Abstract
In this article, we consider the compressible micropolar viscous flow in a bounded or unbounded domain Ω ⊆ ℝ^{3}. We prove the existence of unique local strong solutions for large initial data satisfying some compatibility conditions. The key point here is that the initial density need not be positive and may vanish in an open set.
1 Introduction
Compressible micropolar viscous fluids study the viscous compressible fluids with randomly oriented (or spherical) particles suspended in the medium, when the deformation of fluid particles is ignored. The theory can help us consider some physical phenomena which cannot be treated by the compressible viscous baratropic flows, due to the effect of microparticles. The microstructure of the polar fluids is mechanically significant. The governing system of equations of compressible micropolar viscous fluids expresses the balance of mass, momentum, and moment of momentum see [1,2], that is,
Here the density ρ = ρ(t, x), the velocity u = [u^{1}(t, x), u^{2}(t, x), u^{3}(t, x)], the microrotational velocity w = [w^{1}(t, x), w^{2}(t, x), w^{3}(t, x)], and the pressure p(ρ) = aρ^{γ}(a > 0, γ > 1) are functions of the time t ∈ (0, T) and the spatial coordinate x ∈ Ω, where Ω is either a bounded open subset in ℝ^{3 }with smooth boundary or a usual unbounded domain such as the whole space ℝ^{3}, the half space and an exterior domain with smooth boundary. μ, λ, μ', λ', and ζ are positive constants, which describe the viscosity of the fluids, satisfying the additional condition: μ + λ  ζ ≥ 0.
We prescribe the initial boundary value conditions:
There are lots of literature on the wellposedness and illposedness of the incompressible micropolar viscous fluid, i.e., the system (1) with ρ ≡ Constant. Yamaguchi [3] considered the global strong solution in a bounded domain with small initial data. Lukaszewicz [4] showed the existence theorem for the incompressible micropolar viscous fluid with sufficiently regular initial data. And with the effect of Magnetic fields, the system of magnetomicropolar fluid, in [5], Yuan gave the local smooth solution without the smallness of the initial data; and also gave the blowup criterion for the smooth solutions. Amirat and Hamdache [6] considered the global weak solutions with finite energy and establish the longtime behavior of the solution. And Amirat et al. [7] also proved the global in time weak solution with finite energy of an initialboundary value problem and longtime behavior of such solution with the effect of magnetic field. We also refer the reader to [8] for local strong solution in bounded domain of ℝ^{3 }and references therein.
For the compressible case, in the absence of vacuum, there also have been lots of studies on the full viscous compressible micropolar fluids (which include also the conservation law of energy) since Eringen [1]. The onedimensional problem studied by Mujaković [912], and Dražić and Mujaković [13] and references therein. For the multidimensional case, we refer the readers to [2,1417] and references therein. If the vacuum occur initially, Chen [18,19] studied the global strong solutions of compressible micropolar viscous fluids in 1D. Recently, Amirat and Hamdache [20] studied the micropolar fluids with the effect of magnetic field, they prove the global weak solution in a bounded domain in ℝ^{3 }with initial vacuum.
Classical compressible viscous flows, i.e., w = 0 and ζ = 0 in (1), have been studied by many authors. In [21], Matsumura and Nishida considered the global smooth solutions under the condition that initial data close to a nonvacuum equilibrium. For the arbitrary initial data, the major breakthrough is due to Lions [22], where he established global existence of weak solutions for the whole space, periodic domain or bounded domains with Dirichlet boundary conditions if γ ≥ 9/5. The restriction on γ was improved to γ > 3/2 by Feireisl etal. in [23].
In [24], Choe and Kim established local in time strong solution of isentropic compressible fluids with initial density ρ_{0 }may vanish in an open subset. After that, Cho and Kim [25] studied the local existence of strong solutions of viscous polytropic fluids with vacuum. Cho et al. [26] considered the unique solvability of the initial boundary value problems for compressible viscous fluids that the initial density need not be bounded away from zero.
Hoff and Serre [27] presented an example which showed the failure of continuous dependence of weak solution, so it should be noticed that contrary to the case of strong solutions, weak solutions with vacuum need not depend continuously on their initial data.
Before stating the main result, we introduce the notions used throughout this article. We denote
For 1 ≤ r ≤ ∞, we denote the standard homogeneous and inhomogeneous Sobolev spaces as follows:
In spirit of [26], our aim of this article is to establish the local strong solution of (1)(4). Assume the initial data (ρ_{0}, w_{0}, u_{0}) satisfying the regularity
where 3 < q < ∞, and the compatibilities
and
for some (g_{1}, g_{2}) ∈ L^{2}.
Now, we state our main result in this article:
Theorem 1 Assume the data (ρ_{0}, u_{0}, w_{0}) satisfy the regularity conditions (5) and the compatibility conditions (6) and (7).
Then there exists a time T_{*}∈ (0, T) and a unique strong solutions (ρ, u, w) to the initial boundary value problem (1)(4) such that
where q_{0 }= min{6, q}.
The article is organized as follow. In Section 2, we prove the global existence and regularity of the unique strong solutions to a linearized problem of the nonlinear problem (1)(4). The result is used in Section 3 to construct approximate solutions to the original nonlinear problem. In Section 3, we derive some uniform bounds of the higher derivatives independent of the lower bound of the density. Moreover, we prove the convergence and obtain the local existence of strong solutions. In Section 4, we finish the proof of Theorem 1.
Remark 1 In this article, we use the following fact frequently later:
under the boundary condition of (3) or (4).
2 Global existence for the linearized equations
In this section, we consider the following linearized system:
where v is a known vector fields. If the initial density ρ_{0 }is bounded away from zero, then we can apply standard arguments to prove the global existence of a unique strong solution to the initial boundary value problem (2)(4) and (8), since the system can be uncoupled into a linear transport equation and two linear parabolic equations.
In this section, we prove the following existence result for the general case of nonnegative initial densities.
Theorem 2 Assume that the data (ρ_{0}, u_{0}, w_{0}) satisfies the regularity conditions:
for some 3 < q < ∞ and the compatibility conditions:
and
for some (g_{1}, g_{2}) ∈ L^{2}. If in addition, v satisfies the regularity conditions:
where q_{0 }= min{6, q}. Then there exists a unique strong solution (ρ, u, w) to the initial boundary value problems (2)(4) and (8) such that
Here, we emphasize that we focus on the bounded open domain Ω with smooth boundary condition. As for the unbounded domain, we can deal the same problem with the standard domain expansion technique that derived in [22]. We also refer the reader to [26]. The key of this technique is that the a priori estimates do not depend on the size of the domain. So, we here emphasize that the a priori estimates deduced in this section are independent of the size of the domain.
2.1 Existence of theorem 2
We begin with an existence result for the case of positive initial densities.
Lemma 1 Let Ω be a bounded domain in ℝ^{3 }with smooth boundary, and let (ρ_{0}, u_{0}, w_{0}) be a given data satisfying the regularity condition (9)(11). Assume further that , , ρ_{0 }∈ H^{2 }and ρ_{0 }≥ δ in Ω for some constant δ > 0. Then there exists a unique strong solution (ρ, u, w) to the initial boundary value problems (2), (3) and (8) such that
Proof Due to the classical embedding result, that υ ∈ C([0, T];H^{2}). Hence the existence and regularity of a unique solution of the linearized continuity Equation (8)_{1 }have been wellknown. Moreover, the unique solution ρ can be expressed by:
where U(t, s, x) is the solution to
We refer the readers to [28,29] for a detailed proof. As a consequence of (14) and Sobolev inequality, we have
for . Hence the linearized moment of momentum Equation (8)_{2 }can be written as a linear parabolic system
The existence and regularity of the unique solution w can be proved by applying classical methods, for instance, the method of continuity (see [29]). Similarly, the linearized momentum Equation (8)_{3 }also as a linear parabolic system
and can be solved using the same method.
Now we prove the existence of strong solutions. Then thanks to Lemma 1, there exists a unique strong solution (ρ, u, w) satisfying the regularity (13). To remove the additional hypotheses in Lemma 1, we will derive some uniform estimates independent of δ, , and .
First, we consider the solution ρ of the linearized continuity Equation (8)_{1}. Since (8)_{1 }is a linear transport equation, so we need to prove the estimates. Multiply (8)_{1 }by ρ^{r}^{1}(r = 2 or q_{0}) and integrating (by parts) over Ω, we obtain:
Then, using Sobolev inequality, we get
Differentiating (8)_{1 }with respect to x_{i}, then multiplying the resultant equation by ∂_{i}ρ∂_{i}ρ^{r}^{2}, i = 1, 2, 3 and then integrating over Ω, we have
By virtue of Sobolev inequality, we get:
Using (16) and (17) together with Gronwall's inequality, one yields:
Since ρ_{t }= υ ·∇ρ  ρ div υ, p = p(ρ) and p(0) = 0, we can easily get
Then, we consider the solution w of the linearized Equation (8)_{2}. Rewrite (8)_{2}, with the help of (8)_{1}, as:
Multiplying this equation by w_{t }and integrating (by parts) over Ω, we have
Then, using Young's inequality, we obtain:
Integrating the above inequality over (0, t), we get:
Thus, we have
Therefore, in view of Gronwall's inequality, we have:
To derive higher regularity estimates, we differentiate (19) with respect to t and obtain:
Multiplying the above equation by w_{t}, and integrating over Ω, we have:
Now, we estimate each term on the righthand side of (24)
Choosing ε= μ'/6, substitute the above estimates into (24), we can get
Integrating over (τ, t) for some fixed τ ∈ (0, T), we deduce that
To estimate , due to(19), we see
and thus we have
Therefore, letting τ → 0 in (26), we conclude that
To obtain further estimates, observe that since for each t ∈ [0, T], is a solution of the elliptic system:
Therefore, using the previous estimates, we can deduce from (28) that
Now, we consider the solution u of the linearized Equation (8)_{3}. Rewrite (8)_{3 }as
Multiplying this equation by u_{t }and integrating over Ω, we have
Then, using Young's inequality, we obtain:
Integrating the above inequality over (0, t) using Young's inequality, we get
Therefore, in view of Gronwall's inequality, we have
To derive higher regularity estimates, differentiate (30) with respect to t and obtain:
Multiplying the above equation by u_{t}, integrating (by parts) over Ω, and using (8)_{1}, we obtain:
Using Sobolev inequality, interpolation inequality and Young's inequality, we obtain:
Substituting the above estimates into (32), we can easily show that:
Now, for fixed τ ∈ (0, T). Since the righthand side of (33) is integrable in (0, T), we deduce that
To estimate from (8)_{3}, we see that
and thus
Therefore, letting τ → 0 in (34), we conclude that
To obtain further estimates, observe that for each t ∈ [0, T], is a solution of the following elliptic system:
where . It follows from the elliptic regularity theory, we get
Therefore, using the previous estimates, we get from (36) that
Since the estimates we have derived, we prove the existence result. First, we consider for the case of bounded domain. Using standard regularization techniques, we choose p^{δ }= p^{δ}(⋅) and υ^{δ}, 0 < δ ≪ 1, so that
Then, for each δ ∈ (0, 1), let and let is the solution to the boundary value problem:
It follows from the elliptic regularity result that in as δ → 0. Hence, if we denote by (ρ^{δ}, u^{δ}, w^{δ}) the solution of (8) with the initial data and (p, v) replaced by (p^{δ}, v^{δ}), it satisfies the estimates (35), (37), (31), (23), (18), (27), (29), where , . Therefore, we conclude that a subsequence of solutions (ρ^{δ}, u^{δ}, w^{δ}) converges to a limit (ρ, u, w) in a weak sense. Then it can easy to show that (ρ, u, w) is a weak solution to the original problem (8). Moreover, due to the lower semicontinuity of various norms, we have the following regularity estimates for (ρ, u, w):
2.2 Continuity and uniqueness
Now, turn our attention to the continuity of the solution (ρ, u, w). The continuity of ρ can be proved by a standard argument from the theory of hyperbolic equations. Since ρ satisfies the regularity (38), it follows from a result of DiPerna and Lions [30] and classical embedding results that (See [31]):
To show the strong continuity in , observe from (17), (18) and i = 1, 2, 3,
and thus . Hence it follows from a wellknown criterion on the strong convergence for the space L^{r }(See [32]) that
Therefore, the continuity of ∇ρ in L^{r}(r = 2, q_{0}) follows from the result and the observation that for each fixed t_{0 }∈ [0, t], the function is a unique strong solution to the similar initial value problem:
To show the continuity of w, we first observe that
We now prove the continuity of ρw_{t }in L^{2}. For a.e. t ∈ (0, T) and the from (19), we have
and thus
Using the regularity (38) of (ρ, w), we show that the righthand side of (39) is bounded above by for some positive function A_{1}(t) ∈ L^{2}(0, T). Hence it follows, from the wellknown result (see [31]) that (ρw_{t})_{t}∈ L^{2}(0, T; H^{1}) and for all where 〈⋅, ⋅〉 denotes the dual pairing of H^{1 }and . Then since , it follows from a standard embedding result ρw_{t }∈ C([0, T]; L^{2}). Therefore, we conclude that for each t∈[0, T], is a solution of the elliptic system:
where G_{1 }= ρw_{t}4ζw ∈ C([0, T]; L^{2}). Now we turn to show that w ∈ C([0, T]; D^{2}). In view of the elliptic regularity estimate (36), we obtain
Using the estimate (36), we obtain:
Substituting this into (40), we conclude that
The continuity of ρu_{t }in L^{2 }is similar as the proof of ρw_{t}. From (30), for a.e. t ∈ (0, T), and all , we have
and thus
Using the regularity (38) of (ρ, u, w), we show the righthand side of (42) is bounded above by for some positive function A_{2}(t) ∈ L^{2}(0, T). Hence by the similar argument of ρw_{t}, we see that (ρu_{t})_{t}∈ L^{2}(0, T; H^{1}), and then ρu_{t }∈ C([0, T]; L^{2}). Therefore, we conclude that for each is a solution of the elliptic system:
where G_{2 }= ρu_{t } ∇p(ρ) + 2ζ rot w ∈ C([0, T]; L^{2}).
Now, we will show that u ∈ C([0, T]; D^{2}). In view of the elliptic regularity estimate (36), we have
To estimate the third term in the righthand side of (43), using the estimate (38), similarly as (41), we get:
Substituting this into (43), we conclude the continuity of u in D^{2}. This completes the proof of the continuity.
Finally, we prove the uniqueness of solutions satisfying the regularity (38). Let (ρ_{1}, u_{1}, w_{1}) and (ρ_{2}, u_{2}, w_{2}) be two strong solutions to the problem (8) and (2)(4). Denote
Then, it follows from (8)_{1 }that
Since and , we conclude from Gronwall's inequality that , i.e., ρ_{1 }= ρ_{2 }in (0, T) × Ω. Next, we choose a cutoff function such that
Define φ_{R}(x) = φ(x/R) for x ∈ ℝ^{3}. From (8)_{2 }and using the uniqueness of ρ, we deduce that
Then multiplying the above equation by , integrating over (0, T) × Ω, and letting R → ∞, we easily get
Hence, we deduce that , and in (0, T), due to . Therefore, we conclude that in (0, T) × Ω.
Similarly, from the uniqueness of ρ, and (8)_{3}, we get
Then multiplying it by integrating over (0, T) × Ω, and letting R → ∞, we obtain
Due to ū(0) = 0, we get in (0, T). Then ū = 0 in (0, T) × Ω.
This completes the proof of the Theorem 2.
3 A local existence result for positive densities
In this section, we assume also that Ω is a bounded domain in ℝ^{3 }with smooth boundary and prove a local (in time) existence result on strong solutions with positive densities to the original nonlinear problem (1)(4).
Proposition 1 Assume that p = aρ^{γ }(a > 0, γ > 1), and the data (ρ_{0}, u_{0}, w_{0}) satisfies the regularity conditions:
for some q with 3 < q < ∞ and compatibilities (6)(7). Assume further that ρ_{0 }≥ δ in Ω for some constant δ > 0. Then there exist a time T_{* }∈ (0, T) and a unique strong solution (ρ, u, w) to the nonlinear problem (1)(3) such that
where q0 = min(6, q). Furthermore, we have the following estimates:
The constant C and the local time T_{* }in (44) are independent of δ.
To prove the proposition, we first construct approximate solutions, inductively, as follows:
 first define u^{0 }= 0, and
 assume that u^{k}^{1 }was defined for k ≥ 1, let (ρ^{k}, u^{k}, w^{k}) be the unique solution to the following initial boundary value problem:
The existence of a global strong solution (ρ^{k}, u^{k}, w^{k}) with the regularity (12) to the linearized problem (45)(49) was proved in the previous section.
From now on, we derive uniform bounds on the approximate solutions and then prove the convergence of the approximate solutions to a strong solutions of the original nonlinear problem.
3.1 Uniform bounds
Let K ≥ 1 be a fixed large integer, and let us introduce an auxiliary function Φ_{K}(t), defined by:
Then we estimate each term of Φ_{K }in terms of some integrals of Φ_{K}, apply arguments of Gronwalltype and thus prove that Φ_{K }is locally bounded. First, notice that
Observe that for any t ∈ [0, T], is a solution of the elliptic system:
Hence we deduce from the elliptic regularity result that
Thus we conclude
Similar as the estimate of . We see that for any t ∈ [0, T], is a solution of the elliptic system:
Hence, we have
Thus we conclude
Multiplying (46) by , and integrating over Ω, using Young's inequality we obtain:
Integrating the above inequality we get
To estimate the righthand side of (52), we first observe that , together with the estimate we conclude that:
Differentiating (46) with respect to t, we obtain
Multiplying this by and integrating over Ω, we obtain:
Using Sobolev inequality and Young's inequality (with ε) repeatedly, we have:
Substituting all these estimates into (54) and choosing ε, η > 0 small enough, we obtain that:
Integrating over (τ, t) for fixed τ > 0, we have
To estimate as τ → 0, we multiplying (46) by , integrate over Ω, we have:
Using Young's inequality, together with the compatibility condition (7), we have
Substituting the above estimate into (56), we conclude that
Multiplying (47) by and integrating over Ω, we obtain:
Using (45), we have
Integrating (58) over (0, t), using (59) and Young's inequality, we have
To estimate the righthand side of (60), with the help of Sobolev inequality and Young's inequality (with ε), we have:
Substitute all the above estimates into (60), we obtain:
Differentiating (47) with respect to t, we obtain:
Multiplying the equation by , and integrating over Ω, we get
Using Sobolev inequality and Young's inequality we have
Substituting the above estimates into (63), and choosing ε small enough, we have
For a fixed τ ∈ (0, T), integrating the above inequality over (τ, t), we have:
To estimate as τ → 0, we multiply (47) by , and integrate over Ω. Then we have
Hence, with the help of compatibility condition (6), we get:
Considering this and letting τ → 0 in (64), we finally have:
Thanks to the estimate (65), (61), (57), (53), we get:
If η is sufficient small such that , then from the recursive relation of , it follows from that
Thus we have
Finally, we recall from (18) that
for all k, 1 ≤ k ≤ K. To estimate for 1 ≤ k ≤ K, we invoke the elliptic regularity result (36) and the estimate (51). If 3 < q_{0 }< 6, then we have
for some θ_{1}, θ_{2 }∈ (0, 1). Thus, we conclude that
Similarly, from (28) and the estimate (50), we can deduce that,
for some θ_{3}, θ_{4 }∈ (0, 1). Thus, we deduce that
Thanks to the estimates (68), (69), (67), and (66), we can easily show that there exists a small time T_{1 }∈ (0, T) depending only on the parameters C such that the following uniform estimate:
for all k ≥ 1.
3.2 Convergence
We show that the approximate solutions converge to a solution to the original problem (1)(4) in a strong sense. To prove this, we define:
Then it follows from (45)(47), we have
Multiplying (72) by , integrating over Ω, using (45) and Young's inequality we have
And thus we have
Multiplying (73) by , integrating over Ω, using (45) and Young's inequality, we obtain
And thus
On the other hand, observing that (71), we can easily prove that . Hence, multiplying (71) by sgn and integrating over Ω, we have
Multiplying the above inequality by on both side, we have
Similarly, we get:
Combining (74)(77), we obtain:
for some function with for all 0 ≤ t ≤ T_{1 }and k ≥ 1.
Let us define
and
Then integrating (78) over (0, t) ⊂ (0, T_{1}), we have
which implies, by virtue of Gronwall's inequality, that
Hence, choosing ε > 0 and then T_{* }> 0 so small that 4(T_{* }+ ε)C_{1 }< 1, T_{* }< T_{1 }and exp(C_{2}T_{*}) < 2, we also deduce from Gronwall's inequality that for all K ≥ 1,
Therefore, we conclude that (ρ^{k}, u^{k}, w^{k}) converges to a limit (ρ, u, w) in the following strong sense:
Now it is simple to check that (ρ, u, w) is a weak solution to the original problem (1)(4). Then, by virtue of the lower semicontinuity of norms, we deduce from the uniform bound (70) that (ρ, u, w) satisfies the following regularity estimate:
The timecontinuity of the solution (ρ, u, w) can be proved by the same argument as in Section 2. This completes the proof of Proposition 3.1.
Here we should emphasize that the constant C and the local existence time T_{* }in (79) do not depend on δ and the size of Ω.
4 Proof of theorem 1.1
In this section, we complete the proof of Theorem 1.1.
Assume for the moment that Ω is a bounded domain with smooth boundary. Let (ρ_{0}, u_{0}, w_{0}) be the given data satisfying (5). For each small δ > 0, let and let be the unique smooth solution to the elliptic problem:
Then by virtue of Proposition 3.1, there exist a time T_{* }∈ (0, T) and a unique strong solution (ρ^{δ}, u^{δ}, w^{δ}) in [0, T_{*}] × Ω to the problem (1)(4) with the initial data replaced by . Notice that in H^{2 }as δ → 0, (ρ^{δ}, u^{δ}, w^{δ}) satisfies the bound (44), and the constant T_{*}, C are independent of δ. Hence, following the same argument as in the proof of Theorem 2.1, we prove the existence and regularity of a strong solution to the original problem (1)  (4). Moreover, since the constant C and the local existence T_{* }in (44) are independent of the size of the domain, we also obtain the same existence and regularity results for unbounded domains by means of the domain expansion technique. Finally, the uniqueness can be proved by using the similar methods to the proof of the convergence in Section 3.
This completes the proof of Theorem 1.1.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The author was indebted to the referee for giving valuable suggestions which improve the presentation of the article.
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