The paper studies large time behaviour of solutions to the Keller–Segel system with quadratic degradation in a liquid environment, as given by

under Neumann boundary conditions in a bounded domain Ω ⊂ ℝn, where n ≥ 1 is arbitrary. It is shown that whenever U : Ω × (0,∞) → ℝn is a bounded and sufficiently regular solenoidal vector field any non-trivial global bounded solution of (⋆) approaches the trivial equilibrium at a rate that, with respect to the norm in either of the spaces L1(Ω) and L∞(Ω), can be controlled from above and below by appropriate multiples of 1/(t + 1). This underlines that, even up to this quantitative level of accuracy, the large time behaviour in (⋆) is essentially independent not only of the particular fluid flow, but also of any effect originating from chemotactic cross-diffusion. The latter is in contrast to the corresponding Cauchy problem, for which known results show that in the n = 2 case the presence of chemotaxis can significantly enhance biomixing by reducing the respective spatial L1 norms of solutions.

In this paper the existence and uniqueness of weak and strong solutions for a non-autonomous non-local reaction–diffusion equation is proved. Furthermore, the existence of minimal pullback attractors in the L2-norm in the frameworks of universes of fixed bounded sets and those given by a tempered growth condition is established, along with some relationships between them. Finally, we prove the existence of minimal pullback attractors in the H1-norm and study relationships among these new families and those given previously in the L2 context. We also present new results in the autonomous framework that ensure the existence of global compact attractors as a particular case.

We give a weak-Lp Serrin-type regularity criterion for a weak solution to the three-dimensional magnetohydrodynamics equations in a bounded domain Ω ⊂ ℝ3.

We derive a macroscopic model for biofilm formation in a porous medium reactor to investigate the role of suspended bacteria on reactor performance. The starting point is the mesoscopic one-dimensional Wanner–Gujer biofilm model. The following processes are included: hydrodynamics and transport of substrate in the reactor, biofilm and suspended bacteria growth in the pore space, attachment of suspended cells to the biofilm, and detachment of biofilm cells. The mesoscopic equations are up-scaled from the biofilm scale to the reactor scale, yielding a stiff system of balance laws, which we study numerically. We find that suspended bacteria and attachment can have a significant effect on biofilm reactor performance.

In this paper we extend to Kawahara type equations a uniqueness result obtained by C. E. Kenig, G. Ponce, and L. Vega for KdV type equations. We prove that, under certain decay's conditions, the null solution is the unique solution.

The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

There are many fluid flow problems involving geometries for which a nonorthogonal curvilinear coordinate system may be the most suitable. To the authors’ knowledge, the Navier–Stokes equations for an incompressible fluid formulated in terms of an arbitrary nonorthogonal curvilinear coordinate system have not been given explicitly in the literature in the simplified form obtained herein. The specific novelty in the equations derived here is the use of the general Laplacian in arbitrary nonorthogonal curvilinear coordinates and the simplification arising from a Ricci identity for Christoffel symbols of the second kind for flat space. Evidently, however, the derived equations must be consistent with the various general forms given previously by others. The general equations derived here admit the well-known formulae for cylindrical and spherical polars, and for the purposes of illustration, the procedure is presented for spherical polar coordinates. Further, the procedure is illustrated for a nonorthogonal helical coordinate system. For a slow flow for which the inertial terms may be neglected, we give the harmonic equation for the pressure function, and the corresponding equation if the inertial effects are included. We also note the general stress boundary conditions for a free surface with surface tension. For completeness, the equations for a compressible flow are derived in an appendix.

This paper introduces two novel conformal structure-preserving algorithms for solving the coupled damped nonlinear Schrödinger (CDNLS) system, which are based on the conformal multi-symplectic Hamiltonian formulation and its conformal conservation laws. The proposed algorithms can preserve corresponding conformal multi-symplectic conservation law and conformal momentum conservation law in any local time-space region, respectively. Moreover, it is further shown that the algorithms admit the conformal charge conservation law, and exactly preserve the dissipation rate of charge under appropriate boundary conditions. Numerical experiments are presented to demonstrate the conformal properties and effectiveness of the proposed algorithms during long-time numerical simulations and validate the analysis.

We consider the existence of normalized solutions in H1(ℝN) × H1(ℝN) for systems of nonlinear Schr¨odinger equations, which appear in models for binary mixtures of ultracold quantum gases. Making a solitary wave ansatz, one is led to coupled systems of elliptic equations of the form

and we are looking for solutions satisfying

where a1> 0 and a2> 0 are prescribed. In the system, λ1 and λ2 are unknown and will appear as Lagrange multipliers. We treat the case of homogeneous nonlinearities, i.e. , with positive constants β, μi, pi, ri. The exponents are Sobolev subcritical but may be L2-supercritical. Our main result deals with the case in which in dimensions 2 ≤ N ≤ 4. We also consider the cases in which all of these numbers are less than 2 + 4/N or all are bigger than 2 + 4/N.

Pinning effect on current-induced magnetic transverse domain wall dynamics in nanostrip is studied for its potential application to new magnetic memory devices. In this study, we carry out a series of calculations by solving generalized Landau-Lifshitz equation involving a current spin transfer torque in one and two dimensional models. The critical current for the transverse wall depinning in nanostrip depends on the size of artificial rectangular defects on the edges of nanostrip. We show that there is intrinsic pinning potential for a defect such that the transverse wall oscillates damply around the pinning site with an intrinsic frequency if the applied current is below critical value. The amplification of the transverse wall oscillation for both displacement and maximum value of m3 is significant by applying AC current and current pulses with appropriate frequency. We show that for given pinning potential, the oscillation amplitude as a function of the frequency of the AC current behaves like a Gaussian distribution in our numerical study, which is helpful to reduce strength of current to drive the transverse wall motion.

In this work, a robust, consistent, and coherent approach, termed as Modified Ghost Method (MGM), is developed to deal with the multi-medium interaction with elastic-plastic solid. This approach is simple to implement and keeps the solvers intact, and can handle multi-medium problems which involve various media including gas, liquid and solid. The MGM is first validated by two-dimensional (2D) cases and then is applied to study the interaction between elastic-plastic solid structure and the underwater explosion. The development of the wave system is described and analyzed. Furthermore, two kinds of complex solid structure subjected to underwater explosion are simulated. Finally, a complex solid structure immersed in water subjected to underwater explosion is simulated and analyzed. The numerical experiments show the viability, effectiveness and versatility of the proposed method which is able to accurately predict the wave pattern at various interfaces.

For 2D elastic-plastic flows with the hypo-elastic constitutive model and von Mises’ yielding condition, the non-conservative character of the hypo-elastic constitutive model and the von Mises’ yielding condition make the construction of the solution to the Riemann problem a challenging task. In this paper, we first analyze the wave structure of the Riemann problem and develop accordingly a Four-Rarefaction wave approximate Riemann Solver with Elastic waves (FRRSE). In the construction of FRRSE one needs to use an iterative method. A direct iteration procedure for four variables is complex and computationally expensive. In order to simplify the solution procedure we develop an iteration based on two nested iterations upon two variables, and our iteration method is simple in implementation and efficient. Based on FRRSE as a building block, we propose a 2nd-order cell-centered Lagrangian numerical scheme. Numerical results with smooth solutions show that the scheme is of second-order accuracy. Moreover, a number of numerical experiments with shock and rarefaction waves demonstrate the scheme is essentially non-oscillatory and appears to be convergent. For shock waves the present scheme has comparable accuracy to that of the scheme developed by Maire et al., while it is more accurate in resolving rarefaction waves.

Consider the scattering of a time-harmonic acoustic incident wave by a bounded, penetrable, and isotropic elastic solid, which is immersed in a homogeneous compressible air or fluid. The paper concerns the numerical solution for such an acoustic-elastic interaction problem in three dimensions. An exact transparent boundary condition (TBC) is developed to reduce the problem equivalently into a boundary value problem in a bounded domain. The perfectly matched layer (PML) technique is adopted to truncate the unbounded physical domain into a bounded computational domain. The well-posedness and exponential convergence of the solution are established for the truncated PML problem by using a PML equivalent TBC. An a posteriori error estimate based adaptive finite element method is developed to solve the scattering problem. Numerical experiments are included to demonstrate the competitive behavior of the proposed method.

We study the dynamics of a domain wall under the influence of applied magnetic fields in a one-dimensional ferromagnetic nanowire, governed by the Landau–Lifshitz–Gilbert equation. Existence of travelling-wave solutions close to two known static solutions is proven using implicit-function-theorem-type arguments.

The KP-II equation was derived by Kadmotsev and Petviashvili to explain stability of line solitary waves of shallow water. Recently, Mizumachi proved nonlinear stability of 1-line solitons for exponentially localized perturbations. In this paper, we prove stability of 1-line solitons for perturbations in (1 + x2)−1/2−0H1(ℝ2) and perturbations in H1(ℝ2) ∩ ∂xL2(ℝ2).

This paper presents a heuristic Learning-based Non-Negativity Constrained Variation (L-NNCV) aiming to search the coefficients of variational model automatically and make the variation adapt different images and problems by supervised-learning strategy. The model includes two terms: a problem-based term that is derived from the prior knowledge, and an image-driven regularization which is learned by some training samples. The model can be solved by classical ε-constraint method. Experimental results show that: the experimental effectiveness of each term in the regularization accords with the corresponding theoretical proof; the proposed method outperforms other PDE-based methods on image denoising and deblurring.

A closed curve flow on the 2-sphere evolved by a fourth-order nonlinear dispersive partial differential equation on the one-dimensional flat torus is studied. The governing equation arises in the field of physics in relation to the continuum limit of the Heisenberg spin chain systems or three-dimensional motion of the isolated vortex filament. The main result of the paper gives the local existence and uniqueness of a solution to the initial-value problem by overcoming loss of derivatives in the classical energy method and the absence of the local smoothing effect. The proof is based on the delicate analysis of the lower-order terms to find out the loss of derivatives and on the gauged energy method to eliminate the obstruction.

In this work, we introduce an IMEX discontinuous Galerkin solver for the weakly compressible isentropic Euler equations. The splitting needed for the IMEX temporal integration is based on the recently introduced reference solution splitting [32, 52], which makes use of the incompressible solution. We show that the overall method is asymptotic preserving. Numerical results show the performance of the algorithm which is stable under a convective CFL condition and does not show any order degradation.

An adaptive moving mesh finite difference method is presented to solve two types of equations with dynamic capillary pressure effect in porous media. One is the non-equilibrium Richards Equation and the other is the modified Buckley-Leverett equation. The governing equations are discretized with an adaptive moving mesh finite difference method in the space direction and an implicit-explicit method in the time direction. In order to obtain high quality meshes, an adaptive monitor function with directional control is applied to redistribute the mesh grid in every time step, then a diffusive mechanism is used to smooth the monitor function. The behaviors of the central difference flux, the standard local Lax-Friedrich flux and the local Lax-Friedrich flux with reconstruction are investigated by solving a 1D modified Buckley-Leverett equation. With the moving mesh technique, good mesh quality and high numerical accuracy are obtained. A collection of one-dimensional and two-dimensional numerical experiments is presented to demonstrate the accuracy and effectiveness of the proposed method.