Geometrically Constrained Walls in Two Dimensions

We address the effect of extreme geometry on a non-convex variational problem, motivated by studies on magnetic domain walls trapped by thin necks. The recent analytical results of Kohn and Slastikov (Calc. Var. Partial Differ. Equ. 28:33–57, 2007) revealed a variety of magnetic structures in three-dimensional ferromagnets depending on the size of the constriction. The main purpose of this paper is to study geometrically constrained walls in two dimensions. The analysis turns out to be significantly more challenging and requires the use of different techniques. In particular, the purely variational point of view of Kohn and Slastikov (loc. cit.) cannot be adopted in the present setting and is here replaced by a PDE approach. The existence of local minimizers representing geometrically constrained walls is proven under suitable symmetry assumptions on the domains and an asymptotic characterization of the wall profile is given. The limiting behavior, which depends critically on the scaling of length and height of the neck, turns out to be more complex than in the higher-dimensional case and a richer variety of regimes is shown to exist.     

The Quantum Hydrodynamics System in Two Space Dimensions

In this paper we study global existence of weak solutions for the quantum hydrodynamics system in two-dimensional energy space. We do not require any additional regularity and/or smallness assumptions on the initial data. Our approach replaces the WKB formalism with a polar decomposition theory which is not limited by the presence of vacuum regions. In this way we set up a self consistent theory, based only on particle density and current density, which does not need to define velocity fields in the nodal regions. The mathematical techniques we use in this paper are based on uniform (with respect to the approximating parameter) Strichartz estimates and the local smoothing property.     

Symmetry of Traveling Wave Solutions to the Allen–Cahn Equation in $${{\mathbb R^{2}}}$$

In this paper, we prove even symmetry of monotone traveling wave solutions to the balanced Allen–Cahn equation in the entire plane. Related results for the unbalanced Allen–Cahn equation are also discussed.     

Regularity and Singularities of Optimal Convex Shapes in the Plane

We focus here on the analysis of the regularity or singularity of solutions Ω ₀ to shape optimization problems among convex planar sets, namely:

$J(\Omega_{0})={\rm min} \{J(\Omega), \Omega \quad {\rm convex},\Omega \in \mathcal{S}_{\rm ad}\},$

where \({\mathcal{S}_{\rm ad}}\) is a set of 2-dimensional admissible shapes and \({J:\mathcal{S}_{\rm ad}\rightarrow\mathbb{R}}\) is a shape functional. Our main goal is to obtain qualitative properties of these optimal shapes by using first and second order optimality conditions, including the infinite dimensional Lagrange multiplier due to the convexity constraint. We prove two types of results:

1. i)
   under a suitable convexity property of the functional J, we prove that Ω ₀ is a W ^2,p -set, \({p\in[1, \infty]}\). This result applies, for instance, with p = ∞ when the shape functional can be written as J(Ω) = R(Ω) + P(Ω), where R(Ω) = F(|Ω|, E _f (Ω), λ₁(Ω)) involves the area |Ω|, the Dirichlet energy E _f (Ω) or the first eigenvalue of the Laplace–Dirichlet operator λ₁(Ω), and P(Ω) is the perimeter of Ω;
    

1. ii)
   under a suitable concavity assumption on the functional J, we prove that Ω ₀ is a polygon. This result applies, for instance, when the functional is now written as J(Ω) = R(Ω) ? P(Ω), with the same notations as above.
    

     

Global Classical Solutions of the Relativistic Vlasov–Darwin System with Small Cauchy Data: The Generalized Variables Approach

We show that a smooth, small enough Cauchy datum launches a unique classical solution of the relativistic Vlasov–Darwin (RVD) system globally in time. A similar result is claimed in Seehafer (Commun Math Sci 6:749–769, 2008) following the work in Pallard (Int Mat Res Not 57191:1–31, 2006). Our proof does not require estimates derived from the conservation of the total energy, nor those previously given on the transversal component of the electric field. These estimates are crucial in the references cited above. Instead, we exploit the formulation of the RVD system in terms of the generalized space and momentum variables. By doing so, we produce a simple a priori estimate on the transversal component of the electric field. We widen the functional space required for the Cauchy datum to extend the solution globally in time, and we improve decay estimates given in Seehafer (2008) on the electromagnetic field and its space derivatives. Our method extends the constructive proof presented in Rein (Handbook of differential equations: evolutionary equations, vol 3. Elsevier, Amsterdam, 2007) to solve the Cauchy problem for the Vlasov–Poisson system with a small initial datum.     

Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum

We prove well-posedness for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) =  C _γ ρ ^γ for γ > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein the density vanishes on the free-boundary, the uniform Kreiss–Lopatinskii condition is violated, and manifest derivative loss ensues. Nevertheless, we are able to establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary, and our estimates have no derivative loss with respect to initial data. Our proof is founded on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term, chosen to preserve as much of the geometric structure of the Euler equations as possible. We first construct solutions to this degenerate parabolic regularization using a higher-order version of Hardy’s inequality; we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter. Solutions to the compressible Euler equations are found in the limit as the artificial viscosity tends to zero. Our regular solutions can be viewed as degenerate viscosity solutions. Our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws.     

A Rigorous Derivation of the Coupling of a Kinetic Equation and Burgers’ Equation

In this article, we investigate the kinetic/fluid coupling on a toy model, which we obtain rigorously from a hydrodynamical limit. The idea is that at the level of the full kinetic model, the coupling is obvious. We then investigate the coupling obtained when passing into the limit. We show that, especially in presence of a shock stuck on the interface, the coupling involves a kinetic layer known as the Milne problem. Due to this layer, the limit process is quite delicate and some blow-up techniques are needed to ensure its strong convergence.     

Stochastic Three-Dimensional Rotating Navier–Stokes Equations: Averaging, Convergence and Regularity

We consider stochastic three-dimensional rotating Navier?CStokes equations and prove averaging theorems for stochastic problems in the case of strong rotation. Regularity results are established by bootstrapping from global regularity of the limit stochastic equations and convergence theorems.     

Γ-Convergence Analysis of Systems of Edge Dislocations: the Self Energy Regime

This paper deals with the elastic energy induced by systems of straight edge dislocations in the framework of linearized plane elasticity. The dislocations are introduced as point topological defects of the displacement-gradient fields. Following the core radius approach, we introduce a parameter ${\varepsilon > 0}$ representing the lattice spacing of the crystal, we remove a disc of radius ${\varepsilon}$ around each dislocation and compute the elastic energy stored outside the union of such discs, namely outside the core region. Then, we analyze the asymptotic behaviour of the elastic energy as ${\varepsilon \rightarrow 0}$ , in terms of Γ-convergence. We focus on the self energy regime of order ${\log\frac{1}{\varepsilon}}$ ; we show that configurations with logarithmic diverging energy converge, up to a subsequence, to a finite number of multiple dislocations and we compute the corresponding Γ-limit.     

Statistical Foundations of Liquid-Crystal Theory. I: Discrete Systems of Rod-Like Molecules

We develop a mechanical theory for systems of rod-like particles. Central to our approach is the assumption that the external power expenditure for any subsystem of rods is independent of the underlying frame of reference. This assumption is used to derive the basic balance laws for forces and torques. By considering inertial forces on par with other forces, these laws hold relative to any frame of reference, inertial or noninertial. Finally, we introduce a simple set of constitutive relations to govern the interactions between rods and find restrictions necessary and sufficient for these laws to be consistent with thermodynamics. Our framework provides a foundation for a statistical mechanical derivation of the macroscopic balance laws governing liquid crystals.