On blow-up space jets for the Navier–Stokes equations in R3 with convergence to Euler equations

2008 ◽  
Vol 49 (11) ◽  
pp. 113101 ◽  
Author(s):  
V. A. Galaktionov
Keyword(s):  
Euler Equations ◽  
Blow Up ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations

scholarly journals Turing Universality of the Incompressible Euler Equations and a Conjecture of Moore

2021 ◽  
Author(s):  
Robert Cardona ◽  
Eva Miranda ◽  
Daniel Peralta-Salas
Keyword(s):  
Euler Equations ◽  
Vector Fields ◽  
Blow Up ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Open Set ◽  
Divergence Free Vector ◽  
Divergence Free ◽  
Turing Complete

Abstract In this article, we construct a compact Riemannian manifold of high dimension on which the time-dependent Euler equations are Turing complete. More precisely, the halting of any Turing machine with a given input is equivalent to a certain global solution of the Euler equations entering a certain open set in the space of divergence-free vector fields. In particular, this implies the undecidability of whether a solution to the Euler equations with an initial datum will reach a certain open set or not in the space of divergence-free fields. This result goes one step further in Tao’s programme to study the blow-up problem for the Euler and Navier–Stokes equations using fluid computers. As a remarkable spin-off, our method of proof allows us to give a counterexample to a conjecture of Moore dating back to 1998 on the non-existence of analytic maps on compact manifolds that are Turing complete.

Particle dynamics and mixing in a viscously decaying shear layer

1991 ◽  
Vol 227 ◽  
pp. 211-244 ◽  
Author(s):  
E. Meiburg ◽  
P. K. Newton
Keyword(s):  
Exact Solution ◽  
Blow Up ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Core Size ◽  
Mixing Processes ◽  
Navier Stokes Equations ◽  
Interface Generation ◽  
Self Similar ◽  
Detailed Picture

We study the mixing of fluid in a viscously decaying row of point vortices. To this end, we employ a simplified model based on Stuart's (1967) one-parameter family of solutions to the steady Euler equations. Our approach relates the free parameter to a vortex core size, which grows in time according to the exact solution of the Navier-Stokes equations for an isolated vortex. In this way, we approach an exact solution for small values of t/Re. We investigate how the growing core size leads to a shrinking of the cat's eye and hence to fluid leaking out of the trapped region into the free streams. In particular, we observe that particles initially located close to each other in neighbouring intervals along the streamwise direction escape from the cat's eye near opposite ends. The size of these intervals scales with the inverse square root of the Reynolds number. We furthermore examine the particle escape times and observe a self-similar blow-up for the particles near the border between two adjacent intervals. This can be explained on the basis of a simple stagnation-point flow. An investigation of interface generation shows that viscosity leads to an additional factor proportional to time in the growth rates. Numerical simulations confirm the above results and give a detailed picture of the underlying mixing processes.

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