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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.

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    $\begingroup$ 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 $r < 2$. $\endgroup$
    – Terry Tao
    Commented Sep 14, 2021 at 16:50
  • $\begingroup$ @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 $L^p_tL^q_x$. $\endgroup$
    – Bazin
    Commented Sep 15, 2021 at 12:30
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    $\begingroup$ 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 mathoverflow.net/questions/72195/… $\endgroup$
    – Terry Tao
    Commented Sep 15, 2021 at 19:31
  • $\begingroup$ 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. $\endgroup$
    – Terry Tao
    Commented Sep 15, 2021 at 19:33
  • $\begingroup$ @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. $\endgroup$
    – Bazin
    Commented Sep 16, 2021 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.)

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  • $\begingroup$ Thank you very much. I have changed slightly the formulation. $\endgroup$
    – Bazin
    Commented Sep 14, 2021 at 16:08

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