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SUMMARY:Systematic Search For Singularities in 3D Navier-Stokes Flows - Ba
 rtosz Protas (McMaster University)
DTSTART:20220307T133000Z
DTEND:20220307T143000Z
UID:TALK170663@talks.cam.ac.uk
DESCRIPTION:This investigation concerns a systematic computational search 
 for potentially singular behavior in 3D Navier-Stokes flows. Enstrophy $\\
 mathcal{E}(t)$ serves as a convenient indicator of the regularity of solut
 ions to the Navier Stokes equation --- as long as this quantity remains fi
 nite\, the solutions are guaranteed to be smooth and satisfy the equations
  in the classical (pointwise) sense. Another well-known conditional regula
 rity result are the Ladyzhenskaya-Prodi-Serrin conditions asserting that t
 he quantity $\\mathcal{L}_{q\,p}:=\\int_0^T \\| \\mathbf{u}(t) \\|_{L^q(\\
 Omega)}^p \\\, dt$\, where $2/p+3/q \\le 1$\, $q > 3$\, must remain bounde
 d if the solution $\\mathbf{u}(t)$ is smooth on the interval $[0\,T]$. How
 ever\, there are no finite a priori bounds available for these quantities 
 and hence the regularity problem for the 3D Navier-Stokes system remains o
 pen. To quantify the maximum possible growth of $\\mathcal{E}(T)$ and $\\m
 athcal{L}_{q\,p}$\, we consider families of PDE optimization problems in w
 hich initial conditions are sought subject to certain constraints so that 
 these quantities in the resulting Navier-Stokes flows are maximized. These
  problems are solved computationally using a large-scale adjoint-based gra
 dient approach. By solving these problems for a broad range of parameter v
 alues we demonstrate that the maximum growth of $\\mathcal{E}(T)$ and $\\m
 athcal{L}_{q\,p}$ appears finite and follows well-defined power-law relati
 ons in terms of the size of the initial data. Thus\, in the worst-case sce
 narios the two quantities remain bounded for all times and there is no evi
 dence for singularity formation in finite time. We will also review earlie
 r results where a similar approach allowed us to probe the sharpness of a 
 priori bounds on the growth of enstrophy and palinstrophy in 1D Burgers an
 d 2D Navier-Stokes flows.
LOCATION:Seminar Room 1\, Newton Institute
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