Rectifiability of entropy defect measures in a micromagnetics model

We study the fine properties of a class of weak solutions u of the eikonal equation arising as asymptotic domain of a family of energy functionals introduced in [T. Rivière and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics, Comm. Pure Appl. Math. 54 2001, 3, 294–338]. In particular, we prove that the entropy defect measure associated to u is concentrated on a 1-rectifiable set, which detects the jump-type discontinuities of u.

On convergence of approximate solutions to the compressible Euler system

We consider a sequence of approximate solutions to the compressible Euler system admitting uniform energy bounds and/or satisfying the relevant field equations modulo an error vanishing in the asymptotic limit. We show that such a sequence either (i) converges strongly in the energy norm, or (ii) the limit is not a weak solution of the associated Euler system. This is in sharp contrast to the incompressible case, where (oscillatory) approximate solutions may converge weakly to solutions of the Euler system. Our approach leans on identifying a system of differential equations satisfied by the associated turbulent defect measure and showing that it only has a trivial solution.

Minimizing acceleration on the group of diffeomorphisms and its relaxation

We study a second-order variational problem on the group of diffeomorphisms of the interval [0, 1] endowed with a right-invariant Sobolev metric of order 2, which consists in the minimization of the acceleration. We compute the relaxation of the problem which involves the so-called Fisher–Rao functional, a convex functional on the space of measures. This relaxation enables the derivation of several optimality conditions and, in particular, a sufficient condition which guarantees that a given path of the initial problem is also a minimizer of the relaxed one. Based on these sufficient conditions, the main result is that, when the value of the (minimized) functional is small enough, the minimizers are classical, that is the defect measure vanishes.

Two scale defect measure and linear equations with oscillating coefficients

Microlocal measure ? is associated to a two-scale convergent sequence un over Rd with the limit u ? L2(Rd x Td), Td is a torus, to analyze possible strong limit. ? is an operator valued measure absolutely continuous with respect to the product of scalar microlocal defect measure and a measure on the d-dimensional torus. The result is applied to the first order linear PDE with the oscillating coefficients.

GLOBAL WEAK SOLUTIONS FOR THE RELATIVISTIC WATERBAG CONTINUUM

In this paper we consider the relativistic waterbag continuum which is a useful PDE for collisionless kinetic plasma modeling recently developed in Ref. 11. The waterbag representation of the statistical distribution function of particles can be viewed as a special class of exact weak solution of the Vlasov equation, allowing to reduce this latter into a set of hydrodynamic equations (with the complexity of a multi-fluid model) while keeping its kinetic features (Landau damping and nonlinear resonant wave-particle interaction). These models are very promising because they are very useful for analytical theory and numerical simulations of laser-plasma and gyrokinetic physics.10–16, 56, 57 The relativistic waterbag continuum is derived from two phase-space variable reductions of the relativistic Vlasov–Maxwell equations through the existence of two underlying exact invariants, one coming from physics properties of the dynamics is the canonical transverse momentum, and the second, named the "water-bag" and coming from geometric property of the phase-space is just the direct consequence of the Liouville Theorem. In this paper we prove the existence and uniqueness of global weak entropy solutions of the relativistic waterbag continuum. Existence is based on vanishing viscosity method and bounded variations (BV) estimates to get compactness while proof of uniqueness relies on kinetic formulation of the relativistic waterbag continuum and the associated kinetic entropy defect measure.