Asymptotic Stability of Landau Solutions to Navier–Stokes System Under $L^p$-Perturbations

Abstract

In this paper, we show that Landau solutions to the Navier–Stokes system are asymptotically stable under $L^3$-perturbations. We give the local well-posedness of solutions to the perturbed system with the initial data in the $L_{\sigma }^3$ space and the global well-posedness with the small initial data in the $L_{\sigma }^3$ space, together with a study of the $L^q$ decay for all $q>3$. Moreover, we have also studied the local well-posedness, the global well-posedness and the stability in $L^p$ spaces for $3<p<\infty$.

Appendix

Proof of Lemma 2.6

Set

$$\begin{aligned} \begin{aligned} II&= -r(t)^{2} \int _{{\mathbb {R}}^{3}} |w(\cdot ,t)|^{r(t)-2} w_i\partial _j (w_iw_j)\textrm{d}x,\\ III&= -r(t)^{2} \int _{{\mathbb {R}}^{3}} |w(\cdot ,t)|^{r(t)-2} w_i\partial _j (w_iv_j)\textrm{d}x,\\ IV&= -r(t)^{2} \int _{{\mathbb {R}}^{3}} |w(\cdot ,t)|^{r(t)-2} w_i\partial _j (v_iw_j)\textrm{d}x,\\ V&= -r(t)^{2} \int _{{\mathbb {R}}^{3}} |w(\cdot ,t)|^{r(t)-2}w_i\partial _i\pi \textrm{d}x. \end{aligned} \end{aligned}$$

(8.1)

Thanks to integration by parts, Hölder’s inequality and Sobolev embedding ${\dot{H}}^{1}({\mathbb {R}}^{3})\hookrightarrow L^6({\mathbb {R}}^{3})$ (the best constant can be seen in [44]), we have

$$\begin{aligned} II= & {} -\frac{r(t)^{2} }{2}\int _{{\mathbb {R}}^{3}} |w(\cdot ,t)|^{r(t)-2} w_j\partial _j|w|^2\textrm{d}x\nonumber \\= & {} \frac{r(t)^{2} }{2}\int _{{\mathbb {R}}^{3}} \partial _j(|w(\cdot ,t)|^{r(t)-2}) w_j|w|^2\textrm{d}x\nonumber \\\le & {} r(t)(r(t)-2)\int _{{\mathbb {R}}^{3}} \left| \nabla (|w(\cdot ,t)|^{\frac{r(t)}{2}})\right| |w|^{\frac{r(t)}{2}}|w|\textrm{d}x\nonumber \\\le & {} r(t)(r(t)-2)\left\| \nabla \left( |w(t)|^{\frac{r(t)}{2}}\right) \right\| _{L^{2}} \left\| |w(t)|^{\frac{r(t)}{2}}\right\| _{L^{6}}\Vert w(t)\Vert _{L^{3}}\nonumber \\\le & {} r(t)(r(t)-2)\left\| \nabla \left( |w(t)|^{\frac{r(t)}{2}}\right) \right\| _{L^{2}}^{2}\Vert w(t)\Vert _{L^{3}}. \end{aligned}$$

(8.2)

Combining (1.13) with $\Vert w_0\Vert _{L^3}\le \varepsilon _0,$ there holds

$$\begin{aligned} \begin{aligned} II\le r(t)(r(t)-2)C\varepsilon _0 \left\| \nabla (|w(t)|^{\frac{r(t)}{2}})\right\| ^2_{L^2}. \end{aligned} \end{aligned}$$

(8.3)

According to Hölder’s inequality, the Hardy inequality in Lemmas 2.2 and 2.3, we deduce

$$\begin{aligned} II= & {} \frac{r(t)^{2}}{2} \int _{{\mathbb {R}}^{3}} \partial _j(|w(\cdot ,t)|^{r(t)-2} )|w|^2v_j\textrm{d}x\nonumber \\= & {} r(t)(r(t)-2)\int _{{\mathbb {R}}^{3}} v_c\cdot \nabla (|w(\cdot ,t)|^{\frac{r(t)}{2}})|w(\cdot ,t)|^{\frac{r(t)}{2}}\textrm{d}x\nonumber \\\le & {} r(t)(r(t)-2)\left\| \nabla \left( |w(t)|^{\frac{r(t)}{2}}\right) \right\| _{L^{2}}\left\| \frac{|w(t)|^{\frac{r(t)}{2}}}{|x|}\right\| _{L^{2}}\left\| |x| v_{c}\right\| _{L^{\infty }}\nonumber \\\le & {} 2 r(t)(r(t)-2) K_{c}\left\| \nabla \left( |w(t)|^{\frac{r(t)}{2}}\right) \right\| _{L^{2}}^{2}. \end{aligned}$$

(8.4)

For the term II, using integration by parts, we have

$$\begin{aligned} IV= & {} -r(t)^{2} \int _{{\mathbb {R}}^{3}} |w(\cdot ,t)|^{r(t)-2} w_i\partial _j (v_iw_j)\textrm{d}x\\= & {} r(t)^{2}\int _{{\mathbb {R}}^{3}} \partial _j (|w(\cdot ,t)|^{r(t)-2}) w_iv_iw_j\textrm{d}x + r(t)^{2}\int _{{\mathbb {R}}^{3}} |w(\cdot ,t)|^{r(t)-2} \partial _jw_i v_iw_j\textrm{d}x. \end{aligned}$$

The estimate of the first part is similar to (8.4). We have

$$\begin{aligned} r(t)^{2}\int _{{\mathbb {R}}^{3}} \partial _j (|w(\cdot ,t)|^{r(t)-2}) w_iv_iw_j\textrm{d}x \le 4 r(t)(r(t)-2) K_{c}\left\| \nabla \left( |w(t)|^{\frac{r(t)}{2}}\right) \right\| _{L^{2}}^{2}. \end{aligned}$$

By Lemma 2.2, Cauchy inequality and the Hardy inequality, we can estimate the second part as follows

$$\begin{aligned}{} & {} r(t)^{2}\int _{{\mathbb {R}}^{3}} |w(\cdot ,t)|^{r(t)-2} \partial _jw_i v_iw_j\textrm{d}x\\{} & {} \quad \le r(t)^{2}\int _{{\mathbb {R}}^{3}} |w(\cdot ,t)|^{r(t)-2} |\partial _jw_i | ||x|v_i|\frac{|w_j|}{|x|}\textrm{d}x\\{} & {} \quad \le r(t)^{2}K_c\int _{{\mathbb {R}}^{3}} |w(\cdot ,t)|^{r(t)-2} |\partial _jw_i | \frac{|w_j|}{|x|}\textrm{d}x\\{} & {} \quad \le \frac{r(t)^{2}}{2} K_c\int _{{\mathbb {R}}^{3}} |w(\cdot ,t)|^{r(t)-2} |\nabla w(\cdot ,t)|^{2} dx+ \frac{r(t)^{2}}{2} K_c \int _{{\mathbb {R}}^{3}} |w(\cdot ,t)|^{r(t)-2} \frac{|w(\cdot ,t)|^{2}}{|x|^2} dx\\{} & {} \quad \le \frac{r(t)^{2}}{2} K_c\int _{{\mathbb {R}}^{3}} |w(\cdot ,t)|^{r(t)-2} |\nabla w(\cdot ,t)|^{2} dx+ 2r(t)^{2} K_c\left\| \nabla \left( |w(t)|^{\frac{r(t)}{2}}\right) \right\| _{L^{2}}^{2}. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \begin{aligned} IV\le&\frac{r(t)^{2}}{2} K_c\int _{{\mathbb {R}}^{3}} |w(\cdot ,t)|^{r(t)-2} |\nabla w(\cdot ,t)|^{2} dx\\&+ (4 r(t)(r(t)-2)+2r(t)^{2}) K_c\left\| \nabla \left( |w(t)|^{\frac{r(t)}{2}}\right) \right\| _{L^{2}}^{2}. \end{aligned} \end{aligned}$$

(8.5)

Note that the pressure $\pi =-\frac{\partial _i\partial _j}{\Delta }(w_iw_j+v_iw_j+w_iv_j )$, using integration by parts, we obtain

$$\begin{aligned} V= & {} r(t)^{2} \int _{{\mathbb {R}}^{3}} \partial _i(|w(\cdot ,t)|^{r(t)-2})w_i\pi \textrm{d}x\nonumber \\= & {} r(t)^{2} \int _{{\mathbb {R}}^{3}} \partial _i(|w(\cdot ,t)|^{r(t)-2})w_i\left( -\frac{\partial _i\partial _j}{\Delta }(v_iw_j+w_iv_j+w_iw_j)\right) \textrm{d}x. \end{aligned}$$

(8.6)

This term is more complex to deal with, we will estimate it more carefully. Set

$$\begin{aligned} V_1= r(t)^{2} \int _{{\mathbb {R}}^{3}} \partial _i(|w(\cdot ,t)|^{r(t)-2})w_i\left( -\frac{\partial _i\partial _j}{\Delta }(v_iw_j+w_iv_j)\right) \textrm{d}x, \end{aligned}$$

and

$$\begin{aligned} V_2= r(t)^{2} \int _{{\mathbb {R}}^{3}} \partial _i(|w(\cdot ,t)|^{r(t)-2})w_i\left( -\frac{\partial _i\partial _j}{\Delta }w_iw_j\right) \textrm{d}x. \end{aligned}$$

According to [12], there holds $|x|^{r-2}\in A_r$ with $1<r<\infty$. By Hölder’s inequality, boundedness of the Riesz transforms on weighted $L^p$ spaces (Theorem 9.4.6 in [12]), Lemma 2.2 and the Hardy inequality, there holds

$$\begin{aligned} V_1\le & {} 2r(t)(r(t)-2)\int _{{\mathbb {R}}^{3}} \left| \nabla \left( |w(\cdot , t)|^{\frac{r(t)}{2}}\right) \right| |w(\cdot , t)|^{\frac{r(t)}{2}-1} \left| \frac{\partial _i\partial _j}{\Delta }(v_iw_j+w_iv_j)\right| \textrm{d} x\nonumber \\\le & {} 4 r(t)(r(t)-2) C_{r}\Vert |x|^{\frac{r-2}{r}}\left( v_{c} \otimes w\right) \Vert _{L^{r}}\left\| \frac{w^{\frac{r}{2}-1}}{x^{\frac{r-2}{r}}} \right\| _{L^{\frac{2 r}{r-2}}}\Vert \nabla (|w(\cdot , t)|^{\frac{r}{2}})\Vert _{L^{2}}\nonumber \\\le & {} 4 r(t)(r(t)-2) C_{r}\Vert |x|v_{c} \Vert _{L^{\infty }}\Vert |x|^{-\frac{2}{r}} w\Vert _{L^{r}}\left\| \frac{w}{|x|^{\frac{2}{r}}}\right\| _{L^{r}}^{\frac{r}{2}-1}\Vert \nabla (|w(\cdot , t)|^{\frac{r}{2}})\Vert _{L^{2}}\nonumber \\\le & {} 4 r(t)(r(t)-2) C_{r}K_c\left\| \frac{|w|^{\frac{r}{2}}}{|x|}\right\| _{L^{2}}\Vert \nabla (|w(\cdot , t)|^{\frac{r}{2}})\Vert _{L^{2}}\nonumber \\\le & {} 8 r(t)(r(t)-2) C_{r} K_c\Vert \nabla (|w(\cdot , t)|^{\frac{r}{2}})\Vert _{L^{2}}^2, \end{aligned}$$

(8.7)

where $C_r$ is as in Theorem 9.4.6 in [12]. Thanks to [15], we deduce that $\Vert \frac{\partial _i\partial _j}{\Delta } f\Vert _{L^r} \le H_r\Vert f\Vert _{L^r}$. Combining with Hölder’s inequality and Sobolev embedding ${\dot{H}}^{1}({\mathbb {R}}^{3})\hookrightarrow L^6({\mathbb {R}}^{3})$, we have

$$\begin{aligned} V_2\le & {} 2r(t)(r(t)-2) \Vert \nabla (|w(\cdot , t)|^{\frac{r(t)}{2}})\Vert _{L^2}\Vert |w(\cdot , t)|^{\frac{r(t)}{2}-1} \Vert _{L^{\frac{6r}{r-2}}}\left\| \frac{\partial _i\partial _j}{\Delta }w_iw_j\right\| _{L^{\frac{3r}{r+1}}}\\\le & {} 2r(t)(r(t)-2) H_{\frac{3r}{r+1}}\Vert \nabla (|w(\cdot , t)|^{\frac{r(t)}{2}})\Vert _{L^2}\Vert |w(\cdot , t)|^{\frac{r(t)}{2}-1} \Vert _{L^{\frac{6r}{r-2}}}\left\| w \otimes w\right\| _{L^{\frac{3r}{r+1}}}\\\le & {} 2r(t)(r(t)-2) H_{\frac{3r}{r+1}}\Vert \nabla (|w(\cdot , t)|^{\frac{r(t)}{2}})\Vert _{L^2}\Vert w(\cdot , t) \Vert ^{\frac{r(t)}{2}-1}_{L^{3r}}\left\| w(\cdot , t)\right\| _{L^{3r}}\left\| w(\cdot , t)\right\| _{L^{3}}\\\le & {} 4r(t)(r(t)-2) H_{\frac{3r}{r+1}}\Vert \nabla (|w(\cdot , t)|^{\frac{r(t)}{2}})\Vert _{L^2}\Vert |w(\cdot , t)|^{\frac{r(t)}{2}}\Vert _{L^6}\Vert w(\cdot , t) \Vert _{L^{3}}\\\le & {} 4r(t)(r(t)-2) H_{\frac{3r}{r+1}}\Vert \nabla (|w(\cdot , t)|^{\frac{r(t)}{2}})\Vert ^2_{L^2}\Vert w(\cdot , t) \Vert _{L^{3}}. \end{aligned}$$

According to (1.13), when $\Vert w_0\Vert _{L^3}\le \varepsilon _0,$ there holds

(8.8)

From the above estimates, we can finish the proof of Lemma 2.6. $\square$