In this work, we study the quasineutral limit of the one-dimensional Vlasov-Poisson equation for ions with massless thermalized electrons. We prove new weak-strong stability estimates in the Wasserstein metric that allow us to extend and improve previously known convergence results. In particular, we show that given a possibly unstable analytic initial profile, the formal limit holds for sequences of measure initial data converging sufficiently fast in the Wasserstein metric to this profile. This is achieved without assuming uniform analytic regularity.

In this paper we study the non-relativistic and low Mach num-ber limits of two P1 approximation model arising in radiation hydrodynamics in T^3, i.e. the barotropic model and the Navier-Stokes-Fourier model. For the barotropic model, we consider the case that the initial data is a small perturbation of stable equilbria while for the Navier-Stokes-Fourier model, we consider the case that the initial data is large. For both models, we prove the convergence to the solution of the incompressible Navier-Stokes equations with/without stationary transport equations.

We study Gaussian approximations to the distribution of a diffusion. The approximations are easy to compute: they are defined by two simple ordinary differential equations for the mean and the covariance. Time correlations can also be computed via solution of a linear stochastic differential equation. We show, using the Kullback-Leibler divergence, that the approximations are accurate in the small noise regime. An analogous discrete time setting is also studied. The results provide both theoretical support for the use of Gaussian processes in the approximation of diffusions, and methodological guidance in the construction of Gaussian approximations in applications.

We study the variations of the principal eigenvalue associated to a growth-fragmentation-death equation with respect to a parameter acting on growth and fragmentation. To this aim, we use the probabilistic individual-based interpretation of the model. We study the variations of the survival probability of the stochastic model, using a generation by generation approach. Then, making use of the link between the survival probability and the principal eigenvalue established in a previous work, we deduce the variations of the eigenvalue with respect to the parameter of the model.

This article proposes a derivation of the Vlasov-Navier-Stokes system for spray/aerosol flows. The distribution function of the dispersed phase is governed by a Vlasov-equation, while the velocity field of the propellant satisfies the Navier-Stokes equations for incompressible fluids. The dynamics of the dispersed phase and of the propellant are coupled through the drag force exerted by the propellant on the dispersed phase. We present a formal derivation of this model from a multiphase Boltzmann system for a binary gaseous mixture, involving the droplets/dust particles in the dispersed phase as one species, and the gas molecules as the other species. Under suitable assumptions on the collision kernels, we prove that the sequences of solutions to the multiphase Boltzmann system converge to distributional solutions to the Vlasov-Navier-Stokes equation in some appropriate distinguished scaling limit. Specifically, we assume (a) that the mass ratio of the gas molecules to the dust particles/droplets is small, (b) that the thermal speed of the dust particles/droplets is much smaller than that of the gas molecules and (c) that the mass density of the gas and of the dispersed phase are of the same order of magnitude. The class of kernels modelling the interaction between the dispersed phase and the gas includes, among others, elastic collisions and inelastic collisions of the type introduced in [F. Charles: in “Proceedings of the 26th International Symposium on Rarefied Gas Dynamics”, AIP Conf. Proc. 1084, (2008), 409-414].

In this paper, we investigate the integrability and asymptotic behaviors of positive solutions for a nonlinear integral system in a functional setting. Using the regularity lifting lemma and some delicate analysis techniques, we obtain the optimal integral intervals and the asymptotic estimates for such solutions around the origin and near infinity. Moreover, the index of regular solution is distinct from one of the previous several related systems.

In this paper we consider a tensor version of Cahn-Hilliard system in the context of incompatible linearized elasticity. We interpret the trace of the incompatible strain as the density of point defects and propose a model for point defects collapse into dislocation loops in single crystals as based on Cahn-Hilliard dynamics. By means of a asymptotic analysis, we determine the front dynamics and obtain as a result a tensor version of Mullins-Sekerka law which allows for flat interfaces, i.e., such as polygonization.

Ill-posed inverse problems are ubiquitous in applications. Understanding of algorithms for their solution has been greatly enhanced by a deep understanding of the linear inverse problem. In the applied communities ensemble-based filtering methods have recently been used to solve inverse problems by introducing an artificial dynamical system. This opens up the possibility of using a range of other filtering methods, such as 3DVAR and Kalman based methods, to solve inverse problems, again by introducing an artificial dynamical system. The aim of this paper is to analyze such methods in the context of the linear inverse problem. Statistical linear inverse problems are studied in the sense that the observational noise is assumed to be derived via realization of a Gaussian random variable. We investigate the asymptotic behavior of filter based methods for these inverse problems. Rigorous convergence rates are established for 3DVAR and for the Kalman filters, including minimax rates in some instances. Blowup of 3DVAR and a variant of its basic form is also presented, and optimality of the Kalman filter is discussed. These analyses reveal a close connection between (iterated) regularization schemes in deterministic inverse problems and filter based methods in data assimilation. Numerical experiments are presented to illustrate the theory.

In this paper, the $m$th order infinite dimensional Hilbert tensor (hypermatrix) is intrduced to define an $(m-1)$-homogeneous operator on the spaces of analytic functions, which is called Hilbert tensor operator. The boundedness of Hilbert tensor operator is presented on Bergman spaces $A^p$ ($p>2(m-1)$). On the base of the boundedness, two positively homogeneous operators are introduced to the spaces of analytic functions, and hence the upper bounds of norm of such two operators are found on Bergman spaces $A^p$ ($p>2(m-1)$). In particular, the norms of such two operators on Bergman spaces $A^{4(m-1)}$ are smaller than or equal to $\pi$ and $\pi^\frac1{m-1}$, respectively.

A mathematical model for tissue growth is considered. This model describes the dynamics of the density of cells due to pressure forces and proliferation. It is known that some cell population models of this kind converge at the incompressible limit towards a Hele-Shaw type free boundary problem. The novelty of this work is to impose a non-overlapping constraint. This constraint is important to be satisfied in many applications. One way to guarantee this non-overlapping constraint is to choose a singular pressure law. The aim of this paper is to prove that, although the pressure law has a singularity, the incompressible limit leads to the same Hele-Shaw free boundary problem.

We consider the one-dimensional squared Bessel process given by the stochastic differential equation (SDE): dX_t = 1 dt + 2\sqrt{X_t} dW_t, X+0=x_0, t\in [0,1], and study strong (pathwise) approximation of the solution X at the final time point t=1. This SDE is a particular instance of a Cox-Ingersoll-Ross (CIR) process where the boundary point zero is accessible. We consider numerical methods that have access to values of the driving Brownian motion W at a finite number of time points. We show that the polynomial convergence rate of the n-th minimal errors for the class of adaptive algorithms as well as for the class of algorithms that rely on equidistant grids are equal to infinity and 1/2, respectively. This shows that adaption results in a tremendously improved convergence rate. As a by-product, we obtain that the parameters appearing in the CIR process affect the convergence rate of strong approximation.

We consider the dynamics of nonlinear Schrodinger equations with strong constant magnetic fields. In an asymptotic scaling limit the system exhibits a purely magnetic confinement, based on the spectral properties of the Landau Hamiltonian. Using an averaging technique we derive an associated effective description via an averaged model of nonlinear Schrodinger type. In a special case this also yields a derivation of the LLL equation.

We study the global existence and asymptotic behavior of smooth solutions near a non-flat steady state to the compressible non-isentropic Euler-Poisson system in R^3. Using some concise energy estimates and an interpolation trick, we show that the solution converges to the stationary solution exponentially fast. Here our results can follow from that the H^3 norms of the initial density, velocity and temperature are small. In this sense, we reduce the regularity of the initial temperature in [Y.P. Li, J. Differential Equations 225 (2006) 134-167].

In this study, we reveal an approximate linear relation for Bessel functions of the first kind, based on asymptotic analyses. A set of coefficients are calculated from a linear algebraic system. For any given error tolerance, a Bessel function of an order big enough is approximated by a linear combination of those with neighboring orders using these coefficients. This naturally leads to a class of ALmost EXact (ALEX) boundary conditions in atomic and multiscale simulations.

Parameterization, a process of mapping a complicated domain onto a simple canonical domain, is crucial in different areas such as computer graphics, medical imaging and scientific computing. Conformal parameterization has been widely used since it preserves the local geometry well. However, a major drawback is the area distortion introduced by the conformal parameterization, causing inconvenience in many applications such as texture mapping in computer graphics or visualization in medical imaging. This work proposes a remedy to construct a parameterization that balances between conformality and area distortions. We present a variational algorithm to compute the optimized quasiconformal parameterization with controllable area distortions. The distribution of the area distortion can be prescribed by users according to the application. The main strategy is to minimize a combined energy functional consisting of an area mismatching term and a regularization term involving the Beltrami coefficient of the map. The Beltrami coefficient controls the conformality of the parameterization. Landmark constraints can be incorporated into the model to obtain landmarkaligned parameterization. Experiments have been carried out on both synthetic and real data. Results demonstrate the efficacy of the proposed algorithm to compute the optimized parameterization with controllable area distortion while preserving the local geometry as well as possible.

Metastable dynamics of a hyperbolic variation of the Allen-Cahn equation with homogeneous Neumann boundary conditions are considered. Using the "dynamical approach" proposed by Carr-Pego and Fusco-Hale to study slow-evolution of solutions in the classic parabolic case, we prove existence and persistence of metastable patterns for an exponentially long time. In particular, we show the existence of an "approximately invariant" N-dimensional manifold M_0 for the hyperbolic Allen–Cahn equation: if the initial datum is in a tubular neighborhood of M_0, the solution remains in such neighborhood for an exponentially long time. Moreover, the solution has N transition layers and the transition points move with exponentially small velocity. In addition, we determine the explicit form of a system of ordinary differential equations describing the motion of the transition layers and we analyze the differences with the corresponding motion valid for the parabolic case.

We present results on breather solutions of a discrete nonlinear Schrodinger equation with a cubic Hartree-type nonlinearity that models laser light propagation in waveguide arrays that use a nematic liquid crystal substratum. A recent study of that model by Ben et al showed that nonlocality leads to some novel properties such as the existence of orbitaly stable breathers with internal modes, and of shelf-like configurations with maxima at the interface. In this work we present rigorous results on these phenomena and consider some more general solutions. First, we study energy minimizing breathers, showing existence as well as symmetry and monotonicity properties. We also prove results on the spectrum of the linearization around one-peak breathers, solutions that are expected to coincide with minimizers in the regime of small linear intersite coupling. A second set of results concerns shelf-type breather solutions that may be thought of as limits of solutions examined in Ben et. al. We show the existence of solutions with a non-monotonic front-like shape and justify computations of the essential spectrum of the linearization around these solutions in the local and nonlocal cases.

We prove the existence of weak solutions to a kinetic flocking model with cut-off interaction function by using the weak convergence method. Under the natural assumption that the v-support of the initial distribution function f_0(x,v) is bounded, we show that the v-support of the distribution function f(t,x,v) is uniformly bounded in time. Employing this property, we remove the constraint in the paper of Karper, Mellet and Trivisa[SIAM. J. Math. Anal., (45)2013, pp.215-243] that the initial distribution function should have better integrability for large |x|.