Unforced periodic Leray–Hopf solution

Let $d\ge 2$, $L>0$, $\nu >0$, $u_0\in H$, and use the spaces and weak-time conventions of Periodic solenoidal Fourier spaces and Weak time continuity and energy spaces. A global unforced periodic Leray–Hopf solution with data $(\nu ,u_0)$ is a map $u:[0,\mathrm{\infty })\to H$ such that its restriction to each finite interval is in $C_w([0,T];H)$ and its Bochner equivalence classes satisfy $u\in L^{\mathrm{\infty }}(0,\mathrm{\infty };H)\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(0,\mathrm{\infty };V)$. All clauses involving $u(t)$ at a specified time, including the initial trace and energy inequalities, refer to this weakly continuous map. By the uniqueness result in Weak time continuity and energy spaces, it is the unique weakly continuous representative of those Bochner classes. It must satisfy $u(0)=u_0$ in $H$, $\mathrm{\parallel }u(t)-u_0{\mathrm{\parallel }}_H\to 0$ as $t\downarrow 0$, and for every smooth real periodic divergence-free test field $\varphi :[0,\mathrm{\infty })\times {\mathbb{T}}_L^d\to {\mathbb{R}}^d$ compactly supported in time, ${\int }_0^{\mathrm{\infty }}\text{\hspace{0.17em}⁣}[(u,{\mathrm{\partial }}_t\varphi )+(u\otimes u,\mathrm{\nabla }\varphi )-\nu (\mathrm{\nabla }u,\mathrm{\nabla }\varphi )]dt+(u_0,\varphi (0))=0.$ It must also satisfy, for every $t\ge 0$, $\mathrm{\parallel }u(t){\mathrm{\parallel }}_2^2+2\nu {\int }_0^t\mathrm{\parallel }\mathrm{\nabla }u(s){\mathrm{\parallel }}_2^{2}ds\le \mathrm{\parallel }{u}_0{\mathrm{\parallel }}_2^2,$ and there must be a full-measure set $S\subset (0,\mathrm{\infty })$ such that for every $s\in S\cup \{0\}$ and every $t\ge s$, $\mathrm{\parallel }u(t){\mathrm{\parallel }}_2^2+2\nu {\int }_s^t\mathrm{\parallel }\mathrm{\nabla }u(\tau ){\mathrm{\parallel }}_2^{2}d\tau \le \mathrm{\parallel }u(s){\mathrm{\parallel }}_2^2\mathrm{.}$ For a matrix field $A$, $(\mathrm{div}A{)}_i={\sum }_j{\mathrm{\partial }}_j{A}_{ij}$. The tensor pairing means $(u\otimes u):\mathrm{\nabla }\varphi ={\sum }_{i,j}{u}_i{u}_j{\mathrm{\partial }}_j{\varphi }_i$; the gradient pairing means $\mathrm{\nabla }u:\mathrm{\nabla }\varphi ={\sum }_{i,j}{\mathrm{\partial }}_j{u}_i{\mathrm{\partial }}_j{\varphi }_i$. All derivatives in the weak identity are distributional; the definition asserts neither uniqueness nor smoothness.

Periodic solenoidal Fourier spaces

Leonid Ryzhik, Math 256B Lecture Notes, §5.5, pp. 48–49.

Berselli and Spirito, On the Existence of Leray–Hopf Weak Solutions, Definition 1, §3, p. 6.