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The Navier-Stokes system for incompressible fluids in $\mathbb R^3$ reads as \begin{align} &\frac{\partial v}{\partial t}+\mathbb P\bigl((v\cdot \nabla) v\bigr)-\nu \Delta v=0, \quad \text{div} v=0, \\ &v(t=0)=a,\quad \text{div} a=0, \end{align} where $\mathbb P$ is the Leray projector. Let us assume that $v$ and $w$ are two $C^\infty$ solutions of the above system, having in particular the same initial datum $a$.

Question 1: Is it absolutely clear that $v=w$, if $v,w$ both belong to $L^r_{t,x}$ for some finite $r$? A classical result asserts that uniqueness holds true provided one of the solutions belongs to $L^p_tL^q_x$ with $ \frac2{p}+\frac3{q}=1, $ for instance $L^2_tL^\infty_x$. But assuming smoothness for both $v$ and $w$ does not encompass a global estimate of that kind, so this classical result seems to fall short of providing a proof of the previous uniqueness result.

Question 2: A grand open problem is to ask for the uniqueness of Leray solutions, i.e. belonging to $L^\infty_tL^2_x\cap L^2_t\dot H^1_x$ with an initial datum in $L^2$: if $v,w$ are two Leray solutions which are also $C^\infty$ it does not seem clear that you can prove that $v=w$ although you have made the drastic smoothness assumption and in doing so, ruled out turbulent solutions.

Bazin
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    The regularity hypotheses you impose on $v$ are not sufficient to guarantee that ${\mathbb P}( (v \cdot \nabla) v)$ makes sense, even as a spacetime distribution, if $r < 2$. – Terry Tao Sep 14 '21 at 16:50
  • @Terry Tao Assuming $r\ge 2$ or that $v,w$ are both Leray solutions, my point is that the great problem of uniqueness is not simplified when the solutions are smooth (a local property); in fact to apply the standard uniqueness result stated in my question, you need a global property $L^p_tL^q_x$. – Bazin Sep 15 '21 at 12:30
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    Yes, though this is reasonably well known. For instance even the linear heat equation $\partial v / \partial t = \Delta v$ suffers lack of uniqueness for smooth solutions if one does not impose some growth condition at infinity; see for instance the discussion at https://mathoverflow.net/questions/72195/unconditional-nonexistence-for-the-heat-equation-with-rapidly-growing-data – Terry Tao Sep 15 '21 at 19:31
  • Also, I am a bit confused as to why you consider a bound on the $L^p_t L^q_x$ norm to be a global property while a bound on the $L^r_{t,x}$ norm is not considered global. – Terry Tao Sep 15 '21 at 19:33
  • @TerryTao For the Tychonoff counterexample for the heat equation, the non-null solution increases as $e^{\vert x\vert^2}$ and is not a tempered distribution. To recover uniqueness, it is in fact enough to assume that the solutions live in the space of tempered distributions, a rather mild assumption. You are right the $L^r_{t,x}$ bound is a global one. If you ask for the big problem of uniqueness of Leray solutions (in $L^\infty_tL^2_x\cap L^2_t\dot H^1_x$) which are also smooth, it is not clear that you have ended-up with a much simpler problem although you have ruled out turbulent solutions. – Bazin Sep 16 '21 at 14:33

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As how you formulated: no.

Let $a = (0,0,0)$. Then $v \equiv (0,0,0)$ and $p_v\equiv 0$ solves the Navier-Stokes equation, as does $w = (t,0,0)$ and $p_w \equiv -x_1$.

(This is a bit fake since $p_w$ is not what you'd get if you apply the Leray projection operator. But if you just allow $v,w$ to be arbitrary $C^\infty$, then with sufficiently fast growth rates you can easily get them out of the domain of the Leray projection operator, and in which case such an objection becomes moot.)

Willie Wong
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