Lipid vesicles appear ubiquitously in biological systems. Understanding how the mechanical and intermolecular interactions deform vesicle membranes is a fundamental question in biophysics. In this article we develop a fast algorithm to compute the surface configurations of lipid vesicles by introducing surface harmonic functions to approximate themembrane surface. This parameterization allows an analytical computation of the membrane curvature energy and its gradient for the efficient minimization of the curvature energy using a nonlinear conjugate gradient method. Our approach drastically reduces the degrees of freedom for approximating the membrane surfaces compared to the previously developed finite element and finite difference methods. Vesicle deformations with a reduced volume larger than 0.65 can be well approximated by using as small as 49 surface harmonic functions. The method thus has a great potential to reduce the computational expense of tracking multiple vesicles which deform for their interaction with external fields.

The purpose of this paper is to provide a large class of initial data which generates global smooth solution of the 3D inhomogeneous incompressible Navier–Stokes system in the whole space $\mathbb{R}^{3}$. This class of data is based on functions which vary slowly in one direction. The idea is that 2D inhomogeneous Navier–Stokes system with large data is globally well-posed and we construct the 3D approximate solutions by the 2D solutions with a parameter. One of the key point of this study is the investigation of the time decay properties of the solutions to the 2D inhomogeneous Navier–Stokes system. We obtained the same optimal decay estimates as the solutions of 2D homogeneous Navier–Stokes system.

We study two different heat-type equations. First, global well-posedness in the energy space of some high-order semilinear heat-type equation with exponential nonlinearity is obtained for even space dimensions. Second, a finite-time blow-up result for the critical monomial focusing heat equation with the p-Laplacian is proved.

In this paper, the second order convergence of the interpolation based on -element is derived in the case of d=1, 2 and 3. Using the integral average on each element, the new basis functions of tensor product type is builded up and we can easily extend it to the higher dimensional case. Finally, some numerical tests are made to show the analytical results of the interpolation errors.

This paper is concerned with the modified Wigner (respectively, Wigner–Fokker–Planck) Poisson equation. The quantum mechanical model describes the transport of charged particles under the influence of the modified Poisson potential field without (respectively, with) the collision operator. Existence and uniqueness of a global mild solution to the initial boundary value problem in one dimension are established on a weighted $L^{2}$ -space. The main difficulties are to derive a priori estimates on the modified Poisson equation and prove the Lipschitz properties of the appropriate potential term.

The paper presents the size-dependant behaviors of the carbon nanotubes under electrostatic actuation using the modified couple stress theory and homotopy perturbation method. Due to the less accuracy of the classical elasticity theorems, the modified couple stress theory is applied in order to capture the size-dependant properties of the carbon nanotubes. Both of the static and dynamic behaviors under static DC and step DC voltages are discussed. The effects of various dimensions and boundary conditions on the deflection and pull-in voltages of the carbon nanotubes are to be investigated in detail via application of the homotopy perturbation method to solve the nonlinear governing equations semi-analytically.

Removing geometric details from the computational domain can significantly reduce the complexity of downstream task of meshing and simulation computation, and increase their stability. Proper estimation of the sensitivity analysis error induced by removing such domain details, called defeaturing errors, can ensure that the sensitivity analysis fidelity can still be met after simplification. In this paper, estimation of impacts of removing arbitrarily constrained domain details to the analysis of incompressible fluid flows is studied with applications to fast analysis of incompressible fluid flows in complex environments. The derived error estimator is applicable to geometric details constrained by either Dirichlet or Neumann boundary conditions, and has no special requirements on the outer boundary conditions. Extensive numerical examples were presented to demonstrate the effectiveness and efficiency of the proposed error estimator.

Dynamical system theory is applied to the integrable nonlinear wave equation ut ±(u 3–u 2)x +(u 3)xxx =0. We obtain the single peak solitary wave solutions and compacton solutions of the equation. Regular compacton solution of the equation correspond to the case of wave speed c=0. In the case of c≠0, we find smooth soliton solutions. The influence of parameters of the traveling wave solutions is explored by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for these soliton solutions of the nonlinear wave equation.

In this paper, the idea of a combination of variable separation approach and the extended homoclinic test approach is proposed to seek non-travelling wave solutions of Calogero equation. The equation is reduced to some (1+1)-dimensional nonlinear equations by applying the variable separation approach and solves reduced equations with the extended homoclinic test technique. Based on this idea and with the aid of symbolic computation, some new explicit solutions can be obtained.

In this paper, we study the Cauchy problem for the semilinear heat and Schrödinger equations, with the nonlinear term $f(u)=\unicode[STIX]{x1D706}|u|^{\unicode[STIX]{x1D6FC}}u$ . We show that low regularity of $f$ (i.e., $\unicode[STIX]{x1D6FC}>0$ but small) limits the regularity of any possible solution for a certain class of smooth initial data. We employ two different methods, which yield two different types of results. On the one hand, we consider the semilinear equation as a perturbation of the ODE $w_{t}=f(w)$ . This yields, in particular, an optimal regularity result for the semilinear heat equation in Hölder spaces. In addition, this approach yields ill-posedness results for the nonlinear Schrödinger equation in certain $H^{s}$ -spaces, which depend on the smallness of $\unicode[STIX]{x1D6FC}$ rather than the scaling properties of the equation. Our second method is to consider the semilinear equation as a perturbation of the linear equation via Duhamel’s formula. This yields, in particular, that if $\unicode[STIX]{x1D6FC}$ is sufficiently small and $N$ is sufficiently large, then the nonlinear heat equation is ill-posed in $H^{s}(\mathbb{R}^{N})$ for all $s\geqslant 0$ .

Extended hydrodynamic models for carrier transport are derived from the semiconductor Boltzmann equation with relaxation time approximation of the scattering term, by using the globally hyperbolic moment method and the moment-dependent relaxation time. Incorporating the microscopic relaxation time and the applied voltage bias, a formula is proposed to determine the relaxation time for each moment equation, which sets different relaxation rates for different moments such that higher moments damp faster. The resulting models would give more satisfactory results of macroscopic quantities of interest with a high-order convergence to those of the underlying Boltzmann equation as the involved moments increase, in comparison to the corresponding moment models using a single relaxation time. In order to simulate the steady states efficiently, a multigrid solver is developed for the derived moment models. Numerical simulations of an n +-n-n + silicon diode are carried out to demonstrate the validation of the presented moment models, and the robustness and efficiency of the designed multigrid solver.

We study the exciton diffusion in organic semiconductors from a macroscopic viewpoint. In a unified way, we conduct the equivalence analysis between Monte-Carlo method and diffusion equation model for photoluminescence quenching and photocurrent spectrum measurements, in both the presence and the absence of Förster energy transfer effect. Connections of these two models to Stern-Volmer method and exciton-exciton annihilation method are also specified for the photoluminescence quenching measurement.

This paper develops the theory of multisymplectic variational integrators for nonsmooth continuum mechanics with constraints. Typical problems are the impact of an elastic body on a rigid plate or the collision of two elastic bodies. The integrators are obtained by combining, at the continuous and discrete levels, the variational multisymplectic formulation of nonsmooth continuum mechanics with the generalized Lagrange multiplier approach for optimization problems with nonsmooth constraints. These integrators verify a spacetime multisymplectic formula that generalizes the symplectic property of time integrators. In addition, they preserve the energy during the impact. In the presence of symmetry, a discrete version of the Noether theorem is verified. All these properties are inherited from the variational character of the integrator. Numerical illustrations are presented.

The self-similar growth-fragmentation equation describes the evolution of a medium in which particles grow and divide as time proceeds, with the growth and splitting of each particle depending only upon its size. The critical case of the equation, in which the growth and division rates balance one another, was considered in Doumic and Escobedo (2015) for the homogeneous case where the rates do not depend on the particle size. Here, we study the general self-similar case, using a probabilistic approach based on Lévy processes and positive self-similar Markov processes which also permits us to analyse quite general splitting rates. Whereas existence and uniqueness of the solution are rather easy to establish in the homogeneous case, the equation in the nonhomogeneous case has some surprising features. In particular, using the fact that certain self-similar Markov processes can enter (0,∞) continuously from either 0 or ∞, we exhibit unexpected spontaneous generation of mass in the solutions.

We consider the zero-resistivity limit for Hasegawa–Wakatani equations in a cylindrical domain when the initial data are Stepanov almost-periodic in the axial direction. First, we prove the existence of a solution to Hasegawa–Wakatani equations with zero resistivity; second, we obtain uniform a priori estimates with respect to resistivity. Such estimates can be obtained in the same way as for our previous results; therefore, the most important contribution of this paper is the proof of the existence of a local-in-time solution to Hasegawa–Wakatani equations with zero resistivity. We apply the theory of Bohr–Fourier series of Stepanov almost-periodic functions to such a proof.

We construct and analyze conservative local discontinuous Galerkin (LDG) methods for the Generalized Korteweg-de-Vries equation. LDG methods are designed by writing the equation as a system and performing separate approximations to the spatial derivatives. The main focus is on the development of conservative methods which can preserve discrete versions of the first two invariants of the continuous solution, and a posteriori error estimates for a fully discrete approximation that is based on the idea of dispersive reconstruction. Numerical experiments are provided to verify the theoretical estimates.

Integrable couplings of the Boiti-Pempinelli-Tu hierarchy are constructed by a class of non-semisimple block matrix loop algebras. Further, through using the variational identity theory, the Hamiltonian structures of those integrable couplings are obtained. The method can be applied to obtain other integrable hierarchies.

It is known that large time-stepping method are useful for simulating phase field models. In this work, an adaptive time-stepping strategy is proposed based on numerical energy stability and equi-distribution principle. The main idea is to use the energy variation as an indicator to update the time step, so that the resulting algorithm is free of user-defined parameters, which is different from several existing approaches. Some numerical experiments are presented to illustrate the effectiveness of the algorithms.

In the first part, we study the convergence of discrete solutions to splitting schemes for two-phase flow with different mass densities suggested in [Guillen-Gonzalez, Tierra, J.Comput.Math. (6)2014]. They have been formulated for the diffuse interface model in [Abels, Garcke, Grün, M3AS, 2012, DOI:10.1142/S0218202511500138] which is consistent with thermodynamics. Our technique covers various discretization methods for phase-field energies, ranging from convex-concave splitting to difference quotient approaches for the double-well potential. In the second part of the paper, numerical experiments are presented in two space dimensions to identify discretizations of Cahn-Hilliard energies which are ϕ-stable and which do not reduce the acceleration of falling droplets. Finally, 3d simulations in axial symmetric geometries are shown to underline even more the full practicality of the approach.

Consider the time-harmonic acoustic scattering from an extended elastic body surrounded by a finite number of point-like obstacles in a fluid. We assume point source waves are emitted from arrayed transducers and the signals of scattered near-field data are recorded by receivers not far away from the scatterers (compared to the incident wavelength). The forward scattering can be modeled as an interaction problem between acoustic and elastic waves together with a multiple scattering problem between the extend solid and point scatterers. We prove a necessary and sufficient condition that can be used simultaneously to recover the shape of the extended elastic solid and to locate the positions of point scatterers. The essential ingredient in our analysis is the outgoing-to-incoming (OtI) operator applied to the resulting near-field response matrix (or operator). In the first part, we justify the MUSIC algorithm for locating point scatterers from near-field measurements. In the second part, we apply the factorization method, the continuous analogue of MUSIC, to the two-scale scattering problem for determining both extended and point scatterers. Numerical examples in 2D are demonstrated to show the validity and accuracy of our inversion algorithms.