Approximation of invariant measure for a stochastic population model with Markov chain and diffusion in a polluted environment

1.   Introduction

The paper is devoted to classical solutions of the Dirichlet problem for the Darcy-Forchheimer-Brinkman system

where $\Omega \subset R^m$ is a bounded domain.

Boundary value problems for the Darcy-Forchheimer-Brinkman system have been extensively studied in the recent years. This system describes flows through porous media saturated with viscous incompressible fluids, where the inertia of such fluids is not negligible. The constants $\lambda, b > 0$ are determined by the physical properties of the porous medium. (For further details we refer the reader to the book [1,p.17] and the references therein.)

M. Kohr et al. studied in ^[2] the transmission problem, where the Darcy-Forchheimer-Brinkman system is considered in a bounded domain $\Omega_+ \subset {\mathbb R}^3$ with connected Lipschitz boundary and the Stokes system is given on its complementary domain $\Omega_-$. Solutions belong to the space ${\mathcal H}^1(\Omega_\pm)\times L^2(\Omega_\pm)$, where ${\mathcal H}^1(\Omega) = \{ {\bf u}\in L^2_{\rm loc}(\Omega, {\mathbb R}^3); \partial_j u_i\in L^2(\Omega), (1+|{\bf x}|^2)^{-1/2}u_j({\bf x})\in L^2(\Omega)\}$. The paper ^[3] is concerned with another transmission problem. A bounded domain $\Omega \subset {\mathbb R}^m$ with connected Lipschitz boundary splits into two Lipschitz domains $\Omega_+$ and $\Omega_-$. A solution is found satisfying the homogeneous Darcy-Forchheimer-Brinkman system is in $\Omega_-$ and the homogeneous Navier-Stokes system in $\Omega_+$. The transmission condition on the interface $\partial \Omega_- \cap \partial \Omega_+$ is accompanied by the Robin condition on $\partial \Omega$. The paper ^[4] investigates the Robin problem for the Darcy-Forchheimer-Brinkman system (1.1) with $b = 0$ in the space $H^s(\Omega, {\mathbb R}^m)\times H^{s-1}(\Omega)$, where $1 < s < 3/2$ and $\Omega \subset {\mathbb R}^m$ is a bounded domain with connected Lipschitz boundary, $m\in \{ 2, 3\}$. The mixed Dirichlet-Robin problem and the mixed Dirichlet-Neumann problem for the Darcy-Forchheimer-Brinkman system (1.1) with $b = 0$ are studied in $H^{3/2}(\Omega, {\mathbb R}^3)\times H^{1/2}(\Omega)$ (see ^[4] and ^[5]). Here $\Omega \subset {\mathbb R}^3$ is a bounded creased domain with connected Lipschitz boundary. M. Kohr et al. discussed in ^[4] the problem of Navier's type for the Darcy-Forchheimer-Brinkman system (1.1) with $b = 0$ in $H^1(\Omega, {\mathbb R}^3)\times L^2(\Omega)$, where $\Omega \subset {\mathbb R}^3$ is a bounded domain with connected Lipschitz boundary.

Now we briefly sketch results concerning the Dirichlet problem for the Darcy-Forchheimer-Brinkman system (1.1). It is supposed that $\Omega \subset {\mathbb R}^m$ is a bounded domain with Lipschitz boundary. For ${\bf f}\equiv 0$ and $2\le m\le 3$ solutions of the problem are looked for in $W^{s, 2}(\Omega, {\mathbb R}^m)\times W^{s-1, 2}(\Omega)$ with $1\le s < 3/2$ (see ^[6], ^[7] and ^[8]). The paper ^[9] is devoted to similar problems on compact Riemannian manifolds. ^[10] considers bounded solutions of the problem for $b = 0$ and a domain $\Omega$ with Ljapunov boundary.

This paper begins with the study of classical solutions of the Dirichlet problem for the generalized Brinkman system

If $\Omega \subset {\mathbb R}^m$ is a bounded open set with boundary of class ${\mathcal C}^{k, \alpha }$ we prove the existence of a solution $({\bf v}, p)\in {\mathcal C}^{k, \alpha }(\overline \Omega; {\mathbb R}^m)\times {\mathcal C}^{k-1, \alpha }(\overline \Omega)$. Unlike the previous papers we do not suppose that $\partial \Omega$ is connected. Using the fixed point theorems give the existence of solutions of the Dirichlet problem for the Darcy-Forchheimer-Brinkman system (1.1) in $({\bf v}, p)\in {\mathcal C}^{k, \alpha }(\overline \Omega; {\mathbb R}^m) \times {\mathcal C}^{k-1, \alpha }(\overline \Omega)$. Here $\lambda, a, b\in {\mathcal C}^{\max (k-2, 0), \alpha } (\overline \Omega)$. If $k\le 3$ then $a$ can be arbitrary. If $k > 3$ then there exists ${\bf v}\in {\mathcal C}^\infty (\overline \Omega; {\mathbb R}^m)$ with $\nabla \cdot {\bf v} = 0$ such that $|{\bf v}|{\bf v}\not \in {\mathcal C}^{k-2, \alpha }(\overline \Omega; {\mathbb R}^m)$. (See Remark 3.3.) So, for $k > 3$ we must suppose that $a\equiv 0$.

2.   Dirichlet problem for the Brinkman system

Before we investigate the Dirichlet problem for the Brinkman system we need the following auxiliary lemma.

Lemma 2.1. Let $\Omega \subset {\mathbb R}^m$ be a bounded open set, $0 < \alpha < 1$ and $\lambda \in {\mathcal C}^{0, \alpha } (\overline \Omega)$ be non-negative. If ${\bf f}\in {\mathcal C}^{0, \alpha }(\overline \Omega, {\mathbb R}^m)$ then there exists a solution $({\bf v}, p)\in {\mathcal C}^{2, \alpha }(\overline \Omega; {\mathbb R}^m) \times {\mathcal C}^{1, \alpha }(\overline \Omega)$ of

Proof. Choose a bounded domain $\omega$ with smooth boundary such that $\overline \Omega \subset \omega$. Then we can suppose that ${\bf f}\in {\mathcal C}^{0, \alpha }(\overline \omega, {\mathbb R}^m)$ and $\lambda \in {\mathcal C}^{0, \alpha }(\overline \omega)$. (See [11,Theorem 1.8.3] or [12,Chapter Ⅵ,§2].) Choose $q$ such that $m/(1-\alpha) < q < \infty$. According to Lemma 3.6 in the Appendix there exists a solution $({\bf v}, p)\in W^{1, q}(\omega, {\mathbb R}^m)\times L^q(\omega)$ of

Put ${\bf F} = {\bf f}-\lambda {\bf v}$. Since ${\bf v}\in W^{1, q}(\omega)\hookrightarrow {\mathcal C}^{0, \alpha }(\overline \omega)$ by [11,Theorem 5.7.8], we infer that ${\bf v}, {\bf F}\in {\mathcal C}^{0, \alpha }(\overline \omega)$ by Lemma 3.7 in the Appendix. Then

Choose bounded open sets $\omega_1$ and $\omega_2$ such that $\overline \Omega \subset \omega_1 \subset \overline \omega_1 \subset \omega_2 \subset \overline \omega_2\subset \omega$. Fix $\varphi \in {\mathcal C}^\infty ({\mathbb R}^m)$ such that $\varphi = 1$ on $\omega_1$ and $\varphi = 0$ on ${\mathbb R}^m \setminus \omega_2$. Define $\tilde {\bf F} = \varphi {\bf F}$ in $\omega$ and $\tilde {\bf F} = 0$ in ${\mathbb R}^m \setminus \omega$.

For $x\in {\mathbb R}^m \setminus \{ 0\}$ and $i, j\in \{ 1, 2, \dots, m\}$ define

where $\sigma_m$ is the area of the unit sphere in $R^m$. Then $E = \{ E_{ij}\}$, $Q = (Q_1, \dots, Q_m)$ form a fundamental tensor of the Stokes system, i.e.,

where $\delta_0$ is the Dirac measure. (See for example ^[13].) Define $\tilde {\bf v}: = E*\tilde {\bf F}$ and $\tilde p: = Q*\tilde {\bf F}$. Then

Define

the fundamental solution for the Laplace equation. Since $F_j\in {\mathcal C}^{0, \alpha }_{\mathrm loc} ({\mathbb R}^m)$, [14,Theorem 3.14.2] gives that $h_\Delta *\tilde F_j\in {\mathcal C}^{2, \alpha }_{\mathrm loc} ({\mathbb R}^m)$ for $j = 1, \dots, m$. Since $Q_j = \partial_j h_\Delta$, we infer

Since

we infer that

in the sense of distributions. Thus $p-\tilde p\in {\mathcal C}^2(\omega_1)$ by [14,Theorem 2.18.2]. Since $\tilde p\in {\mathcal C}^{1, \alpha }(\omega)$ we infer that $p\in {\mathcal C}^{1, \alpha }_{\mathrm loc} (\omega_1)$. Thus $\Delta {\bf v} = \nabla p-{\bf F}\in {\mathcal C}^{0, \alpha }_{\mathrm loc}(\omega_1;{\mathbb R}^m)$. According to [14,Proposition 3.18.1] we obtain that ${\bf v}\in {\mathcal C}^{2, \alpha }_{\mathrm loc} (\omega_1;{\mathbb R}^m)$. (We can prove that ${\bf v}\in {\mathcal C}^{2, \alpha }_{\mathrm loc}(\omega_1;{\mathbb R}^m)$ also using results in ^[15].)

Theorem 2.2. Let $0 < \beta < \alpha < 1$. Suppose that $\Omega \subset {\mathbb R}^m$ is a bounded domain with boundary of class ${\mathcal C}^{1, \alpha }$ and $\lambda \in {\mathcal C}^{0, \beta }(\overline \Omega)$ is non-negative. Let ${\bf f}\in {\mathcal C}^{0, \beta }(\overline \Omega, {\mathbb R}^m)$, ${\bf g} \in {\mathcal C}^{1, \beta }(\partial \Omega, {\mathbb R}^m)$. Then there exist ${\bf v}\in {\mathcal C}^{1, \beta }(\overline \Omega; {\mathbb R}^m)\cap {\mathcal C}^2(\Omega; {\mathbb R}^m)$ and $p\in {\mathcal C}^{0, \beta }(\overline \Omega) \cap {\mathcal C}^1(\Omega)$ solving

if and only if

A velocity ${\bf v}$ is unique and a pressure $p$ is unique up to an additive constant. Moreover,

Proof. If ${\bf v}\in {\mathcal C}^{1, \beta }(\overline \Omega; {\mathbb R}^m)\cap {\mathcal C}^2(\Omega; {\mathbb R}^m)$, $p\in {\mathcal C}^{0, \beta }(\overline \Omega) \cap {\mathcal C}^1(\Omega)$ solve (2.2) then (2.3) holds by the Green formula.

Suppose now that (2.3) holds. According to Lemma 2.1 there exists $(\tilde {\bf v}, \tilde p) \in {\mathcal C}^{2, \beta }(\overline \Omega; {\mathbb R}^m)\times {\mathcal C}^{1, \beta }(\overline \Omega)$ such that

Put $\hat {\bf g}: = {\bf g}-\tilde {\bf v}$. Then $\hat {\bf g}\in {\mathcal C}^{1, \beta }(\partial \Omega, {\mathbb R}^m)$. Choose $q$ such that $m/(1-\beta) < q < \infty$. According to Lemma 3.6 there exists a solution $(\hat {\bf v}, \hat p)\in W^{1, q}(\Omega, {\mathbb R}^m)\times L^q(\Omega)$ of

A velocity $\hat {\bf v}$ is unique and a pressure $\hat p$ is unique up to an additive constant. Define ${\bf v}: = \tilde {\bf v}+\hat {\bf v}$, $p: = \tilde p+\hat p$. Then $({\bf v}, p)$ is a solution of (2.2).

If $\lambda \equiv 0$ then $\hat {\bf v}\in {\mathcal C}^{1, \beta }(\overline \Omega; {\mathbb R}^m)$, $\hat p \in {\mathcal C}^{0, \beta }(\overline \Omega)$ by [16,Theorem 5.2]. Moreover, $\hat {\bf v}\in {\mathcal C}^\infty (\Omega; {\mathbb R}^m)$, $\hat p\in {\mathcal C}^\infty (\Omega)$. (See for example [17,§1.2].) Thus ${\bf v}\in {\mathcal C}^{1, \beta }(\overline \Omega; {\mathbb R}^m)\cap {\mathcal C}^2(\Omega; {\mathbb R}^m)$, $p\in {\mathcal C}^{0, \beta }(\overline \Omega)\cap {\mathcal C}^1(\Omega)$.

Let now $\lambda$ be general. Since ${\bf v}\in W^{1, q}(\Omega)\hookrightarrow {\mathcal C}^{0, \beta } (\overline \Omega)$ by [11,Theorem 5.7.8], Lemma 3.7 in the Appendix gives that $\lambda {\bf v}\in {\mathcal C}^{0, \beta }(\overline \Omega)$. Therefore

We have proved that ${\bf v}\in {\mathcal C}^{1, \beta }(\overline \Omega; {\mathbb R}^m)\cap {\mathcal C}^2(\Omega; {\mathbb R}^m)$, $p\in {\mathcal C}^{0, \beta }(\overline \Omega)\cap {\mathcal C}^1(\Omega)$.

Denote by $Y$ the set of all ${\bf g}\in W^{1-1/q, q}(\partial \Omega; {\mathbb R}^m)$ satisfying (2.3). Define $X: = \{ {\bf v}\in W^{1, q}(\Omega; {\mathbb R}^m); \nabla \cdot {\bf v} = 0$ in $\Omega, {\bf v}|_{\partial \Omega } \in Y\}$,

Then $U_\lambda :X\times L^q(\Omega)\to W^{-1, q}(\Omega; {\mathbb R}^m)\times Y \times {\mathbb R}^1$ is an isomorphism by Lemma 3.6. Denote $Z = [X\cap {\mathcal C}^{1, \beta }(\overline \Omega; {\mathbb R}^m)] \times {\mathcal C}^{0, \beta }(\overline \Omega)$, $W = {\mathcal C}^{0, \beta }(\overline \Omega, {\mathbb R}^m)\times [Y\cap {\mathcal C}^{1, \beta }(\partial \Omega, {\mathbb R}^m)]\times {\mathbb R}^1$. We have proved that $U_\lambda^{-1}(W)\subset Z$. Since $U_\lambda^{-1}:W\to Z$ is closed, it is continuous by the Closed graph theorem.

Theorem 2.3. Let $0 < \alpha < 1$ and $k\in {\mathbb N}_0$. Suppose that $\Omega \subset {\mathbb R}^m$ is a bounded domain with boundary of class ${\mathcal C}^{k+2, \alpha }$ and $\lambda \in {\mathcal C}^{k, \alpha }(\overline \Omega)$ is non-negative. Let ${\bf f}\in {\mathcal C}^{k, \alpha }(\overline \Omega, {\mathbb R}^m)$, ${\bf g} \in {\mathcal C}^{k+2, \alpha }(\partial \Omega, {\mathbb R}^m)$. Then there exists a solution $({\bf v}, p)\in {\mathcal C}^{k+2, \alpha }(\overline \Omega; {\mathbb R}^m)\times {\mathcal C}^{k+1, \alpha }(\overline \Omega)$ of (2.2) if and only if (2.3) holds. A velocity ${\bf v}$ is unique and a pressure $p$ is unique up to an additive constant. Moreover,

Proof. (2.3) is a necessary condition for the solvability of the problem (2.2) by Theorem 2.2.

Denote by $Y$ the set of all ${\bf g}\in {\mathcal C}^{k+2, \alpha }(\partial \Omega; {\mathbb R}^m)$ satisfying (2.3). Define $X: = \{ {\bf v}\in {\mathcal C}^{k+2, \alpha }(\overline \Omega; {\mathbb R}^m); \nabla \cdot {\bf v} = 0$ in $\Omega, {\bf v}|_{\partial \Omega }\in Y\}$,

Then $U_0 :X\times {\mathcal C}^{k+1, \alpha }(\overline \Omega)\to {\mathcal C}^{k, \alpha }(\overline \Omega; {\mathbb R}^m)\times Y\times {\mathbb R}^1$ is an isomorphism by Theorem 2.2 and [18,Theorem IV.7.1]. Since ${\bf u}\mapsto \lambda {\bf u}$ is a bounded operator on ${\mathcal C}^{k, \alpha }(\overline \Omega, {\mathbb R}^m)$ by Lemma 3.7 and ${\mathcal C}^{k+2, \alpha }(\overline \Omega, {\mathbb R}^m) \hookrightarrow {\mathcal C}^{k, \alpha }(\overline \Omega, {\mathbb R}^m)$ is compact by [19,Lemma 6.36], the operator $U_\lambda -U_0 :X\times {\mathcal C}^{k+1, \alpha }(\overline \Omega) \to {\mathcal C}^{k, \alpha }(\overline \Omega; {\mathbb R}^m)\times Y\times {\mathbb R}^1$ is compact. Hence the operator $U_\lambda :X\times {\mathcal C}^{k+1, \alpha }(\overline \Omega)\to {\mathcal C}^{k, \alpha }(\overline \Omega; {\mathbb R}^m)\times Y\times {\mathbb R}^1$ is Fredholm with index $0$. The kernel of $U_\lambda$ is trivial by Theorem 2.2. Therefore, $U_\lambda :X\times {\mathcal C}^{k+1, \alpha }(\overline \Omega)\to {\mathcal C}^{k, \alpha }(\overline \Omega; {\mathbb R}^m) \times Y\times {\mathbb R}^1$ is an isomorphism.

3.   Darcy-Forchheimer-Brinkman system

We prove the existence of a classical solution of the Dirichlet problem for the Darcy-Forchheimer-Brinkman system using fixed point theorems. The following two lemmas are crucial for it.

Lemma 3.1. Let $\Omega \subset {\mathbb R}^m$ be open, $0 < \alpha \le 1$ and $k\in {\mathbb N}$. Put $l = \max (k-2, 0)$. Let $b \in {\mathcal C}^{l, \alpha }({\overline \Omega; \mathbb R}^m)$. Define

Then there exists $C_1\in (0, \infty)$ such that if ${\bf u}, {\bf v}\in {\mathcal C}^{k, \alpha } (\overline \Omega; {\mathbb R}^m)$, then $L_b({\bf u}, {\bf v})\in {\mathcal C}^{l, \alpha } (\overline \Omega; {\mathbb R}^m)$ and

Proof. Lemma 3.7 in the Appendix forces that $L_b({\bf u}, {\bf v})\in {\mathcal C}^{l, \alpha } (\overline \Omega; {\mathbb R}^m)$ and the estimate (3.2) holds true. Since

the estimate (3.3) is a consequence of the estimate (3.2).

Lemma 3.2. Let $\Omega \subset {\mathbb R}^m$ be a bounded domain with boundary of class ${\mathcal C}^{1, \alpha }$ and $0 < \beta \le \alpha < 1$. Suppose that $a\in {\mathcal C}^{0, \beta }(\overline \Omega)$. For ${\bf v}\in {\mathcal C}(\overline \Omega; {\mathbb R}^m)$ define

1. Then there exists a constant $C_1$ such that for ${\bf u}, {\bf v}\in {\mathcal C}^{1, 0}(\overline \Omega; {\mathbb R}^m)$ it holds

2. If $a\in {\mathcal C}^{1, \beta }(\overline \Omega)$, then there exists a positive constant $C_2$ such that $A_a:{\mathcal C}^{2, 0}(\overline \Omega; {\mathbb R}^m)\to {\mathcal C}^{1, \beta } (\overline \Omega; {\mathbb R}^m)$ is a compact continuous mapping and

Proof. Easy calculation yields that

So, according to Lemma 3.7 in the Appendix

Since ${\mathcal C}^{1, 0}(\overline \Omega; {\mathbb R}^m)\hookrightarrow {\mathcal C}^{0, \beta } (\overline \Omega; {\mathbb R}^m)$ by [19,Lemma 6.36], we obtain the estimate (3.5).

Clearly

Hence there exists a constant $c_1$ such that

We now calculate derivatives of $A_1{\bf v}$. If $x\in \Omega$ and ${\bf v}(x)\ne 0$ then

Hence

Let now $x\in \Omega$ and ${\bf v}(x) = 0$. Denote ${\bf e}_j = (\delta_{1j}, \dots, \delta_{mj})$. Then

and (3.10) holds too. Suppose now that $x\in \partial \Omega$. If ${\bf v}(x)\ne 0$ then $\partial_j[|{\bf v}(x)|v_k(x)]$ can be continuously extended to $x$ by (3.9). If ${\bf v}(x) = 0$ then (3.10) gives that $\nabla A_1{\bf v}$ can be continuously extended to $x$ by $\nabla A_1{\bf v}(x) = 0$. The estimates (3.5) and (3.10) give that there exists a constant $c_2$ such that

Suppose that ${\bf u}, {\bf v}\in {\mathcal C}^{1, 0}(\overline \Omega; {\mathbb R}^m)$. Let $x\in \Omega$. Suppose first that ${\bf u}(x) = 0$. Since $\nabla A_1{\bf u}(x) = 0$, (3.10) gives

Let now $|{\bf v}(x)|\ge |{\bf u}(x)| > 0$. According to (3.9)

This inequality, (3.12) and (3.8) give that there exists a constant $c_3$ such that

Since ${\mathcal C}^{1, 0}(\overline \Omega; {\mathbb R}^m)\hookrightarrow {\mathcal C}^{0, \beta } (\overline \Omega; {\mathbb R}^m)$ by [19,Lemma 6.36], there exists a constant $c_4$ such that

Since $A_a{\bf v} = aA_1{\bf v}$, Lemma 3.7 gives that there exists a constant $C_1$ such that (3.6) holds.

Let now ${\bf v}\in {\mathcal C}^{2, 0}(\overline \Omega; {\mathbb R}^m)$. We are going to calculate $\nabla^2 A_1{\bf v}$. If $x\in \Omega$ and ${\bf v}(x)\ne 0$, then we obtain from (3.9)

So,

Now we calculate $\partial_l \partial_j [|{\bf v}(x)|v_k(x)]$ in the sense of distributions. For $\epsilon \ge 0$ denote $\Omega (\epsilon): = \{ x\in \Omega; |{\bf v}(x)| > \epsilon \}$, $V(\epsilon): = \Omega \setminus \overline{\Omega (\epsilon)}$. Suppose that $\varphi \in {\mathcal C}^\infty ({\mathbb R}^m)$ has compact support in $\Omega$. We have proved that if ${\bf v}(x) = 0$, then $\partial_j [|{\bf v}(x)|v_k(x)] = 0$. Thus

According to the Green formula

If $x\in \partial \Omega (\epsilon)\cap \Omega$ then $|{\bf v}(x)| = \epsilon$. If $x\in \partial \Omega (\epsilon)\setminus \Omega$ then $\varphi (x) = 0$. Thus we obtain by (3.14) and (3.9)

According to the Green formula

This and (3.15) give

So, if we define $\partial_l \partial_j [|{\bf v}|v_k](x)$ by (3.13) for ${\bf v}(x)\ne 0$, and $\partial_l \partial_j [|{\bf v}|v_k](x) = 0$ for ${\bf v}(x) = 0$, then this function is $\partial_l \partial_j [|{\bf v}|v_k]$ in sense of distributions. (3.14) forces $A_1 {\bf v}\in W^{2, \infty }(\Omega; {\mathbb R}^m)$ and

$W^{2, \infty }(\Omega)\hookrightarrow {\mathcal C}^{1, \beta }(\overline \Omega)$ compactly by [20,Theorem 6.3]. So,

Since $A_1$ maps bounded subsets of ${\mathcal C}^{2, 0}(\overline \Omega; {\mathbb R}^m)$ to bounded subsets of $W^{2, \infty }(\Omega; {\mathbb R}^m)$ and $W^{2, \infty }(\Omega) \hookrightarrow {\mathcal C}^{1, \beta }(\overline \Omega)$ compactly, the mapping $A_1: {\mathcal C}^{2, 0}(\overline \Omega; {\mathbb R}^m)\to {\mathcal C}^{1, \beta }(\overline \Omega; {\mathbb R}^m)$ is compact. We now show that the mapping $A_1:{\mathcal C}^{2, 0} (\overline \Omega; {\mathbb R}^m)\to {\mathcal C}^{1, \beta }(\overline \Omega; {\mathbb R}^m)$ is continuous. Suppose the opposite. Then there exist $\{ {\bf v}_k\} \subset {\mathcal C}^{2, 0}(\overline \Omega; {\mathbb R}^m)$ and $\epsilon > 0$ such that ${\bf v}_k \to {\bf v}$ in ${\mathcal C}^{2, 0}(\overline \Omega; {\mathbb R}^m)$ and $\| A_1{\bf v}_k -A_1 {\bf v} \|_{{\mathcal C}^{1, \beta }(\overline \Omega)} > \epsilon$. Since $A_1:{\mathcal C}^{2, 0} (\overline \Omega; {\mathbb R}^m)\to {\mathcal C}^{1, \beta }(\overline \Omega; {\mathbb R}^m)$ is compact, there exists a sub-sequence $\{ {\bf v}_{k(n)}\}$ of $\{ {\bf v}_k\}$ and ${\bf w}\in {\mathcal C}^{1, \beta }(\overline \Omega)$ such that $A_1{\bf v}_{k(n)}\to {\bf w}$ in ${\mathcal C}^{1, \beta }(\overline \Omega)$. Since $A_1:{\mathcal C}^{2, 0}(\overline \Omega; {\mathbb R}^m)\to {\mathcal C}^{0, \beta }(\overline \Omega; {\mathbb R}^m)$ is continuous, it must be ${\bf w} = A_1{\bf v}$. That is a contradiction.

Since $A_a{\bf v} = aA_1{\bf v}$, Lemma 3.7 and (3.16) give that $A_a:{\mathcal C}^{2, 0}(\overline \Omega; {\mathbb R}^m)\to {\mathcal C}^{1, \beta } (\overline \Omega; {\mathbb R}^m)$ is a compact continuous mapping and (3.7) holds.

Remark 3.3. Let $A_a$ be from Lemma 3.2 with $a\in {\mathcal C}^\infty (\overline \Omega)$. We cannot show that $A_a: {\mathcal C}^{k, \beta }(\overline \Omega; {\mathbb R}^m)\to {\mathcal C}^{k-2, \beta } (\overline \Omega; {\mathbb R}^m)$ for $k\ge 4$ as shows the following example. Fix $z\in \Omega$ and define ${\bf v}(x): = (x_2-z_2, 0, \dots, 0)$. Then ${\bf v}\in {\mathcal C}^\infty (R^m; R^m)$ but $|{\bf v}|{\bf v}\not \in {\mathcal C}^2(\Omega; R^m)$.

Theorem 3.4. Let $0 < \beta \le \alpha < 1$ and $k\in {\mathbb N}$. If $k = 1$ suppose that $\beta < \alpha$. Put $l = \max (k-2, 0)$. Let $\Omega \subset {\mathbb R}^m$ be a bounded domain with boundary of class ${\mathcal C}^{k, \alpha }$. Let $a, b, \lambda \in {\mathcal C}^{l, \beta }(\overline \Omega)$ and $\lambda \ge 0$. If $k\ge 3$ suppose that $a\equiv 0$. Then there exist $\delta, \epsilon, C\in (0, \infty)$ such that the following holds: If ${\bf g}\in {\mathcal C}^{k, \beta } (\partial \Omega; {\mathbb R}^m)$ satisfying (2.3), ${\bf F}\in {\mathcal C}^{l, \beta } (\overline \Omega; {\mathbb R}^m)$ and

then there exists a unique solution $({\bf u}, p)\in [{\mathcal C}^{k, \beta }(\overline \Omega; {\mathbb R}^m) \cap {\mathcal C}^2 (\Omega; {\mathbb R}^m)]\times [{\mathcal C}^{k-1, \beta }(\overline \Omega) \cap {\mathcal C}^1 (\Omega)]$ of the Dirichlet problem for the Darcy-Forchheimer-Brinkman system

such that

and

Moreover,

If $\tilde {\bf g}\in {\mathcal C}^{k, \beta }(\partial \Omega; {\mathbb R}^m)$, $\tilde {\bf F}\in {\mathcal C}^{l, \beta }(\overline \Omega; {\mathbb R}^m)$, $\tilde {\bf u}\in {\mathcal C}^{k, \beta }(\overline \Omega; {\mathbb R}^m)\cap {\mathcal C}^2 (\Omega; {\mathbb R}^m)$ and $\tilde p\in {\mathcal C}^{k-1, \beta }(\overline \Omega)\cap {\mathcal C}^1 (\Omega)$,

and $\| \tilde {\bf u}\|_{{\mathcal C}^{k, \beta }(\overline \Omega)} < \epsilon$, then

Proof. Let $L_b$ be defined by (3.1), $A_a$ be given by (3.4). Put $D_{ab}{\bf u}: = L_b({\bf u}, {\bf u})+A_a{\bf u}$. According to Lemma 3.1 and Lemma 3.2 there exists a constant $C_1$ such that

By Theorem 2.2 and Theorem 2.3 there exists a constant $C_2$ such that for each ${\bf g}\in {\mathcal C}^{k, \beta }(\partial \Omega; {\mathbb R}^m)$ satisfying (2.3) and ${\bf f}\in {\mathcal C}^{l, \beta }(\overline \Omega; {\mathbb R}^m)$ there exists a unique solution $({\bf u}, p)\in [{\mathcal C}^{k, \beta }(\overline \Omega; {\mathbb R}^m)\cap {\mathcal C}^2 (\Omega; {\mathbb R}^m)]\times [{\mathcal C}^{k-1, \beta }(\overline \Omega) \cap {\mathcal C}^1 (\Omega)]$ of the Dirichlet problem (2.2), (3.19). Moreover,

Remark that $({\bf u}, p)$ is a solution of (3.18) if $({\bf u}, p)$ is a solution of (2.2) with ${\bf f} = {\bf F}-D_{ab}{\bf u}$. Put

If $({\bf u}, p), (\tilde {\bf u}, \tilde p)\in [{\mathcal C}^{k, \beta }(\overline \Omega; {\mathbb R}^m) \cap {\mathcal C}^2 (\Omega; {\mathbb R}^m)]\times [{\mathcal C}^{k-1, \beta }(\overline \Omega) \cap {\mathcal C}^1 (\Omega)]$ are solutions of (3.18), (3.19) and (3.22) with (3.20) and $\| \tilde {\bf u}\|_{{\mathcal C}^{k, \beta }(\overline \Omega)} < \epsilon$, then

Since $2C_1C_2\epsilon < 1/2$ we get subtracting $2\epsilon C_1C_2\| {\bf u}-\tilde {\bf u} \|_{{\mathcal C}^{k, \beta }(\overline \Omega)}$ from the both sides

Therefore a solution of (3.18) satisfying (3.19) and (3.20) is unique. Putting $\tilde p\equiv 0$, $\tilde {\bf u}\equiv 0$, $\tilde {\bf F}\equiv 0$ and $\tilde {\bf g}\equiv 0$, we obtain (3.21) with $C = 2C_2$.

Denote $X: = \{ {\bf v}\in {\mathcal C}^{k, \beta }(\overline \Omega, {\mathbb R}^m); \| {\bf v}\|_{{\mathcal C}^{k, \beta }(\overline \Omega)}\le \epsilon \}$. Fix ${\bf g}\in {\mathcal C}^{k, \beta }(\partial \Omega; {\mathbb R}^m)$ and ${\bf F}\in {\mathcal C}^{l, \beta }(\overline \Omega; {\mathbb R}^m)$ satisfying (2.3) and (3.17). For ${\bf v}\in X$ there exists a unique solution $({\bf u}^{\bf v}, p^{\bf v})\in [{\mathcal C}^{k, \beta }(\overline \Omega; {\mathbb R}^m)\cap {\mathcal C}^2 (\Omega; {\mathbb R}^m)] \times [{\mathcal C}^{k-1, \beta }(\overline \Omega)\cap {\mathcal C}^1 (\Omega)]$ of the Dirichlet problem (2.2), (3.19) with ${\bf f} = {\bf F}-D_{ab}{\bf v}$. Remember that $({\bf u}^{\bf v}, p^{\bf v})$ is a solution of (3.18) if and only if ${\bf u}^{\bf v} = {\bf v}$. According to (3.26), (3.17) and (3.24)

As $C_2 \delta +C_2 C_1 \epsilon^2 < \epsilon$, we infer ${\bf u}^{\bf v}\in X$. If ${\bf w}\in X$ then

by (3.26) and (3.25). Since $C_2C_1 2\epsilon < 1$, the Fixed point theorem ([21, Satz 1.24]) gives that there exists ${\bf v}\in X$ such that ${\bf u}^{\bf v} = {\bf v}$. So, $({\bf u}^{\bf v}, p^{\bf v})$ is a solution of (3.18), (3.19) in $[{\mathcal C}^{k, \beta }(\overline \Omega; {\mathbb R}^m)\cap {\mathcal C}^2 (\Omega; {\mathbb R}^m)] \times [{\mathcal C}^{k-1, \beta }(\overline \Omega)\cap {\mathcal C}^1 (\Omega)]$ satisfying (3.20).

In Theorem 3.4 we suppose that $a\equiv 0$ for $k\ge 3$. This assumption cannot be removed for $k > 3$ as Remark 3.3 shows. The following theorem is devoted to the case $k = 3$.

Theorem 3.5. Let $0 < \beta \le \alpha < 1$ and $\Omega \subset {\mathbb R}^m$ be a bounded domain with boundary of class ${\mathcal C}^{3, \alpha }$. Let $a, b, \lambda \in {\mathcal C}^{1, \beta }(\overline \Omega)$ and $\lambda \ge 0$. Then there exist $\delta, \epsilon \in (0, \infty)$ such that the following holds: If ${\bf g}\in {\mathcal C}^{3, \beta }(\partial \Omega; {\mathbb R}^m)$ satisfies (2.3), ${\bf F}\in {\mathcal C}^{1, \beta }(\overline \Omega; {\mathbb R}^m)$ and

then there exists a unique solution $({\bf u}, p)\in {\mathcal C}^{3, \beta }(\overline \Omega; {\mathbb R}^m) \times {\mathcal C}^{2, \beta }(\overline \Omega)$ of the Dirichlet problem for the Darcy-Forchheimer-Brinkman system (3.18), (3.19) such that

Proof. Let $L_b$ be defined by (3.1), $A_a$ be given by (3.4). We conclude from Lemma 3.1 and Lemma 3.2 that there exists a constant $C_1$ such that

According to Theorem 2.3 there exists a constant $C_2$ such that for each ${\bf g}\in {\mathcal C}^{3, \beta }(\partial \Omega; {\mathbb R}^m)$ satisfying (2.3) and ${\bf f}\in {\mathcal C}^{1, \beta }(\overline \Omega; {\mathbb R}^m)$ there exists a unique solution $({\bf u}, p)\in {\mathcal C}^{3, \beta }(\overline \Omega; {\mathbb R}^m)\times {\mathcal C}^{2, \beta }(\overline \Omega)$ of the Dirichlet problem (2.2), (3.19). Moreover,

Suppose now that

Put $X_\epsilon : = \{ {\bf v}\in {\mathcal C}^{3, \beta }(\overline \Omega, {\mathbb R}^m); \| {\bf v}\|_{{\mathcal C}^{3, \beta }(\overline \Omega)}\le \epsilon \}$. Fix ${\bf g}\in {\mathcal C}^{3, \beta }(\partial \Omega; {\mathbb R}^m)$ and ${\bf F}\in {\mathcal C}^{1, \beta }(\overline \Omega; {\mathbb R}^m)$ satisfying (2.3) and (3.27). For ${\bf v}\in X_\epsilon$ there exists a unique solution $({\bf u}^{\bf v}, p^{\bf v}) \in {\mathcal C}^{3, \beta }(\overline \Omega; {\mathbb R}^m)\times {\mathcal C}^{2, \beta }(\overline \Omega)$ of the Dirichlet problem (2.2), (3.19) with ${\bf f} = {\bf F}-L_b({\bf v}, {\bf v})$. Moreover, there is a unique solution $(\tilde {\bf u}^{\bf v}, \tilde p^{\bf v})\in {\mathcal C}^{3, \beta }(\overline \Omega; {\mathbb R}^m)\times {\mathcal C}^{2, \beta }(\overline \Omega)$ of

If ${\bf u}^{\bf v}+\tilde {\bf u}^{\bf v} = {\bf v}$ then $({\bf v}, p^{\bf v}+\tilde p^{\bf v})$ is a solution of the problem (3.18), (3.19). If ${\bf w}\in X_\epsilon$ then

by (3.31), (3.27), (3.29) and (3.32). So, ${\bf u}^{\bf v}+\tilde {\bf u}^{\bf w} \in X_\epsilon$. According to (3.31), (3.30) and (3.32)

So ${\bf v}\mapsto {\bf u}^{\bf v}$ is a contractive mapping on $X_\epsilon$. Since $A_a:{\mathcal C}^{3, \beta }(\overline \Omega; {\mathbb R}^m)\to {\mathcal C}^{1, \beta } (\overline \Omega; {\mathbb R}^m)$ is a compact continuous mapping by Lemma 3.2, the mapping ${\bf v}\mapsto \tilde {\bf u}^{\bf v}$ is a compact continuous mapping on $X_\epsilon$. Lemma 3.8 forces that there exists ${\bf v}\in X_\epsilon$ such that ${\bf u}^{\bf v}+\tilde {\bf u}^{\bf v} = {\bf v}$. Hence $({\bf v}, p^{\bf v}+\tilde p^{\bf v})$ is a solution of the problem (3.18), (3.19), (3.28) in ${\mathcal C}^{3, \beta } (\overline \Omega; {\mathbb R}^m)\times {\mathcal C}^{1, \beta }(\overline \Omega)$. By Theorem 3.4, for sufficiently small $\epsilon$ and $\delta$ there exists at most one solution of the problem (3.18), (3.19), (3.28) in ${\mathcal C}^{3, \beta } (\overline \Omega; {\mathbb R}^m)\times {\mathcal C}^{1, \beta }(\overline \Omega)$.

Acknowledgments

Supported by RVO: 67985840 and GAČR grant GA17-01747S.

Conflict of interest

The author declares no conflict of interest in this paper.