Decay estimates of linearized micropolar fluid flows in R³ space with applications to L₃-strong solutions

Abstract

Through analytical argument, the L_p − L_q estimate of a three-dimensional linearized micropolar fluid flow in the whole space R³ is established. This estimate is used to show the existence and uniqueness of small L₃-strong solutions of the micropolar fluid motion system. Sharp time decay estimates of the L₃-strong solutions are derived.

Introduction

The micropolar fluid motion model developed by Eringen [5] accounts for micro-rotational effects and micro-rotational inertia in a fluid motion system. For such fluid particles contained in a small volume element, the fluid motion is described by an unknown velocity field v = (v₁, v₂, v₃), an unknown scalar pressure field π, and an unknown gyration field w = (w₁, w₂, w₃), which describes rotation about the centroid of the volume element in an average sense. As is explained in the seminal investigation of Eringen [5], in a physical sense, micropolar fluids represent fluids consisting of bar-like elements. For example, certain anisotropic fluids such as liquid crystals which are made up of dumbbell molecules.

This fluid motion model differs from the Navier–Stokes equations describing a purely viscous fluid flow because it incorporates a coupling stress tensor connecting the unknowns v, π and w. Eringen [5, Eqs. (4.15), (6.9) and (6.10)] developed a set of coupled equations describing the incompressible micropolar fluid motion in the whole space R³ with spatial variable x = (x₁, x₂, x₃) in the form

$$\left.\begin{array}{c}\nabla ·\mathbf{v}=0\text{,}\hfill \\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathbf{v}-\left(\nu +\frac{\kappa }{2}\right)\mathrm{\Delta }\mathbf{v}-\kappa \nabla \times \mathbf{w}+\nabla \pi +\mathbf{v}·\nabla \mathbf{v}=0\text{,}\hfill \\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathbf{w}-\gamma \mathrm{\Delta }\mathbf{w}-(\alpha +\beta )\nabla \nabla ·\mathbf{w}+2\kappa \mathbf{w}-\kappa \nabla \times \mathbf{v}+\mathbf{v}·\nabla \mathbf{w}=0\text{,}\hfill \end{array}\right\}$$

where the differential operators

$$\nabla =({\mathrm{\partial }}_1\text{,}{\mathrm{\partial }}_2\text{,}{\mathrm{\partial }}_3)\text{,}\mathrm{\Delta }={\mathrm{\partial }}_1^2+{\mathrm{\partial }}_2^2+{\mathrm{\partial }}_3^2\text{where}{\mathrm{\partial }}_j=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\text{for}j=1\text{,}2\text{,}3\text{.}$$

The Newtonian kinetic viscosity ν, gyration viscosity κ and other constant viscosity coefficients α, β and γ satisfy the conditions

$$\nu >0\text{,}\kappa ⩾0\text{,}\gamma >0\text{,}\gamma +\alpha +\beta >0\text{.}$$

When micro-rotation effects are neglected, i.e., κ = 0 and w = 0, Eq. (1) reduces to the three-dimensional incompressible Navier–Stokes equations.

There exists an extensive literature associated with the mathematical theory of (1). For example, to name but a few, we draw the reader’s attention to the existence and uniqueness of solutions of (1) examined by Galdi and Rionero [6], Sava [21] and Łukaszewicz [14]. These are further supported by studies of Bayada et al. [1], Chiriţa [3], Durán et al. [4], Gawinecki [7], and Reséndiz and Rojas-Medar [19]. Łukaszewicz [16] and Sadowski [20] investigated the existence of attractors and Kamal and Siddiqui [12] describe numerical computations. One may also consult with Łukaszewicz [15] for detailed analysis and application of the micropolar flow theory. These studies assume the partial differential equation form (1) directly and such analyses give rise to a generalized energy inequality and, additionally, the Poincaré inequality if the fluid domain is bounded. However, the Poincaré inequality is not valid for the unbounded fluid domain R³, and the time decay properties of micropolar fluid flows and strong solutions, when the initial vector field is only in the Lebesgue space L₃(R³)⁶, are not fully covered by these approaches. In contrast, the present study deals with the integral form of (1) by using the L_p − L_q estimate of linearized micropolar flows rather than the generalized energy inequality and the Poincaré inequality.

To explain more specifically, we apply the divergence operator ∇· to the second equation of (1) to obtain the nonlinear term

$$\mathrm{\Delta }\pi =-\nabla ·(\mathbf{v}·\nabla \mathbf{v})$$

and this permits representation of the pressure force in the form

$$\nabla \pi =-\nabla {\mathrm{\Delta }}^{-1}\nabla ·(\mathbf{v}·\nabla \mathbf{v})\text{.}$$

For simplicity of notation, we let δ = α + β,

$$\mathbf{u}=\left(\begin{array}{c}\hfill v_1\hfill \\ \hfill v_2\hfill \\ \hfill v_3\hfill \\ \hfill w_1\hfill \\ \hfill w_2\hfill \\ \hfill w_3\hfill \end{array}\right)\text{,}{\mathbf{u}}_0=\left(\begin{array}{c}\hfill {\mathbf{v}}_0\hfill \\ \hfill {\mathbf{w}}_0\hfill \end{array}\right)=\left(\begin{array}{c}\hfill v_{10}\hfill \\ \hfill v_{20}\hfill \\ \hfill v_{30}\hfill \\ \hfill w_{10}\hfill \\ \hfill w_{20}\hfill \\ \hfill w_{30}\hfill \end{array}\right)\text{,}D=\left(\begin{array}{c}\hfill {\mathrm{\partial }}_1\hfill \\ \hfill {\mathrm{\partial }}_2\hfill \\ \hfill {\mathrm{\partial }}_3\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right)$$

and

$$P\mathbf{u}=\mathbf{u}-D{\mathrm{\Delta }}^{-1}D·\mathbf{u}\text{.}$$

This definition implies (see [25]) that P is a bounded projector mapping L_r(R³)⁶ onto the space

$$L_{\sigma \text{,}r}(R^3{)}^6=\left\{\mathbf{u}\in L_r(R^3{)}^6\text{;}D·\mathbf{u}=0\text{in the sense of distributions}\right\}\text{,}$$

for 1 < r < ∞. This allows (1) to be expressed as

$$\frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathbf{u}+A\mathbf{u}+P(\mathbf{v}·\nabla \mathbf{u})=0\text{,}$$

together with the initial condition

$$\mathbf{u}(\mathbf{x}\text{,}0)={\mathbf{u}}_0(\mathbf{x})\text{,}$$

where A is the linear differential operator represented by

$$\left(\begin{array}{cccccc}\hfill -\left(\nu +\frac{\kappa }{2}\right)\mathrm{\Delta }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\kappa {\mathrm{\partial }}_3\hfill & \hfill \kappa {\mathrm{\partial }}_2\hfill \\ \hfill 0\hfill & \hfill -\left(\nu +\frac{\kappa }{2}\right)\mathrm{\Delta }\hfill & \hfill 0\hfill & \hfill \kappa {\mathrm{\partial }}_3\hfill & \hfill 0\hfill & \hfill -\kappa {\mathrm{\partial }}_1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -\left(\nu +\frac{\kappa }{2}\right)\mathrm{\Delta }\hfill & \hfill -\kappa {\mathrm{\partial }}_2\hfill & \hfill \kappa {\mathrm{\partial }}_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\kappa {\mathrm{\partial }}_3\hfill & \hfill \kappa {\mathrm{\partial }}_2\hfill & \hfill 2\kappa -\gamma \mathrm{\Delta }-\delta {\mathrm{\partial }}_1^2\hfill & \hfill -\delta {\mathrm{\partial }}_1{\mathrm{\partial }}_2\hfill & \hfill -\delta {\mathrm{\partial }}_1{\mathrm{\partial }}_3\hfill \\ \hfill \kappa {\mathrm{\partial }}_3\hfill & \hfill 0\hfill & \hfill -\kappa {\mathrm{\partial }}_1\hfill & \hfill -\delta {\mathrm{\partial }}_2{\mathrm{\partial }}_1\hfill & \hfill 2\kappa -\gamma \mathrm{\Delta }-\delta {\mathrm{\partial }}_2^2\hfill & \hfill -\delta {\mathrm{\partial }}_2{\mathrm{\partial }}_3\hfill \\ \hfill -\kappa {\mathrm{\partial }}_2\hfill & \hfill \kappa {\mathrm{\partial }}_1\hfill & \hfill 0\hfill & \hfill -\delta {\mathrm{\partial }}_3{\mathrm{\partial }}_1\hfill & \hfill -\delta {\mathrm{\partial }}_3{\mathrm{\partial }}_2\hfill & \hfill 2\kappa -\gamma \mathrm{\Delta }-\delta {\mathrm{\partial }}_3^2\hfill \end{array}\right)\text{.}$$

This differential operator A is self-conjugate and the linearized system of (3), (4) becomes

$$\frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathbf{u}+A\mathbf{u}=0\text{,}\mathbf{u}(0)={\mathbf{u}}_0\text{.}$$

We assume u = e^−tAu₀ is a solution of this equation and $(\frac{\mathrm{\partial }}{\mathrm{\partial }t}+A{)}^{-1}f$ the solution of the non-homogeneous equation

$$\frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathbf{u}+A\mathbf{u}=f\text{,}\mathbf{u}(0)=0\text{.}$$

The integral equation form of (3), (4) is expressed as

$$\mathbf{u}={\mathrm{e}}^{-\mathit{tA}}{\mathbf{u}}_0-{\left(\frac{\mathrm{\partial }}{\mathrm{\partial }t}+A\right)}^{-1}P(\mathbf{v}·\nabla \mathbf{u})$$

or

$$\mathbf{u}={\mathrm{e}}^{-\mathit{tA}}{\mathbf{u}}_0-{\int }_0^t{\mathrm{e}}^{-(t-s)A}P(\mathbf{v}·\nabla \mathbf{u})(s)\mathrm{d}s\text{.}$$

To examine the integral Eq. (6), it is essential to establish fundamental estimates of the semigroup e^−tA. The objective of this paper is twofold. Namely, to provide the knowledge base for an analytic study of the integral Eq. (6) by determining the L_p − L_q estimate for the semigroup e^−tA and secondary, through application of the estimate to derive the existence of L³-strong solutions with sharp time decay rates.

The main results of the present study are summarized as follows:

Theorem 1.1

The L_p − L_q estimate

$$\Vert {\nabla }^k{\text{e}}^{-\mathit{tA}}{\mathbf{u}}_0{\Vert }_{L_q}+\Vert {\nabla }^k{\text{e}}^{-\mathit{tA}}P{\mathbf{u}}_0{\Vert }_{L_q}⩽{\mathit{ct}}^{-\frac{3}{2}\left(\frac{1}{p}-\frac{1}{q}\right)-\frac{k}{2}}\Vert {\mathbf{u}}_0{\Vert }_{L_p}\text{,}$$

holds true for k = 0,1, 2, … , u₀ ∈ L_p(R³)⁶ and

$$\text{<mi mathvariant="italic" is="true">either</mi>}1⩽p<q⩽\infty \text{<mi mathvariant="italic" is="true">or</mi>}1<p⩽q<\infty \text{,}$$

where c denotes a generic constant independent of time t and the initial vector field u₀.

Here we use the norm definition

$$\Vert f{\Vert }_{L_p}={\left({\int }_{R^3}|f{|}^p\mathrm{d}{x}_1\mathrm{d}{x}_2\mathrm{d}{x}_3\right)}^{\frac{1}{p}}\text{.}$$

As an application of this L_p − L_q estimate, the following result defines decay estimates of strong solutions of (1), (4) with an initial vector field in the Lebesgue space L₃(R³)⁶.

Theorem 1.2

Let us assume u₀ ∈ L_σ,3(R³)⁶. Then there exists a small constant ϵ > 0 such that, for $\Vert {\mathbf{u}}_0{\Vert }_{L_3}⩽ϵ$, Eq. (6) admits a unique solution u satisfying the strong continuity condition

$$\Vert \mathbf{u}(t)-{\mathbf{u}}_0{\Vert }_{L_3}\to 0\text{,}\text{<mi mathvariant="italic" is="true">as</mi>}t\to 0\text{,}$$

the bound

$$\Vert \mathbf{u}(t){\Vert }_{L_3}+t^{\frac{1}{2}}\Vert \mathbf{u}(t){\Vert }_{L_{\infty }}+t^{\frac{1}{2}}\Vert \nabla \mathbf{u}(t){\Vert }_{L_3}+t^{1-\frac{1}{2q}}\Vert \nabla \mathbf{u}(t){\Vert }_{L_q}⩽c\Vert {\mathbf{u}}_0{\Vert }_{L_3}\text{,}$$

the time decay estimate

$$\underset{t\to \infty }{\lim }\left(\Vert \mathbf{u}(t){\Vert }_{L_3}+t^{\frac{1}{2}}\Vert \mathbf{u}(t){\Vert }_{L_{\infty }}+t^{\frac{1}{2}}\Vert \nabla \mathbf{u}(t){\Vert }_{L_3}+t^{1-\frac{1}{2q}}\Vert \nabla \mathbf{u}(t){\Vert }_{L_q}\right)=0\text{,}$$

for 3 < q < ∞, and continuity with respect to initial vector fields as expressed by

$$\Vert \mathbf{u}(t)-{\mathbf{u}}^{\prime }(t){\Vert }_{L_3}+t^{\frac{1}{2}}\Vert \mathbf{u}(t)-{\mathbf{u}}^{\prime }(t){\Vert }_{L_{\infty }}+t^{\frac{1}{2}}\Vert \nabla \mathbf{u}(t)-\nabla {\mathbf{u}}^{\prime }(t){\Vert }_{L_3}⩽c\Vert {\mathbf{u}}_0-{\mathbf{u}}_0^{\prime }{\Vert }_{L_3}\text{,}$$

where u′ denotes the solution initially evolving from u₀′ ∈ L_σ,3(R³)⁶ with $\Vert {\mathbf{u}}_0^{\prime }{\Vert }_{L_3}⩽ϵ$.

If additionally u₀ ∈ L₁(R³)⁶, a higher time decay property is given by

$$(t+1{)}^{\frac{1}{2}}\Vert \mathbf{u}(t)-{\text{e}}^{-\mathit{tA}}{\mathbf{u}}_0{\Vert }_{L_3}+\Vert \mathbf{u}(t){\Vert }_{L_3}+t^{\frac{1}{2}}\Vert \mathbf{u}(t){\Vert }_{L_{\infty }}+t^{\frac{1}{2}}\Vert \nabla \mathbf{u}(t){\Vert }_{L_3}+t^{1-\frac{1}{2q}}\Vert \nabla \mathbf{u}(t){\Vert }_{L_q}⩽\frac{c}{t+1}\text{,}$$

for 3 < q < ∞, where the constant c dependents only on ϵ, q and $\Vert {\mathbf{u}}_0{\Vert }_{L_1}$.

Since (1) with w = 0 and κ = 0 reduces to the three-dimensional Navier–Stokes viscous fluid flow system and it remains generally not known whether this simplified system has a unique solution for a given smooth and large initial velocity, thus Theorem 1.2 deals with small classical solutions.

Theorem 1.1 is associated with the linearized problem of (1). It is essentially based on the special decomposition of the differential operator A derived in Section 2 and the Mihlin–Hörmander Fourier multiplier theorem (see [11], [18], [25]), which is a fundamental theorem of singular integral theory. The proof of Theorem 1.1 is given in Section 3. Theorem 1.2 is concerned with the nonlinear problem (1) and is proved in Section 4. The existence of small L₃-strong solutions of a three-dimensional Navier–Stokes system is well established since the studies of Kato [13] and Giga and Miyakawa [8]. However, the existence of L₃-strong solutions of the micropolar fluid motion system described by (1), (4) is not known. The proof of Theorem 1.2 is developed from the investigations of Kato [13], Schonbek [24] and Chen and Xin [2]. Theorem 1.2 also includes the higher decay rate problem described by (11) because

$$\Vert \mathbf{u}(t){\Vert }_{L_3}⩽c(t+1{)}^{-\frac{3}{2}}\text{,}\text{whenever}\Vert {\mathrm{e}}^{-\mathit{tA}}{\mathbf{u}}_0{\Vert }_{L_3}⩽c(t+1{)}^{-\frac{3}{2}}\text{.}$$

Similar problems with respect to Navier–Stokes equations were examined by Schonbek [24] and Miyakawa [17]. One may also consult the celebrated work of Schonbek [22], [23] for L₂ decay estimates of Navier–Stokes flows. It is natural to impose a higher decay assumption for linearized Navier–Stokes flows since the initial velocity field v₀ is subject to the divergence free condition ∇ · v₀ = 0 and so ${\int }_{R^3}{\mathbf{v}}_0=0$ (see [17]) due to v₀ ∈ L₁(R³)³, and therefore a tendency to higher decay rates for e^tΔv₀ (see [17], [24]). However, a difference arises in micropolar fluid flows if the initial gyration field w₀ does not satisfy the divergence free condition and so does not satisfy the restriction ${\int }_{R^3}{\mathbf{w}}_0=0$. A discussion of the higher decay rate problem of Navier–Stokes equations is provided by He and Miyakawa [9], [10] for the case of an exterior domain.

If we take κ = 0 and w₀ = 0 in (1) the solution evolving from such an initial field u₀ is nothing more than the three-dimensional Navier–Stokes flow, and the semigroup e^−tA reduces to the heat semigroup e^νtΔ. The L_p − L_q estimate of the heat semigroup is well known (see, for example, [26], [27]). For similar decay results of the L₃-strong Navier–Stokes solutions, one may refer to Kato [13].

We are gratefull to a referee who informed us the study by Yamaguchi [28] who discusses a similar problem involving a three-dimensional bounded domain. This investigation shows decay estimates of the micropolar flow based on the resolvent estimate and the compactness of the Laplacian operator −Δ in a bounded domain. It should be noted that the decay problem of micropolar flows in bounded domains and the decay problem of the micropolar problem in the whole space are very different. In a similar manner to the Navier–Stokes flow, the micropolar flow decays in an analogous form to the heat flow e^Δtu₀. However, for the case of a bounded domain, the heat flow decays in an exponential form e^−δt since the compactness of the Laplacian ensures the positivity of the first eigenvalue of the Lapalacian operator, whereas for the case of the whole space, the heat flow decays in an algebraic form characterized by the L_p − L_q estimate, because the noncompactness of the Laplacian operator implies the absence of eigenvalues and the spectrum of the Lapalacian operator containing the half line [0, ∞).