Timeline for A simple question on the Navier-Stokes system
Current License: CC BY-SA 4.0
10 events
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| Sep 16, 2021 at 14:53 | history | edited | Bazin | CC BY-SA 4.0 |
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| Sep 16, 2021 at 14:33 | comment | added | Bazin | @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. | |
| Sep 15, 2021 at 19:33 | comment | added | Terry Tao | 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. | |
| Sep 15, 2021 at 19:31 | comment | added | Terry Tao | 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/… | |
| Sep 15, 2021 at 12:30 | comment | added | Bazin | @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 $L^p_tL^q_x$. | |
| Sep 14, 2021 at 16:50 | comment | added | Terry Tao | 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 $r < 2$. | |
| Sep 14, 2021 at 16:32 | history | edited | YCor |
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| Sep 14, 2021 at 16:06 | history | edited | Bazin | CC BY-SA 4.0 |
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| Sep 14, 2021 at 15:42 | answer | added | Willie Wong | timeline score: 1 | |
| Sep 14, 2021 at 14:54 | history | asked | Bazin | CC BY-SA 4.0 |