Luc TARTAR Compensated Compactness with more ... - ICMS

In 3-dimensional space-time one has H 1 ⊂ L 6 , so that a cubic
term belongs to L 2 ; in 4-dimensional space-time one has H 1 ⊂
L 4 , so that a quadratic term belongs to L 2 . It then suggests
that an existence theorem should involve a solution being in
H 1 in (x, t), which is not usual for a semi-group approach to
semi-linear systems, which typically uses bounded functions in
t with values in a Sobolev space of functions in x.
In the beginning of 1985, BOSTICK (1916–1991) published an
article on a conjectured toroidal shape for his “living electrons”.
He used electromagnetism and the de Broglie’s wavelength of
an electron, and a current having swirl.
I thought that the coupled system of Dirac’s equation and the
Maxwell–Heaviside equation might support a solution (exact
or approximate) having such a toroidal structure.
Since such a toroidal solution comes from dressing a particular
geometrical curve (a circle) as a first term of an expansion, I
thought that one could create other “particles” by starting from
knotted curves; however, it would not be a problem of topology
(of the embedding of the curve in R 3 ) but of geometry, the
current going through the curve creating strong forces pushing
the curve to prefer a particular geometrical pattern.
It then reminds of an idea of THOMSON (1824–1907), known as
Lord Kelvin after 1892, who wanted to describe the whole world
with vortices, but replacing the equations of fluid dynamics by
more basis hyperbolic systems. Of course, the (slightly silly)
program of string theorists is also a revival of this dream, but
my idea is to discover what type of solutions with oscillations
are compatible with the coupled Dirac / Maxwell–Heaviside
system, or more general hyperbolic systems, and not to invent
games with geometrical objects and pretend that it is physics.
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