Local existence of smooth Navier–Stokes solutions on $\mathbb{R}^3 \times [0,T)$ (Leray 1934)

Read-back of local_existence_R3.

Throughout, $\mathbb{R}^3$ denotes Euclidean 3-space with its standard inner product and norm $\|\cdot\|$, and $e_1,e_2,e_3$ its standard basis vectors. For a map $v:\mathbb{R}^3\to\mathbb{R}^3$ and a point $x$, the custom divergence used below is

i.e. $\sum_i \partial_i v_i(x)$, where $Dv(x)$ is the (Fréchet) derivative of $v$ at $x$ — with the convention that if $v$ is not differentiable at $x$ then $Dv(x)$ is taken to be the zero map, so $\operatorname{div}v(x)=0$ there by fiat. The same "junk value $0$ at non-differentiable points" convention applies to every derivative symbol below ($\partial_t$, $D$, $\Delta$, $\nabla$).

The statement is: let $\nu$ be a real number with $\nu>0$, and let $u_0:\mathbb{R}^3\to\mathbb{R}^3$ be a vector field satisfying the three "initial data" conditions

1. $u_0$ is $C^\infty$ on all of $\mathbb{R}^3$ (here "$\infty$" is the smooth class: all finite-order derivatives exist and are continuous; it is not the analytic class);
2. $\operatorname{div}u_0(x)=0$ for every $x\in\mathbb{R}^3$ (since $u_0$ is smooth, this is the genuine divergence);
3. for every pair of natural numbers $k,K$ there is a real constant $C_{k,K}$ such that

where $\|D^k u_0(x)\|$ is the operator norm of the $k$-th derivative as a $k$-linear map ($k=0$ gives $\|u_0(x)\|$ itself). So every derivative of $u_0$ decays faster than every polynomial (Schwartz-type decay). These hypotheses are not vacuous: $u_0\equiv 0$ satisfies them, as do e.g. curls of Schwartz fields.

Then there exists a real number $T>0$ (no lower bound on $T$, no stated dependence on $\nu$ or $u_0$) and there exist functions $u:\mathbb{R}\times\mathbb{R}^3\to\mathbb{R}^3$ and $p:\mathbb{R}\times\mathbb{R}^3\to\mathbb{R}$ (arbitrary functions, no measurability or regularity assumed a priori, defined for all real times) such that, writing $S=[0,T)$ (half-open: $0\in S$, $T\notin S$), the following six conditions hold:

• Smoothness of $u$. The joint map $(t,x)\mapsto u(t,x)$ is $C^\infty$ on the set $S\times\mathbb{R}^3=[0,T)\times\mathbb{R}^3$, in the "differentiable within the set" sense: at every point of the slab, derivatives of all orders in $(t,x)$ exist as derivatives taken within the slab and are continuous on the slab. At interior points ($0<t<T$) this is ordinary joint $C^\infty$ smoothness; at the boundary face $t=0$ it means one-sided (from the right in $t$) smoothness. Nothing whatsoever is asserted about $u(t,\cdot)$ for $t<0$ or $t\ge T$.
• Smoothness of $p$. The same, verbatim, for $(t,x)\mapsto p(t,x)$ on $[0,T)\times\mathbb{R}^3$.
• Momentum equation. For every $t\in S$ with $0<t$ — i.e. for every $t\in(0,T)$, excluding $t=0$ — and for every $x\in\mathbb{R}^3$,

Here $\partial_t u(t,x)$ is the ordinary two-sided derivative of $s\mapsto u(s,x)$ at $s=t$; $(D_x u(t,\cdot))(x)[u(t,x)]$ is the spatial Jacobian of $u(t,\cdot)$ at $x$ applied to the vector $u(t,x)$, i.e. $(u\cdot\nabla)u$; $\Delta(u(t,\cdot))(x)=\sum_{i=1}^3 D^2_x u(t,\cdot)(x)[e_i,e_i]$ is the componentwise (vector) Laplacian built from the second Fréchet derivative; and $\nabla(p(t,\cdot))(x)\in\mathbb{R}^3$ is the gradient vector representing the differential of $p(t,\cdot)$ at $x$ via the inner product. This is a pointwise (classical) equation at every $x\in\mathbb{R}^3$, not a weak formulation. Because each such $(t,x)$ is an interior point of the slab where $u$ and $p$ are smooth, all derivatives in this equation are genuine (no junk values arise). No equation is imposed at $t=0$.

• Divergence-free. For every $t\in[0,T)$ (this time including $t=0$) and every $x$, $\operatorname{div}\big(u(t,\cdot)\big)(x)=0$.
• Initial condition. $u(0,\cdot)=u_0$ as functions, i.e. $u(0,x)=u_0(x)$ for all $x$. (Note $0\in S$ because $T>0$.)
• Bounded energy. There exists a real number $C$ such that for every $t\in[0,T)$,

where the left side is the lower Lebesgue integral (lintegral) with respect to Lebesgue measure on $\mathbb{R}^3$ of the $[0,\infty]$-valued function $x\mapsto\|u(t,x)\|^2$: it is always defined (as a supremum over simple functions below the integrand, so no measurability is needed) and takes values in $[0,\infty]$, and the right side is the real $C$ clipped to $[0,\infty)$. Since the inequality is strict against a finite quantity, this forces $\int\|u(t,\cdot)\|^2<\infty$ for every $t\in[0,T)$ — so $u(t,\cdot)\in L^2$ uniformly in $t$ — and it forces $C>0$ (if $C\le 0$ the right side is $0$ and the condition is unsatisfiable). The constant $C$ is a single constant for all $t\in[0,T)$; it is not related to $\|u_0\|_{L^2}$, to $\nu$, or to $T$, and no energy inequality or monotonicity in $t$ is asserted.

What is not asserted. Nothing is claimed about uniqueness of $(u,p)$, about the behaviour of $u$ or $p$ at or after time $T$ or before time $0$, about any decay, integrability, or normalisation of $p$ (so $p$ may be altered by any function of $t$ alone without affecting any condition), about $T$ being maximal or bounded below in terms of the data, or about continuity of $t\mapsto u(t,\cdot)$ in any $L^2$ or other norm beyond what joint smoothness on the slab gives. The conclusion is trivially satisfiable when $u_0\equiv0$ (take $u\equiv0$, $p\equiv0$, any $T>0$); for nonzero $u_0$ the initial condition $u(0,\cdot)=u_0$ rules out the zero solution. The pressure and velocity in the theorem statement are existentially quantified only (there is no "$\exists!$"), and the dimension is fixed at $n=3$.