Well-posedness for the Navier-Stokes equations with datum in Sobolev-Fourier-Lorentz spaces

详细信息    查看全文

• 作者：D.Q. Khai ; N.M. Tri ; ^{triminh@math.ac.vn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Navier&ndash ; Stokes equations ; Existence and uniqueness of local and global mild solutions ; Critical Sobolev spaces
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 May 2016
• 年：2016
• 卷：437
• 期：2
• 页码：754-781
• 全文大小：527 K

文摘

In this note, for s∈R$s\in \mathbb{R}$ and 1≤p,r≤∞$1\le p,r\le \infty$, we introduce and study Sobolev–Fourier–Lorentz spaces ${\dot{H}}_{{\mathcal{L}}^{p,r}}^s({\mathbb{R}}^d)$. In the family spaces ${\dot{H}}_{{\mathcal{L}}^{p,r}}^s({\mathbb{R}}^d)$, the critical invariant spaces for the Navier–Stokes equations correspond to the value $s=\frac{d}{p}-1$. When the initial datum belongs to the critical spaces ${\dot{H}}_{{\mathcal{L}}^{p,r}}^{\frac{d}{p}-1}({\mathbb{R}}^d)$ with d≥2$d\ge 2$, 1≤p<∞$1\le p<\infty$, and 1≤r<∞$1\le r<\infty$, we establish the existence of local mild solutions to the Cauchy problem for the Navier–Stokes equations in spaces $L^{\infty }([0,T];{\dot{H}}_{{\mathcal{L}}^{p,r}}^{\frac{d}{p}-1}({\mathbb{R}}^d))$ with arbitrary initial value, and existence of global mild solutions in spaces $L^{\infty }([0,\infty );{\dot{H}}_{{\mathcal{L}}^{p,r}}^{\frac{d}{p}-1}({\mathbb{R}}^d))$ when the norm of the initial value in the Besov spaces ${\dot{B}}_{{\mathcal{L}}^{\tilde{p},\infty }}^{\frac{d}{\tilde{p}}-1,\infty }({\mathbb{R}}^d)$ is small enough, where $\tilde{p}$ may take some suitable values.