1− χ0(∂x)− χ1(∂x))f(x). Here χ0(∂x) = F−1(χ0(ξ)) and χ1(∂x) = F−1(χ1(ξ)) satisfy 0 ≤ χ0(ξ), χ1(ξ) ≤ 1 and χ0(ξ) = { 1, |ξ| < r0/2, 0, |ξ| > r0, χ1(ξ) = { 0, |ξ| < R0, 1, |ξ| > R0 + 1, EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 31 for some fixed r0 and R0. Therefore, it is easy to see that f(x) = f l(x) + fm(x) + fh(x) =: fL(x) + fh(x) =: f l(x) + fH(x), (6.1) where fL(x) = f l(x) + fm(x) and fH(x) = fm(x) + fh(x). Lemma 6.1 ([38]). For f(x) ∈ Hs(R3) and any given integers k, k0, k1 with k0 ≤ k ≤ k1 ≤ s, it holds that ∥∂k xf l∥L2(R3) ≤ rk−k0 0 ∥∂k0 x f l∥L2(R3), ∥∂k xf l∥L2(R3) ≤ ∥∂k1 x f∥L2(R3), ∥∂k xf h∥L2(R3) ≤ 1 Rk1−k 0 ∥∂k1 x fh∥L2(R3), ∥∂k xf h∥L2(R3) ≤ ∥∂k1 x f∥L2(R3), rk0∥fm∥L2(R3) ≤ ∥∂k xf m∥L2(R3) ≤ Rk 0∥fm∥L2(R3). Acknowledgement. The research was supported by the National Natural Science Foundation of China (11971100) and by the Natural Science Foundation of Shanghai (22ZR1402300). References [1] S. Abe, S. Thurner; Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion, Physica A 356 (2005), 403-407. [2] H. Bahouri, J. Y. Chemin, R. Danchin; Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Vol. 343. Berlin, Heidelberg: Springer-Verlag, 2011. [3] L. A. Caffarelli, A. Vasseur; Drift diffusion equations with fractional diffusion and the quasi- geostrophic equation, Ann. Math. 171 (2010), 1903-1930. [4] F. Charve, R. Danchin; A global existence result for the compressible Navier-Stokes equations in the critical Lp framework, Arch. Ration. Mech. Anal. 198 (2010), 233-271. [5] J. Y. Chemin; Fluides parfaits incompressibles, Ast’erisque, 230, 1995. [6] L. Chen, C. Tan, L. Tong; On the global classical solution to compressible Euler system with singular velocity alignment, Methods Appl. Anal. 28 (2021), 153-172. [7] Q. Chen, C. Miao, Z. Zhang; Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure. Appl. Math. 63 (2010), 1173-1224. [8] Q. Chen, C. Miao, Z. Zhang; On the ill-posedness of the compressible Navier-Stokes equations in the critical Besov spaces, Rev. Mat. Iberoam. 31 (2015), 1375-1402. [9] N. Chikami, R. Danchin; On the well-posedness of the full compressible Navier-Stokes system in critical Besov spaces, J. Differential Equations 258 (2015), 3435-3467. [10] R. Danchin; Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math. 141 (2000), 579-614. [11] R. Danchin; Global existence in critical spaces for flows of compressible viscous and heat- conductive gases, Arch. Rational Mech. Anal. 160 (2001), 1-39. [12] R. Danchin, L. He; The incompressible limit in Lp type critical spaces, Math. Ann. 366 (2016), 1365-1402. [13] R. J. Duan, S. Ukai, T. Yang, H. J. Zhao; Optimal convergence rate for the compressible Navier-Stokes equations with potential force, Math. Models Methods Appl. Sci. 17 (2007), 737-758. [14] E. Feireisl; Compressible Navier-Stokes equations with a non-monotone pressure law, J. Dif- ferential Equations 184 (2002), 97-108. [15] E. Feireisl, A. Novotný, H. Petzeltová; On the global existence of globally defined weak solu- tions to the Navier-Stokes equations of isentropic compressible fluids, J. Math. Fluid Mech. 3 (2001), 358-392. [16] Y. Guo, Y. J. Wang; Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations 37 (2012), 2165-2208. [17] B. Haspot; Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces, J. Differential Equations 251 (2011), 2262-2295. [18] L. He, J. Huang, C. Wang; Global stability of large solutions to the 3D compressible Navier- Stokes equations, Arch. Rational Mech. Anal. 234 (2019), 1167-1222. 32 M. LIU, L. NIU, Z. WU EJDE-2025/35 [19] D. Hoff, K. Zumbrun; Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J. 44 (1995), 603-676. [20] D. Hoff, K. Zumbrun; Pointwise decay estimates for multidimensional Navier-Stokes diffu- sion waves, Z. angew. Math. Phys. 48 (1997), 597-614. [21] X. D. Huang, J. Li, Z. P. Xin; Global well-posedness of classical solutions with large oscilla- tions and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure. Appl. Math. 65 (2012), 549-585. [22] M. Jara; Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math. 62 (2009), 198-214. [23] Q. S. Jiu, C. X. Miao, J. H. Wu, Z. F. Zhang; The two-dimensional incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal. 46 (2014), 3426-3454. [24] A. Kiselev, F. Nazarov, A. Volberg; Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 (2007), 445-453. [25] H. L. Li, T. Zhang; Large time behavior of isentropic compressible Navier-Stokes system in R3, Math. Meth. Appl. Sci. 34 (2011), 670-682. [26] J. Li, Z. Xin; Entropy bounded solutions to the one-dimensional compressible Navier-Stokes equations with zero heat conduction and far field vacuum, Adv. Math. 361 (2020), 106923. [27] J. L. Li, Z. Y. Yin, X. P. Zhai; Global well-posedness for the full compressible Navier-Stokes equations, Acta Math. Sci. 42B (2022), 2131-2148. [28] T. P. Liu, W. K. Wang; The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimension, Comm. Math. Phys. 196 (1998), 145-173. [29] A. Matsumura, T. Nishida; The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ. 20 (1980), 67-104. [30] A. Matsumura, T. Nishida; The initial value problem for the equations of motion of com- pressible viscous and heat-conductive fluids, J. Math. Kyoto Univ. 20 (1980), 17408. [31] A. Mellet, S. Mischler, C. Mouhot; Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal. 199 (2011), 493-525. [32] A. Mellet, A. Vasseur; On the barotropic compressible Navier-Stokes equations, Comm. Par- tial Differential Equations 32 (2007), 431-452. [33] H. Y. Peng, X. P. Zhai; The Cauchy problem for the N-dimensional compressible Navier- Stokes equations without heat conductivity, SIAM J. Math. Anal. 55 (2023), 1439-1463. [34] M. E. Schonbek, T. P. Schonbek; Asymptotic behavior to dissipative quasi-geostrophic flows, SIAM J. Math. Anal. 35 (2003), 357-375. [35] E. M. Stein; Singular Integrals and Differentiability Properties of Functions, Princeton Uni- versity Press, 1970. [36] S. Wang, S. Z. Zhang; The initial value problem for the equations of motion of fractional compressible viscous fluids, J. Differential Equations 377 (2023), 369-417. [37] S. Wang, S. Z. Zhang; The global classical solution to compressible system with fractional viscous term, Nonlinear Anal. Real World Appl. 75 (2024), 103963. [38] W. J. Wang, H. Y. Wen; Global well-posedness and time-decay estimates for compressible Navier-Stokes equations with reaction diffusion, Sci. China Math. 65 (2022), 1199-1228. [39] H. Wen, C. Zhu; Global solutions to the three-dimensional full compressible Navier-Stokes equations with vacuum at infinity in some classes of large data, SIAM J. Math. Anal. 49 (2017), 162-221. [40] Z. Xin; Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math. 51 (1998), 229-240. [41] Z. Xin, J. Xu; Optimal decay for the compressible Navier-Stokes equations without additional smallness assumptions, J. Differential Equations 274 (2021), 543-575. [42] X. P. Zhai, Y. Li, F. Zhou; Global large solutions to the three dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal. 52 (2020), 1806-1843. Mengqian Liu School of Mathematics and Statistics, Donghua University, Shanghai 201620, China Email address: hbulmq@163.com Lei Niu (corresponding author) School of Mathematics and Statistics, Donghua University, Shanghai 201620, China Email address: lei.niu@dhu.edu.cn EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 33 Zhigang Wu School of Mathematics and Statistics, Donghua University, Shanghai 201620, China Email address: zhigangwu@hotmail.com 1. Introduction 2. Preliminaries 3. A priori estimates 4. Proof of Theorem 1.1 5. Optimal decay rates 5.1. Energy estimates on the highest-order derivative 5.2. Cancellation of a low-frequency part 5.3. Decay rates for the nonlinear system 6. Appendix Acknowledgement References