20 research outputs found
Numerical investigations of non-uniqueness for the Navier-Stokes initial value problem in borderline spaces
We consider the Cauchy problem for the incompressible Navier-Stokes equations
in for a one-parameter family of explicit scale-invariant
axi-symmetric initial data, which is smooth away from the origin and invariant
under the reflection with respect to the -plane. Working in the class of
axi-symmetric fields, we calculate numerically scale-invariant solutions of the
Cauchy problem in terms of their profile functions, which are smooth. The
solutions are necessarily unique for small data, but for large data we observe
a breaking of the reflection symmetry of the initial data through a
pitchfork-type bifurcation. By a variation of previous results by Jia &
\v{S}ver\'ak (2013) it is known rigorously that if the behavior seen here
numerically can be proved, optimal non-uniqueness examples for the Cauchy
problem can be established, and two different solutions can exists for the same
initial datum which is divergence-free, smooth away from the origin, compactly
supported, and locally -homogeneous near the origin. In particular,
assuming our (finite-dimensional) numerics represents faithfully the behavior
of the full (infinite-dimensional) system, the problem of uniqueness of the
Leray-Hopf solutions (with non-smooth initial data) has a negative answer and,
in addition, the perturbative arguments such those by Kato (1984) and Koch &
Tataru (2001), or the weak-strong uniqueness results by Leray, Prodi, Serrin,
Ladyzhenskaya and others, already give essentially optimal results. There are
no singularities involved in the numerics, as we work only with smooth profile
functions. It is conceivable that our calculations could be upgraded to a
computer-assisted proof, although this would involve a substantial amount of
additional work and calculations, including a much more detailed analysis of
the asymptotic expansions of the solutions at large distances.Comment: 31 pages, 19 figure
Liouville theorems in unbounded domains for the time-dependent Stokes system
In this paper, We characterize bounded ancient solutions to the
time-dependent Stokes system with zero boundary value in various domains,
including the half space.Comment: 11 pages; final versio
Local structure of the set of steady-state solutions to the 2D incompressible Euler equations
It is well known that the incompressible Euler equations can be formulated in
a very geometric language. The geometric structures provide very valuable
insights into the properties of the solutions. Analogies with the
finite-dimensional model of geodesics on a Lie group with left-invariant metric
can be very instructive, but it is often difficult to prove analogues of
finite-dimensional results in the infinite-dimensional setting of Euler's
equations. In this paper we establish a result in this direction in the simple
case of steady-state solutions in two dimensions, under some non-degeneracy
assumptions. In particular, we establish, in a non-degenerate situation, a
local one-to-one correspondence between steady-states and co-adjoint orbits.Comment: 81 page