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## Publications of Walter Noll

### Archive for Rational Mechanics and Analysis (1986-12-01) 93: VI-387 , December 01, 1986

## Maximum properties of Cauchy's problem in three-dimensional space-time

### Archive for Rational Mechanics and Analysis (1965-01-01) 18: 14-26 , January 01, 1965

## Partial Regularity for Holonomic Minimisers of Quasiconvex Functionals

### Archive for Rational Mechanics and Analysis (2016-10-01) 222: 91-141 , October 01, 2016

We prove partial regularity for local minimisers of certain strictly quasiconvex integral functionals, over a class of Sobolev mappings into a compact Riemannian manifold, to which such mappings are said to be holonomically constrained. Our approach uses the lifting of Sobolev mappings to the universal covering space, the connectedness of the covering space, an application of Ekeland’s variational principle and a certain tangential $${\mathbb{A}}$$ -harmonic approximation lemma obtained directly via a Lipschitz approximation argument. This allows regularity to be established directly on the level of the gradient. Several applications to variational problems in condensed matter physics with broken symmetries are also discussed, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals.

## Classical Limit for a System of Non-Linear Random Schrödinger Equations

### Archive for Rational Mechanics and Analysis (2013-07-01) 209: 321-364 , July 01, 2013

This work is concerned with the semi-classical analysis of mixed state solutions to a Schrödinger–Position equation perturbed by a random potential with weak amplitude and fast oscillations in time and space. We show that the Wigner transform of the density matrix converges weakly and in probability to solutions of a Vlasov–Poisson–Boltzmann equation with a linear collision kernel.Aconsequence of this result is that a smooth non-linearity such as the Poisson potential (repulsive or attractive) does not change the statistical stability property of the Wigner transform observed in linear problems.We obtain, in addition, that the local density and current are self-averaging, which is of importance for some imaging problems in random media. The proof brings together the martingale method for stochastic equations with compactness techniques for non-linear PDEs in a semi-classical regime. It relies partly on the derivation of an energy estimate that is straightforward in a deterministic setting but requires the use of a martingale formulation and well-chosen perturbed test functions in the random context.

## A macroscopic non-linear theory of magnetothermoelastic continua

### Archive for Rational Mechanics and Analysis (1977-03-01) 65: 1-24 , March 01, 1977

## Orientability and Energy Minimization in Liquid Crystal Models

### Archive for Rational Mechanics and Analysis (2011-11-01) 202: 493-535 , November 01, 2011

Uniaxial nematic liquid crystals are modelled in the Oseen–Frank theory through a unit vector field *n*. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which *n* should be equivalent to −*n*. This symmetry is preserved in the constrained Landau–de Gennes theory that works with the tensor
$${Q=s \left(n\otimes n-\frac{1}{3} Id\right)}$$
. We study the differences and the overlaps between the two theories. These depend on the regularity class used as well as on the topology of the underlying domain. We show that for simply-connected domains and in the natural energy class *W*^{1,2} the two theories coincide, but otherwise there can be differences between the two theories, which we identify. In the case of planar domains with holes and various boundary conditions, for the simplest form of the energy functional, we completely characterise the instances in which the predictions of the constrained Landau–de Gennes theory differ from those of the Oseen–Frank theory.

## Axisymmetric Solutions of the Euler Equations for Polytropic Gases

### Archive for Rational Mechanics and Analysis (1998-07-01) 142: 253-279 , July 01, 1998

### Abstract.

We construct rigorously a three‐parameter family of self‐similar, globally bounded, and continuous weak solutions in two space dimensions for all positive time to the Euler equations with axisymmetry for polytropic gases with a quadratic pressure‐density law. We use the axisymmetry and self‐similarity assumptions to reduce the equations to a system of three ordinary differential equations, from which we obtain detailed structures of solutions besides their existence. These solutions exhibit familiar structures seen in hurricanes and tornadoes. They all have finite local energy and vorticity with well‐defined initial and boundary values. These solutions include the one‐parameter family of explicit solutions reported in a recent article of ours.