Convex integration for Lipschitz mappings and counterexamples to regularity

We study Lispchitz solutions of partial differential relations $\nabla u\in K$, where $u$ is a vector-valued function in an open subset of $R^n$. In some cases the set of solutions turns out to be surprisingly large. The general theory is then used to construct counter-examples to regularity of solutions of Euler-Lagrange systems satisfying classical ellipticity conditions.Comment: 28 pages published versio

On Type I singularities of the local axi-symmetric solutions of the Navier-Stokes equations

Local regularity of axially symmetric solutions to the Navier-Stokes equations is studied. It is shown that under certain natural assumptions there are no singularities of Type I

Shape optimization for the generalized Graetz problem

We apply shape optimization tools to the generalized Graetz problem which is a convection-diffusion equation. The problem boils down to the optimization of generalized eigen values on a two phases domain. Shape sensitivity analysis is performed with respect to the evolution of the interface between the fluid and solid phase. In particular physical settings, counterexamples where there is no optimal domains are exhibited. Numerical examples of optimal domains with different physical parameters and constraints are presented. Two different numerical methods (level-set and mesh-morphing) are show-cased and compared

Whirl mappings on generalised annuli and the incompressible symmetric equilibria of the dirichlet energy

In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions.Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation u=(u1,…,uN) : EL[u,X]=⎧⎩⎨⎪⎪Δu=div(P(x)cof∇u)det∇u=1u≡φinX,inX,on∂X, where X is a finite, open, symmetric N -annulus (with N≥2 ), P=P(x) is an unknown hydrostatic pressure field and φ is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when N=3 , the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when N=2 , the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions N≥4 and discuss a number of closely related issues

Dynamics of Geodesic Flows with Random Forcing on Lie Groups with Left-Invariant Metrics

We consider stochastic perturbations of geodesic flow for left-invariant metrics on finite-dimensional Lie groups and study the Hörmander condition and some properties of the solutions of the corresponding Fokker-Planck equations

On Type I Singularities of the Local Axi-Symmetric Solutions of the Navier-Stokes Equations

Local regularity of axially symmetric solutions to the Navier-Stokes equations is studied. It is shown that under certain natural assumptions there are no singularities of Type I. © Taylor and Francis Group, LLC