Singularity formation?

Adding gradients to J2 theory: Jean-Philippe Bouchaud

Hughes et al.

Cell Structures in Stage III Hardening

y Develops Walls, Large Gradients

?y/?t = fy(scalars) sij ?sij/?t

+ g(scalars) ?i?jy ?sij/?t

+ h(scalars) ?iy?jy ?sij/?t

Landau: f, g, h smooth functions.

Lowest order, constants.

Reversible f.

Diffusion / antidiffusion g.

Burgers equation h.

Burgers equation:

?y/?s = (?y/?x)2 + a ?2y/?x2

Develops shocks = walls in finite time