Acta Applicandae Mathematicae最新文献

The time-dependent Ginzburg-Landau (TDGL) model requires the choice of a gauge for the problem to be mathematically well-posed. In the literature, three gauges are commonly used: the Coulomb gauge, the Lorenz gauge and the temporal gauge. It has been noticed (J. Fleckinger-Pellé et al. in Dynamics of the Ginzburg-Landau equations of superconductivity, Technical report, Argonne National Lab. (ANL), Argonne, IL, United States, 1997) that these gauges can be continuously related by a single parameter considering the more general (omega )-gauge, where (omega ) is a non-negative real parameter. In this article, we study the influence of the gauge parameter (omega ) on the convergence of numerical simulations of the TDGL model using finite element schemes. A classical benchmark is first analysed for different values of (omega ) and artefacts are observed for lower values of (omega ). Then, we relate these observations with a systematic study of convergence orders in the unified (omega )-gauge framework. In particular, we show the existence of a tipping point value for (omega ), separating optimal convergence behaviour and a degenerate one. We find that numerical artefacts are correlated to the degeneracy of the convergence order of the method and we suggest strategies to avoid such undesirable effects. New 3D configurations are also investigated (the sphere with or without geometrical defect).

Linear response theory is a fundamental framework studying the macroscopic response of a physical system to an external perturbation. This paper focuses on the rigorous mathematical justification of linear response theory for Langevin dynamics. We give some equivalent characterizations for reversible overdamped/underdamped Langevin dynamics, which is the unperturbed reference system. Then we clarify sufficient conditions for the smoothness and exponential convergence to the invariant measure for the overdamped case. We also clarify those sufficient conditions for the underdamped case, which corresponds to hypoellipticity and hypocoercivity. Based on these, the asymptotic dependence of the response function on the small perturbation is proved in both finite and infinite time horizons. As applications, Green-Kubo relations and linear response theory for a generalized Langevin dynamics are also proved in a rigorous fashion.

We study a reduced order model (ROM) based waveform inversion method applied to a Helmholtz problem with impedance boundary conditions and variable refractive index. The first goal of this paper is to obtain relations that allow the reconstruction of the Galerkin projection of the continuous problem onto the space spanned by solutions of the Helmholtz equation. The second goal is to study the introduced nonlinear optimization method based on the ROM aimed to estimate the refractive index from reflection and transmission data. Finally we compare numerically our method to the conventional least squares inversion based on minimizing the distance between modelled to measured data.

In this paper we consider radially symmetric solutions of the following parabolic-elliptic cross-diffusion system

in (Omega times (0,infty )), with (Omega ) a ball in (mathbb{R}^{N}), (Ngeq 1) under homogeneous Neumann boundary conditions, (g(u)) a regular function with the prototype (g(u)= u^{k}), (ugeq 0), (k>0). The function (f(xi ) = k_{f} (1+ xi )^{-alpha }), (k_{f} >0), describes gradient-dependent limitation of cross diffusion fluxes. Under suitable conditions on the data, we prove that the solution is global in time. If (Ngeq 3), under conditions on (f), (g) and initial data, we prove that if the solution (u(x,t)) blows up in (L^{infty })-norm at finite time (T_{max}) then for some (p>1) it blows up also in (L^{p})-norm. Moreover a lower bound of blow-up time is derived.

In this paper we propose some Harris-like criteria in order to study the long time behavior of general positive and periodic semiflows. These criteria allow us to obtain new existence results of principal eigenelements, and their exponential attractiveness. We present applications to two biological models in a space-time varying environment: a non local selection-mutation equation and a growth-fragmentation equation. The particularity of this article is to study some inhomogeneous problems that are periodic in time, as it appears for instance when the environment changes, due for instance to the seasonal cycle or circadian rhythms.

In this paper, we consider Cauchy problem for the 2D incompressible Keller-Segel-Navier-Stokes equations with the fractional diffusion

where (wedge :=(-Delta )^{frac{1}{2}}) and (alpha in [frac{1}{2},1]). We get the global well-posedness for the above system with the rough initial data by a new priori estimate of the solutions.

When taking a regular planar polygon of (M) sides and length (2pi ) as the initial datum of the vortex filament equation, (mathbf{X}_{t}= mathbf{X}_{s}wedge mathbf{X}_{ss}), the solution becomes polygonal at times of the form (t_{pq} = (p/q)(2pi /M^{2})), with (gcd (p,q)=1), and the corresponding polygon has (Mq) sides, if (q) is odd, and (Mq/2) sides, if (q) is even. Moreover, that polygon is skew (except when (q = 1) or (q = 2), where the initial shape is recovered), and the angle (rho ) between two adjacent sides is a constant. In this paper, we give a rigorous proof of the conjecture that states that, at a time (t_{pq}), (cos ^{q}(rho /2) = cos (pi /M)), if (q) is odd, and (cos ^{q}(rho /2) = cos ^{2}(pi /M)), if (q) is even. Since the transition of one side of the polygon to the next one is given by a rotation in (mathbb{R}^{3}) determined by a generalized Gauss sum, the idea of the proof consists in showing that a certain product of those rotations is a rotation of angle (2pi /M), which is equivalent to proving that some exponential sums with arithmetic content are purely imaginary.

Total (absolute) curvature is defined for any curve in a metric space. Its properties, finiteness, local boundedness, Lipschitz continuity, depending whether there are satisfied or not, permit a classification of curves alternative to the classical regularity classes. In this paper, we are mainly interested in the total curvature estimation. Under the sole assumption of curve simpleness, we prove the convergence, as (epsilon to 0), of the naive turn estimators which are families of polygonal lines whose vertices are at distance at most (epsilon ) from the curve and whose edges are in (Omega (epsilon ^{alpha })cap text{O}(epsilon ^{beta })) with (0<beta le alpha <frac{1}{2}). Besides, we give lower bounds of the speed of convergence under an additional assumption that can be summarized as being “convex-or-Lipschitz”.

The multispecies Landau collision operator describes the two-particle, small scattering angle or grazing collisions in a plasma made up of different species of particles such as electrons and ions. Recently, a structure preserving deterministic particle method (Carrillo et al. in J. Comput. Phys. 7:100066, 2020) has been developed for the single species spatially homogeneous Landau equation. This method relies on a regularization of the Landau collision operator so that an approximate solution, which is a linear combination of Dirac delta distributions, is well-defined. Based on a weak form of the regularized Landau equation, the time dependent locations of the Dirac delta functions satisfy a system of ordinary differential equations. In this work, we extend this particle method to the multispecies case, and examine its conservation of mass, momentum, and energy, and decay of entropy properties. We show that the equilibrium distribution of the regularized multispecies Landau equation is a Maxwellian distribution, and state a critical condition on the regularization parameters that guarantees a species independent equilibrium temperature. A convergence study comparing an exact multispecies Bobylev-Krook-Wu (BKW) solution to the particle solution shows approximately 2nd order accuracy. Important physical properties such as conservation, decay of entropy, and equilibrium distribution of the particle method are demonstrated with several numerical examples.

Recently, lots of studies demonstrated that the signals are not only sparse in some system (e.g. shearlets) but also reveal a certain structure such as sparsity in levels. Therefore, sampling strategy is designed as a variable subsampling strategy in order to use this extra structure, for example magnetic resonance imaging (MRI) and etc. In this paper, we investigate the uniform recovery guarantees on the signals which possess sparsity in levels with respect to a general dual frame. First, we prove that the stable and robust recovery is possible when the weighted (l^{2} )-robust null space property in levels is satisfied. Second, we establish sufficient conditions under which subsampled isometry satisfies the weighted (l^{2} )-robust null space property in levels.