On Slightly Compressible Ideal Flow¶in the Half-Plane

We consider the Euler equations of barotropic inviscid compressible fluids in the half-plane. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In 2D (two dimensions) such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial data. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small. We decompose the solution as the sum of the irrotational part, the incompressible part and the remainder, which describes the interaction between the first two components. First we study the life span of smooth irrotational solutions, i.e., the largest time interval T(?) of existence of classical solutions, when the initial data are a small perturbation of size ? from a constant state. Related to this is a decay property for the irrotational part. Then, we study the interaction between the two components and show the existence on any arbitrary time interval, for any Mach number sufficiently small. This yields the existence of smooth compressible flow on any arbitrary time interval. For the proofs we use a combination of energy and decay estimates.     

2D Slightly Compressible Ideal Flow in an Exterior Domain

We consider the Euler equations of barotropic inviscid compressible fluids in the exterior domain. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In dimension 2 such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial data. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small. First we study the life span of smooth irrotational solutions, i.e. the largest time interval of existence of classical solutions, when the initial data are a small perturbation of size from a constant state. Then, we study the nonlinear interaction between the irrotational part and the incompressible part of a general solution. This analysis yields the existence of smooth compressible flow on any arbitrary time interval and with no restriction on the size of the initial velocity, for any Mach number sufficiently small. Finally, the approach is applied to the study of the incompressible limit. For the proofs we use a combination of energy estimates and a decay estimate for the irrotational part.     

On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions

We study the nonlinear hyperbolic partial differential equation, (u _t+uu_x)_x=1/2u _x ² . This partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles. In particular, it describes waves in a massive director field of a nematic liquid crystal.Global smooth solutions of the partial differential equation do not exist, since their derivatives blow up in finite time, while weak solutions are not unique. We therefore define two distinct classes of admissible weak solutions, which we call dissipative and conservative solutions. We prove the global existence of each type of admissible weak solution, provided that the derivative of the initial data has bounded variation and compact support. These solutions remain continuous, despite the fact that their derivatives blow up.There are no a priori estimates on the second derivatives in any L ^p space, so the existence of weak solutions cannot be deduced by using Sobolev-type arguments. Instead, we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation.We also describe the qualitative properties of the partial differential equation, including a comparison with the Burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions. We show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics, and we prove that the large-time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink-wave.     

Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations

We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves. We study the dynamics of weakly nonlinear waves, and using asymptotic multiple-scale expansions and averaging with respect to the fast time, we obtain a hierarchy of asymptotically exact coupled and uncoupled Ostrovsky equations for unidirectional waves. We then construct a weakly nonlinear solution of the initial-value problem in terms of solutions of the derived Ostrovsky equations within the accuracy of the governing equations, and show that there are no secular terms. When coupling parameters are equal to zero, our results yield a weakly nonlinear solution of the initial-value problem for the Boussinesq equation in terms of solutions of the initial-value problems for two Korteweg-de Vries equations, integrable by the Inverse Scattering Transform. We also perform relevant numerical simulations of the original unapproximated system of Boussinesq equations to illustrate the difference in the behaviour of its solutions for different asymptotic regimes.     

On the self-similar solution to the Euler equations for an incompressible fluid in three dimensions

The equations for a self-similar solution to an inviscid incompressible fluid are mapped into an integral equation that hopefully can be solved by iteration. It is argued that the exponents of the similarity are ruled by Kelvin's theorem of conservation of circulation. The end result is an iteration with a nonlinear term entering a kernel given by a 3D integral for a swirling flow, likely within reach of present-day computational power. Because of the slow decay of the similarity solution at large distances, its kinetic energy diverges, and some mathematical results excluding non-trivial solutions of the Euler equations in the self-similar case do not apply.     

Gas Flows with Several Thermal Nonequilibrium Modes

We study gas flows with any finite number of thermal nonequilibrium modes. The equations describing such flows are a hyperbolic system with several relaxation equations. An important feature is entropy increase dictated by physics for any irreversible process. Under physical assumptions we obtain properties of thermodynamic variables relevant to stability. By the energy method we prove global existence and uniqueness for the Cauchy problem when the initial data are small perturbations of constant equilibrium states. We give a precise formulation of the fundamental solution for the linearized system, and use it to obtain large time behavior of solutions to the nonlinear system. In particular, we show that the entropy increases but stays bounded. The resulting asymptotic picture of nonequilibrium flows is in a pointwise sense both in space and in time.     

Threshold Solutions and Sharp Transitions for Nonautonomous Parabolic Equations on $${\mathbb{R}^N}$$

This paper is devoted to a class of nonautonomous parabolic equations of the form u _t = Δu + f(t, u) on \mathbbR^N{\mathbb{R}^N} . We consider a monotone one-parameter family of initial data with compact support, such that for small values of the parameter the corresponding solutions decay to zero, whereas for large values they exhibit a different behavior (either blowup in finite time or locally uniform convergence to a positive constant steady state). We are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs. We refer to such values as threshold values and to the corresponding solutions as threshold solutions. We prove that the transition from decay to the other behavior is sharp: there is just one threshold value. We also describe the behavior of the threshold solution: it is global, bounded, and asymptotically symmetric in the sense that all its limit profiles, as t → ∞, are radially symmetric about the same center. Our proofs rely on parabolic Liouville theorems, asymptotic symmetry results for nonlinear parabolic equations, and theorems on exponential separation and principal Floquet bundles for linear parabolic equations.     

Internal and inertial waves in a viscous rotating stratified fluid

The effects of viscosity on the propagation of a St. Andrew's cross wave which is generated by a simple-harmonic localized disturbance in a rotating stratified fluid are considered. A similarity solution of the linearised equations shows that the velocities decay and that the wave width increases away from the disturbance. Previous solutions in a stratified non-rotating fluid are recovered by letting the rotation tend to zero. The solutions are also valid in the limit of a homogeneous rotating fluid. Further solutions for waves in a realistic ocean and in an isothermal atmosphere on a rotating Earth are also included.     

Stability of travelling fronts in a model for flame propagation part I: Linear analysis

We investigate the stability of travelling wave solutions of the multidimensional thermodiffusive model for flame propagation with unit Lewis number. This model consists in a system of two nonlinear parabolic equations posed in an infinite cylinder, with Neumann boundary conditions. In this paper, we prove that every travelling wave solution of this model is linearly stable. Our tools are exponential decay estimates for solutions of elliptic equations in a cylinder, and the Maximum Principle for parabolic equations.     

A new numerical model for simulations of wave transformation,breaking and long‐shore currents in complex coastal regions

In this paper, we propose a model based on a new contravariant integral form of the fully nonlinear Boussinesq equations in order to simulate wave transformation phenomena, wave breaking, and nearshore currents in computational domains representing the complex morphology of real coastal regions. The aforementioned contravariant integral form, in which Christoffel symbols are absent, is characterized by the fact that the continuity equation does not include any dispersive term. A procedure developed in order to correct errors related to the difficulties of numerically satisfying the metric identities in the numerical integration of fully nonlinear Boussinesq equation on generalized boundary‐conforming grids is presented. The Boussinesq equation system is numerically solved by a hybrid finite volume–finite difference scheme. The proposed high‐order upwind weighted essentially non‐oscillatory finite volume scheme involves an exact Riemann solver and is based on a genuinely two‐dimensional reconstruction procedure, which uses a convex combination of biquadratic polynomials. The wave breaking is represented by discontinuities of the weak solution of the integral form of the nonlinear shallow water equations. The capacity of the proposed model to correctly represent wave propagation, wave breaking, and wave‐induced currents is verified against test cases present in the literature. The results obtained are compared with experimental measures, analytical solutions, or alternative numerical solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

Analytical Solution for Multi-Species Contaminant Transport in Finite Media with Time-Varying Boundary Conditions

Most analytical solutions available for the equations governing the advective–dispersive transport of multiple solutes undergoing sequential first-order decay reactions have been developed for infinite or semi-infinite spatial domains and steady-state boundary conditions. In this study, we present an analytical solution for a finite domain and a time-varying boundary condition. The solution was found using the Classic Integral Transform Technique (CITT) in combination with a filter function having separable space and time dependencies, implementation of the superposition principle, and using an algebraic transformation that changes the advection–dispersion equation for each species into a diffusion equation. The analytical solution was evaluated using a test case from the literature involving a four radionuclide decay chain. Results show that convergence is slower for advection-dominated transport problems. In all cases, the converged results were identical to those obtained with the previous solution for a semi-infinite domain, except near the exit boundary where differences were expected. Among other applications, the new solution should be useful for benchmarking numerical solutions because of the adoption of a finite spatial domain.     

Kink wave determined by parabola solution of a nonlinear ordinary differential equation

By finding a parabola solution connecting two equilibrium points of a planar dynamical system,the existence of the kink wave solution for 6 classes of nonlinear wave equations is shown.Some exact explicit parametric representations of kink wave solutions are given.Explicit parameter conditions to guarantee the existence of kink wave solutions are determined.     

Submodels of model of nonlinear diffusion with non-stationary absorption

We study the model, describing a nonlinear diffusion process (or a heat propagation process) in an inhomogeneous medium with non-stationary absorption (or source). We found tree submodels of the original model of the nonlinear diffusion process (or the heat propagation process), having different symmetry properties. We found all invariant submodels. All essentially distinct invariant solutions describing these invariant submodels are found either explicitly, or their search is reduced to the solution of the nonlinear integral equations. For example, we obtained the invariant solution describing the nonlinear diffusion process (or the heat distribution process) with two fixed "black holes", and the invariant solution describing the nonlinear diffusion process (or the heat distribution process) with the fixed "black hole" and the moving "black hole". The presence of the arbitrary constants in the integral equations, that determine these solutions provides a new opportunities for analytical and numerical study of the boundary value problems for the received submodels, and, thus, for the original model of the nonlinear diffusion process (or the heat distribution process). For the received invariant submodels we are studied diffusion processes (or heat distribution process) for which at the initial moment of the time at a fixed point are specified or a concentration (a temperature) and its gradient, or a concentration (a temperature) and its rate of change. Solving of boundary value problems describing these processes are reduced to the solving of nonlinear integral equations. We are established the existence and uniqueness of solutions of these boundary value problems under some additional conditions. The obtained results can be used to study the diffusion of substances, diffusion of conduction electrons and other particles, diffusion of physical fields, propagation of heat in inhomogeneous medium.     

Issue Information

This paper describes a numerical discretization of the compressible Euler equations with a gravitational potential. A pertinent feature of the solutions to these inhomogeneous equations is the special case of stationary solutions with zero velocity, described by a nonlinear partial differential equation, whose solutions are called hydrostatic equilibria. We present a well‐balanced method, meaning that besides discretizing the complete equations, the method is also able to maintain all hydrostatic equilibria. The method is a finite volume method, whose Riemann solver is approximated by a so‐called relaxation Riemann solution that takes all hydrostatic equilibria into account. Relaxation ensures robustness, accuracy, and stability of our method, because it satisfies discrete entropy inequalities. We will present numerical examples, illustrating that our method works as promised. Copyright © 2015 John Wiley & Sons, Ltd.     

Elliptic balance solution of two-degree-of-freedom, undamped, forced systems with cubic nonlinearity

The solution of a system of two coupled, nonhomogeneous undamped, ordinary differential equations with cubic nonlinearity and sinusoidal driving force is obtained by the use of Jacobian elliptic functions and the elliptic balance method. To assess the accuracy of our proposed solution, we consider an example that arises in the study of the finite amplitude, nonlinear vibration of a simple shear suspension system. It is shown that the analytical results exhibit good agreement with the numerical integration solutions even for moderate values of the system parameters.     

Similarity temperature profiles for some nonlinear reaction - diffusion equations

A relevant tool in the study of the closed - form solutions of reaction - diffusion equations is the concept of phase - plane. The aim of this paper is to apply this approach to some simplified models of microwave heating problems. We investigate two models: one is power law dependence and the other is exponential dependence. In both cases, we assume a heat source term with spatial polynomial decay but increasing with temperature. The spatial polynomial decay is known to have been applied earlier under some physically reasonable assumptions. In particular, the solutions obtained permit us to compare the contribution of the heat source term and geometry. The results of this analysis are in keeping with what others have observed with nonlinear diffusion; they also apply to numerous equations which have hitherto not been studied.     

A fully implicit scheme for simulating ionized gas flows using the gas dynamics electrodynamics coupled system

The flow of ionized gases under the influence of electromagnetic fields is governed by the coupled system of the compressible flow equations and the Maxwell equations. In this system, coupling of the flow with the electromagnetic field is obtained through nonlinear and stiff source terms, which may cause difficulties with the numerical solution of the coupled system. The discontinuous Galerkin finite element method is used for the numerical solution of this system. For the magnetic field vector, discontinuous Galerkin discretization is performed using a divergence‐free vector base for the magnetic field to preserve zero divergence in the element and retain the implicit constraint of a divergence‐free magnetic field vector down to very low level both globally and locally. To circumvent difficulties resulting from the presence of the stiff source terms, implicit time marching is used for the fully coupled system to avoid wrong wave shapes and propagation speeds that are obtained when the coupling source terms are lagged in time or by using splitting iterative schemes. Numerical solutions for benchmark problems computed on collocated meshes for the flow and electromagnetic field variables with this fully coupled monolithic approach showed good agreement with other numerical solutions and exact results. Copyright © 2014 John Wiley & Sons, Ltd.     

A case of strong nonlinearity: Intermittency in highly turbulent flows

It has long been suspected that flows of incompressible fluids at large or infinite Reynolds number (namely at small or zero viscosity) may present finite time singularities. We review briefly the theoretical situation on this point. We discuss the effect of a small viscosity on the self-similar solution to the Euler equations for inviscid fluids. Then we show that single-point records of velocity fluctuations in the Modane wind tunnel display correlations between large velocities and large accelerations in full agreement with scaling laws derived from Leray's equations (1934) for self-similar singular solutions to the fluid equations. Conversely, those experimental velocity–acceleration correlations are contradictory to the Kolmogorov scaling laws.     

The stability of travelling waves induced by crossed electric and magnetic fields in nematic liquid crystals

A theoretical study is carried out into the stability of travelling wave solutions to an approximate dynamic equation for the problem in which a nematic liquid crystal is subjected to crossed electric and magnetic fields. The authors recently found three types of travelling wave solutions for this problem [2], each characterised by the control parameter which describes the relationship between the magnitudes of the fields and their crossed angle. Two types of stability are ex amined: the first considers perturbations which vanish outside some finite interval in the moving coordinate of the travelling wave, while the second considers quite general perturbations belonging to a weighted space, the weighting function being determined by the particular solution and the control parameter . When the first type of stability occurs, perturbations decay to zero as time increases. In the second type of stability perturbations may eith er decay to zero or induce a small phase shift to the original travelling wave. Both these versions of stability depend crucially on and on the type of travelling wave solution being considered. Received April 15, 1997