Asymptotic stability of stationary solutions to the drift-diffusion model with the fractional dissipation

Abstract

We study the drift-diffusion equation with fractional dissipation $(-\varDelta )^{\theta /2}$ arising from a model of semiconductors. First, we prove the existence and uniqueness of the small solution to the corresponding stationary problem in the whole space. Moreover, it is proved that the unique solution of non-stationary problem exists globally in time and decays exponentially, if initial data are suitably close to the stationary solution and the stationary solution is sufficiently small.

Proof of nonlinear estimates in the Besov space

1.1 Proofs of Lemma 2

We only prove (5). (6) can be proved in completely same way as in the proof of (5). The paraproduct between f and g is defined by

$$\begin{aligned} T_f g =\sum _{j \in {\mathbb {Z}}} S_{j-1} f \varDelta _j g, \end{aligned}$$

where $S_{j} f =\sum _{k \le j-1} \varDelta _k f$. We have the following Bony’s decomposition:

$$\begin{aligned} fg=T_f g + T_g f + R(f,g), \end{aligned}$$

where $R(f,g)=\sum _{j\in {\mathbb {Z}}} \varDelta _j f {\tilde{\varDelta }}_j g$ and ${\tilde{\varDelta }}_j =\varDelta _{j-1} + \varDelta _j + \varDelta _{j+1}$. From the quasiorthogonal property that

$$\begin{aligned} \varDelta _j \varDelta _k f=0 \ \ \text{ if } \ |j-k| \ge 2, \end{aligned}$$

we see that ${\mathcal {F}}[S_{k-1} f \varDelta _k g ]$ and ${\mathcal {F}} [\varDelta _k f {\tilde{\varDelta }}_k ]$ are supported by $2^{k-2} \le |\xi | \le 2^{k+2}$ and $|\xi | \le 2^{k+3}$, respectively. Hence, we have

$$\begin{aligned} \varDelta _j (fg)&= \sum _{|k-j| \le 3} \varDelta _j (S_{k-1} f) \varDelta _k g) + \sum _{|k-j| \le 3} \varDelta _j (S_{k-1} g \varDelta _k f) \nonumber \\&\quad + \sum _{k\ge j-4} \varDelta _j (\varDelta _k f {\tilde{\varDelta }}_k g). \end{aligned}$$

(68)

Since $\varDelta _j$ is bounded, we obtain

$$\begin{aligned} \Vert \varDelta _j (S_{k-1}f \varDelta _k g) \Vert _{L^p}&\le \Vert S_{k-1}f \Vert _{L^\infty } \Vert \varDelta _k g \Vert _{L^p}\\&\le C \Vert \varDelta _k g \Vert _{L^p} \sum _{k' \le k} 2^{\left( \frac{n}{p}-s_1\right) k'} 2^{s_1 k'} \Vert \varDelta _{k'} f\Vert _{L^p} \\&\le C \Vert \varDelta _k g \Vert _{L^p} \Vert f \Vert _{{\dot{B}}^{s_1} _{p,q}} \sum _{k' \le k} 2^{\left( \frac{n}{p}-s_1\right) k'} \\&\le C 2^{(\frac{n}{p}-s_1) k} \Vert f \Vert _{{\dot{B}}^{s_1} _{p,q}} \Vert \varDelta _k g \Vert _{L^p}, \end{aligned}$$

here we used $s_1 < n/p$ in the computation of $\sum _{k' \le k} 2^{(\frac{n}{p}-s_1) k'}$. In the case that $s_1 =n/p$, we estimate $\sum _{k' \le k} 2^{(\frac{n}{p}-s_1) k'} 2^{s_1 k'} \Vert \varDelta _{k'} f\Vert _{L^p}$ by $\Vert f \Vert _{{\dot{B}}^{s_1} _{p,1}}$ in the second line of the above inequalities. Thus, we have for the first term of the right-hand side in (68)

$$\begin{aligned} 2^{\left( s_1 + s_2 -\frac{n}{p}\right) j} \left\| \sum _{|k-j| \le 3} \varDelta _j (S_{k-1} f) \varDelta _k g) \right\| _{L^p}&\le C \Vert f \Vert _{{\dot{B}}^{s_1} _{p,q}} \sum _{|j-k|\le 3} 2^{s_2 k} \Vert \varDelta _k g \Vert _{L^p} \\&\le C c_{j,q} \Vert f \Vert _{{\dot{B}}^{s_1} _{p,q}} \Vert g \Vert _{{\dot{B}}^{s_2} _{p,q}}, \end{aligned}$$

where $\{c_{j,q} \}_{j \in {\mathbb {Z}}}$ is a positive sequence such that $\sum _{j \in {\mathbb {Z}}} c_{j,q} ^q \le 1$. Hence, we have

$$\begin{aligned} \Vert T_f g\Vert _{{\dot{B}}^{s_1 + s_2 - n/p}_{p,q}} \le C \Vert f \Vert _{{\dot{B}}^{s_1} _{p,q}}\Vert g \Vert _{{\dot{B}}^{s_2} _{p,q}}. \end{aligned}$$

In the same way as above, we can show that

$$\begin{aligned} \Vert T_g f\Vert _{{\dot{B}}^{s_1 + s_2 - n/p}_{p,q}} \le C \Vert f \Vert _{{\dot{B}}^{s_1} _{p,q}}\Vert g \Vert _{{\dot{B}}^{s_2} _{p,q}}. \end{aligned}$$

We estimate R(f, g). Since $p \ge 2$, from (3) in Lemma 1, we obtain that

$$\begin{aligned} \Vert \varDelta _j (\varDelta _k f {\tilde{\varDelta }}_k g) \Vert _{L^p} \le 2^{\frac{n}{p}j} \Vert \varDelta _j (\varDelta _k f {\tilde{\varDelta }}_k g)\Vert _{L^{p/2}} \le 2^{\frac{n}{p}j} \Vert \varDelta _k f\Vert _{L^p} \Vert {\tilde{\varDelta }}_k g\Vert _{L^{p}}, \end{aligned}$$

which implies that the third term of the right-hand side in (68) is estimated as follows

$$\begin{aligned} 2^{\left( s_1 + s_2 -\frac{n}{p}\right) j} \left\| \sum _{k \ge j- 4} \varDelta _j (\varDelta _k f {\tilde{\varDelta }}_k g) \right\| _{L^p}&\le C \sum _{k\ge j-4} 2^{(s_1 + s_2)(j- k)} 2^{s_1 k} \\&\quad \times \Vert \varDelta _k f\Vert _{L^p} 2^{s_2 k} \Vert {\tilde{\varDelta }}_k g\Vert _{L^{p}} \\&\le C \Vert f \Vert _{{\dot{B}}^{s_1}_{p,q}} \sum _{k\ge j-4} 2^{(s_1 + s_2)(j- k) } 2^{s_2 k} \\&\quad \times \Vert {\tilde{\varDelta }}_k g\Vert _{L^{p}}. \end{aligned}$$

Taking $l^q$-norm with $j \in {\mathbb {Z}}$, since $s_1 + s_2 >0$, we have from Young’s inequality for the sum

$$\begin{aligned} \Vert R (f,g)\Vert _{{\dot{B}}^{s_1 + s_2 -\frac{n}{p}} _{p,q}} \le C \Vert f \Vert _{{\dot{B}}^{s_1} _{p,q}}\Vert g \Vert _{{\dot{B}}^{s_2} _{p,q}}. \end{aligned}$$

(69)

We note that the assumption that $s_1, s_2 \le n/2$ is not required in the proof of (69). We complete the proof of Lemma 2.

1.2 Proofs of Lemma 3

From Bony’s decomposition, the commutator is written by

$$\begin{aligned} {[}f, \varDelta _j] g= [T_f, \varDelta _j ]g +T_{\varDelta _j g} f + R(\varDelta _j g, f)-\varDelta _j T_g f -\varDelta _j R(f,g). \end{aligned}$$

If f is a solenoidal vector filed and $g=\nabla a$, the last term $\varDelta _j R(f,g)$ is equal to $\varDelta _j \nabla \cdot R(f, a)$. The solenoidal property is only used in the estimate of $\varDelta _j \nabla \cdot R(f, a)$. Here we only give an estimate of $\varDelta _j R(f,g)$. Since $p\ge 2$ and $\rho -\gamma +\frac{2n}{p} >1$, applying the estimate (69) with $s_1 =\frac{n}{p} + \rho$ and $s_2 = \frac{n}{p}-\gamma -1$, we have that

$$\begin{aligned} \Vert \varDelta _j R(f,g)\Vert _{L^p}&\le C c_{j,q} 2^{-j\left( \frac{n}{p}+\rho -1-\gamma \right) } \Vert f \Vert _{ {\dot{B}}^{\frac{n}{p}+\rho } _{p,q}} \Vert g \Vert _{ {\dot{B}}^{\frac{n}{p}-\gamma -1} _{p,q}} \\&\le C c_{j,q} 2^{-j\left( \frac{n}{p}+\rho -1-\gamma \right) } \Vert \nabla f \Vert _{ {\dot{B}}^{\frac{n}{p}+\rho -1} _{p,q}} \Vert g \Vert _{ {\dot{B}}^{\frac{n}{p}-\gamma -1} _{p,q}}. \end{aligned}$$