Approximate solutions and chaotic motions of a piecewise nonlinear–linear oscillator

A global symmetric period-1 approximate solution is analytically constructed for the non-resonant periodic response of a periodically excited piecewise nonlinear–linear oscillator. The approximate solutions are found to be in good agreement with the exact solutions that are obtained from the numerical integration of the original equations. In addition, the dynamic behaviour of the oscillator is numerically investigated with the help of bifurcation diagrams, Lyapunov exponents, Poincare maps, phase portraits and basins of attraction. The existence of subharmonic and chaotic motions and the coexistence of four attractors are observed for some combinations of the system parameters.     

Steady states in plasma physics—the Vlasov–Fokker–Planck equation

In this paper we investigate the non-linear Vlasov–Fokker–Planck (VFP) equation, a both physically and mathematically interesting modification of Vlasov's equation, which describes a plasma in a thermal bath. We prove existence, uniqueness and representation results for steady states of the VFP equation both in the case of a mollified interaction potential and for the VFP–Poisson system. The uniqueness and representation results are of special interest since they distinguish special solutions of the Vlasov equation.     

New analysis and application of fractional order Schrödinger equation using with Atangana–Batogna numerical scheme

In this work, an analytical approximation to the solution of Schrodinger equation has been provided. The fractional derivative used in this equation is the Caputo derivative. The existence and uniqueness conditions of solutions for the proposed model are derived based on the power law. While solving the fractional order Schrodinger equation, Atangana–Batogna numerical method is presented for fractional order equation. We obtain an efficient recurrence relation for solving these kinds of equations. To illustrate the usefulness of the numerical scheme, the numerical simulations are presented. The results show that the numerical scheme is very effective and simple.     

Navier–Stokes equations with slip boundary conditions

In this paper, we consider incompressible viscous fluid flows with slip boundary conditions. We first prove the existence of solutions of the unsteady Navier–Stokes equations in n‐spacial dimensions. Then, we investigate the stability, uniqueness and regularity of solutions in two and three spacial dimensions. In the compactness argument, we construct a special basis fulfilling the incompressibility exactly, which leads to an efficient and convergent spectral method. In particular, we avoid the main difficulty for ensuring the incompressibility of numerical solutions, which occurs in other numerical algorithms. We also derive the vorticity‐stream function form with exact boundary conditions, and establish some results on the existence, stability and uniqueness of its solutions. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimal Hölder Continuity for a Class of Degenerate Elliptic Problems with an Application to Subsonic-Sonic Flows

In this paper, we first study a class of elliptic equations with anisotropic boundary degeneracy. Besides establishing the existence, uniqueness and comparison principle, we obtain the optimal Hölder estimates for weak solutions by the estimates in the Campanato space. Based on such Hölder estimates, we then investigate subsonic-sonic flows with singularities at the sonic curves in a symmetric convergent nozzle with straight wall for an approximate model of the potential flow equation. It is proved that the perturbation problem of the symmetric subsonic-sonic flow is solvable and the symmetric subsonic-sonic flow is stable.     

A note on spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients

In this note, by constructing suitable approximate solutions, we prove the existence of global weak solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients in the whole space or exterior domain, when the initial data are spherically symmetric. In particular, we prove the existence of spherically symmetric solutions to the Saint-Venant model for shallow water in the whole space (or exterior domain).     

Asymptotic stability of stationary solutions to the compressible bipolar Navier–Stokes–Poisson equations

In this paper, we consider the compressible bipolar Navier–Stokes–Poisson equations with a non‐flat doping profile in three‐dimensional space. The existence and uniqueness of the non‐constant stationary solutions are established when the doping profile is a small perturbation of a positive constant state. Then under the smallness assumption of the initial perturbation, we show the global existence of smooth solutions to the Cauchy problem near the stationary state. Finally, the convergence rates are obtained by combining the energy estimates for the nonlinear system and the L²‐decay estimates for the linearized equations. Copyright © 2017 John Wiley & Sons, Ltd.     

Stationary solutions of Euler–Poisson equations for non‐isentropic gaseous stars

The motion of the self‐gravitational gaseous stars can be described by the Euler–Poisson equations. For some velocity fields and entropy functions that solve the conservation of mass and energy, we consider the existence of stationary solutions of Euler–Poisson equations. Under various restriction to the strength of velocity field, different assumptions on the isentropic function and adiabatic exponent, we get the existence, multiplicity and uniqueness of the stationary solutions to the Euler–Poisson system, respectively. Copyright © 2012 John Wiley & Sons, Ltd.     

The Cauchy–Neumann problem for a multi‐dimensional isentropic hydrodynamic model for semiconductors

We investigate a multi‐dimensional isentropic hydrodynamic (Euler–Poisson) model for semiconductors, where the energy equation is replaced by the pressure–density relation p(n) . We establish the global existence of smooth solutions for the Cauchy–Neumann problem with small perturbed initial data and homogeneous Neumann boundary conditions. We show that, as t→+∞, the solutions converge to the non‐constant stationary solutions of the corresponding drift–diffusion equations. Moreover, we also investigate the existence and uniqueness of the stationary solutions for the corresponding drift–diffusion equations. Copyright © 2005 John Wiley & Sons, Ltd.     

Global weak solutions for a periodic two‐component μ‐Camassa–Holm system

In this paper, we consider the global existence of weak solutions for a two‐component μ‐Camassa–Holm system in the periodic setting. Global existence for strong solutions to the system with smooth approximate initial value is derived. Then, we show that the limit of approximate solutions is a global‐in‐time weak solution of the two‐component μ‐Camassa–Holm system. Copyright © 2013 John Wiley & Sons, Ltd.     

A comparison of the plate theories in the sense of Kirchhoff–Love and Reissner–Mindlin

In this article we compare the two plate theories in the sense of Kirchhoff–Love and Reissner–Mindlin for several different settings of the physical system. We establish existence, uniqueness and regularity of solutions to the respective boundary and initial boundary value problems. Moreover, we give asymptotic expansions of the solutions in the limit of a vanishing plate thickness, ϵ→0, whenever this is possible. Finally, we compare the solutions in the sense of Kirchhoff–Love and Reissner–Mindlin in that very limit. Copyright © 1999 John Wiley & Sons, Ltd.     

Global existence of the radially symmetric solutions of the Navier–Stokes equations for the isentropic compressible fluids

We study the isentropic compressible Navier–Stokes equations with radially symmetric data in an annular domain. We first prove the global existence and regularity results on the radially symmetric weak solutions with non‐negative bounded densities. Then we prove the global existence of radially symmetric strong solutions when the initial data ρ₀, u ₀ satisfy the compatibility condition for some radially symmetric g ∈ L². The initial density ρ₀ needs not be positive. We also prove some uniqueness results on the strong solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Global classical solutions of viscous liquid–gas two‐phase flow model

In this paper, we consider the global existence and uniqueness of the classical solutions for the three‐dimensional where the existence of global classical solutions to the compressible Navier–Stokes equations was obtained by using the continuity methods under the assumption that the initial energy is sufficiently small. Copyright © 2012 John Wiley & Sons, Ltd.     

A Crank–Nicolson collocation spectral method for the two‐dimensional viscoelastic wave equation

In this paper, we first establish the Crank–Nicolson collocation spectral (CNCS) method for two‐dimensional (2D) viscoelastic wave equation by means of the Chebyshev polynomials. And then, we analyze the existence, uniqueness, stability, and convergence of the CNCS solutions. Finally, we use some numerical experiments to verify the correctness of theoretical analysis. This implies that the CNCS model is very effective for solving the 2D viscoelastic wave equations.     

Existence and uniqueness of solutions to a model describing miscible liquids

The existence and the uniqueness of solutions to a problem of miscible liquids are investigated in this note. The model consists of Navier–Stokes equations with Korteweg stress terms coupled with the reaction–diffusion equation for the concentration. We assume that the fluid is incompressible and the Boussinesq approximation is adopted. The global existence and uniqueness of solutions is established for some optimal conditions on the reaction source term and the external force functions.     

Modeling and analysis of reactive solute transport in deformable channels with wall adsorption–desorption

We show well posedness for a model of nonlinear reactive transport of chemical in a deformable channel. The channel walls deform due to fluid–structure interaction between an unsteady flow of an incompressible, viscous fluid inside the channel and elastic channel walls. Chemical solutes, which are dissolved in the viscous, incompressible fluid, satisfy a convection–diffusion equation in the bulk fluid, while on the deforming walls, the solutes undergo nonlinear adsorption–desorption physico‐chemical reactions. The problem addresses scenarios that arise, for example, in studies of drug transport in blood vessels. We show the existence of a unique weak solution with solute concentrations that are non‐negative for all times. The analysis of the problem is carried out in the context of semi‐linear parabolic PDEs on moving domains. The arbitrary Lagrangian–Eulerian approach is used to address the domain movement, and the Galerkin method with the Picard–Lindelöf theorem is used to prove existence and uniqueness of approximate solutions. Energy estimates combined with the compactness arguments based on the Aubin–Lions lemma are used to prove convergence of the approximating sequences to the unique weak solution of the problem. It is shown that the solution satisfies the positivity property, that is, that the density of the solute remains non‐negative at all times, as long as the prescribed fluid domain motion is ‘reasonable’. This is the first well‐posedness result for reactive transport problems defined on moving domains of this type. Copyright © 2015 John Wiley & Sons, Ltd.     

Characteristic Symmetric Hyperbolic Systems with Dissipation: Global Existence and Asymptotics

We consider the initial-boundary value problem for quasi-linear symmetric hyperbolic systems with dissipation and characteristic boundary of constant multiplicity. We investigate the global existence and decay property of small regular solutions in suitable functions spaces which take into account the loss of regularity in the normal direction to the characteristic boundary. We also show the existence, uniqueness and stability of the time periodic solutions. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Existence and uniqueness theorem for the Chan–Bodner–Lindholm model

In this paper we consider a problem of non‐linear inelasticity. The global in the time existence and uniqueness for the Chan–Bodner–Lindholm model is proved. The idea of the proof is based on the non‐linear semigroup method. Copyright © 1999 John Wiley & Sons, Ltd.     

Degenerate nonlocal Cahn-Hilliard equations: Well-posedness,regularity and local asymptotics

Existence and uniqueness of solutions for nonlocal Cahn-Hilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence theory. A convection term is also taken into account. Building upon this novel existence result, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts, as the nonlocal convolution kernels approximate a Dirac delta. Eventually, we show that, under suitable assumptions on the data, the solutions to the nonlocal Cahn-Hilliard equations exhibit further regularity, and the nonlocal-to-local convergence is verified in a stronger topology.     

Decay of the energy for the Benjamin–Bona–Mahony equation posed on bounded intervals and on a half‐line

This paper concerns nonlinear dissipative initial boundary value problems for the Benjamin–Bona–Mahony equation posed on a half‐line and on bounded intervals. We prove the existence and uniqueness of global solutions and decay of the energy as time tends to infinity as well as convergence of solutions on bounded intervals to a solution on a half‐line. Copyright © 2012 John Wiley & Sons, Ltd.