Mathematical Models and Methods in Applied Sciences

We describe the asymptotic states for the solutions of a nonlocal equation of evolutionary type, which have the physical meaning of the atom dislocation function in a periodic crystal. More precisely, we can describe accurately the “smoothing effect” on the dislocation function occurring slightly after a “particle collision” (roughly speaking, two opposite transitions layers average out) and, in this way, we can trap the atom dislocation function between a superposition of transition layers which, as time flows, approaches either a constant function or a single heteroclinic (depending on the algebraic properties of the orientations of the initial transition layers). The results are endowed with explicit and quantitative estimates and, as a byproduct, we show that the ODE systems of particles that govern the evolution of the transition layers does not admit stationary solutions (i.e. roughly speaking, transition layers always move).

For time-harmonic electromagnetic waves scattered by either perfectly conducting or dielectric bounded obstacles, we show that the fields depend holomorphically on the shape of the scatterer. In the presence of random geometrical perturbations, our results imply strong measurability of the fields, in weighted spaces in the exterior of the scatterer. These findings are key to prove dimension-independent convergence rates of sparse approximation techniques of polynomial chaos type for forward and inverse computational uncertainty quantification. Also, our shape-holomorphy results imply parsimonious approximate representations of the corresponding parametric solution families, which are produced, for example, by greedy strategies such as model order reduction or reduced basis approximations. Finally, the presently proved shape holomorphy results imply convergence of shape Taylor expansions of far-field patterns for fixed amplitude domain perturbations in a vicinity of the nominal domain, thereby extending the widely used asymptotic linearizations employed in first-order, second moment domain uncertainty quantification.

The incompressible Navier–Stokes–Fourier (INSF) system with viscous heating was first derived from the Boltzmann equation in the form of the diffusive scaling by Bardos–Levermore–Ukai–Yang [Kinetic equations: Fluid dynamical limits and viscous heating, Bull. Inst. Math. Acad. Sin.$(N.S.)$ 3 (2008) 1–49]. The purpose of this paper is to justify such an incompressible hydrodynamic approximation to the Boltzmann equation in $L^2\cap L^{\infty }$ setting in a periodic box. Based on an odd–even expansion of the solution with respect to the microscopic velocity, the diffusive coefficients are determined by the INSF system with viscous heating and the super-Burnett functions. More importantly, the remainder of the expansion is proven to decay exponentially in time via an $L^2-L^{\infty }$ approach on the condition that the initial data satisfies the mass, momentum and energy conversation laws.

We study the problem of optimal mixing of a passive scalar $\rho$ advected by an incompressible flow on the two-dimensional unit square. The scalar $\rho$ solves the continuity equation with a divergence-free velocity field $u$ with uniform-in-time bounds on the homogeneous Sobolev semi-norm ${Ẇ}^{s,p}$, where $s>1$ and $1<p\le \infty$. We measure the degree of mixedness of the tracer $\rho$ via the two different notions of mixing scale commonly used in this setting, namely the functional and the geometric mixing scales. For velocity fields with the above constraint, it is known that the decay of both mixing scales cannot be faster than exponential. Numerical simulations suggest that this exponential lower bound is in fact sharp, but so far there is no explicit analytical example which matches this result. We analyze velocity fields of $cellulartype$, which is a special localized structure often used in constructions of explicit analytical examples of mixing flows and can be viewed as a generalization of the self-similar construction by Alberti, Crippa and Mazzucato [Exponential self-similar mixing by incompressible flows, arXiv:1605.02090]. We show that for any velocity field of cellular type both mixing scales cannot decay faster than polynomially.

We study the nonlinear stability of a composite wave pattern, which is a combination of a viscous contact wave with a rarefaction wave, to the Cauchy problem of one-dimensional compressible Navier–Stokes equations for a viscous and heat conducting ideal polytropic gas with large initial perturbation when the transport coefficients depend on both temperature and density. Our main idea is to use the “smallness mechanism” induced by the structures of the equations under consideration and the smallness of the strengths of the two elementary waves to control the possible growth of the solutions caused by the nonlinearities of the equations, the interactions between the solutions themselves and the wave pattern, and the interactions of waves between different families. The main ingredient in the analysis is to derive the uniform positive lower and upper bounds on the specific volume and the temperature.