Bifurcations of Heteroclinic Orbits

The search for traveling wave solutions of a semilinear diffusion partial differential equation can be reduced to the search for heteroclinic solutions of the ordinary differential equation ü − cu̇ + f(u) = 0, where c is a positive constant and f is a nonlinear function. A heteroclinic orbit is a solution u(t) such that u(t) → γ ₁ as t → −∞ and u(t) → γ ₂ as t → ∞ where γ ₁, γ ₂ are zeros of f. We study the existence of heteroclinic orbits under various assumptions on the nonlinear function f and their bifurcations as c is varied. Our arguments are geometric in nature and so we make only minimal smoothness assumptions. We only assume that f is continuous and that the equation has a unique solution to the initial value problem. Under these weaker smoothness conditions we reprove the classical result that for large c there is a unique positive heteroclinic orbit from 0 to 1 when f(0) = f(1) = 0 and f(u) > 0 for 0 < u < 1. When there are more zeros of f, there is the possibility of bifurcations of the heteroclinic orbit as c varies. We give a detailed analysis of the bifurcation of the heteroclinic orbits when f is zero at the five points −1 < −θ < 0 < θ < 1 and f is odd. The heteroclinic orbit that tends to 1 as t → ∞ starts at one of the three zeros, −θ, 0, θ as t → −∞. It hops back and forth among these three zeros an infinite number of times in a predictable sequence as c is varied.     

Time Periodic Solutions to the One-Dimensional Nonlinear Wave Equation

This paper is concerned with the time periodic solutions to the one-dimensional nonlinear wave equation with either variable or constant coefficients. By adjusting the basis of L ² function space, we can circumvent the difficulties caused by η _u  = 0 and obtain the existence of a weak periodic solution, which was posed as an open problem by Baubu and Pavel in (Trans Am Math Soc 349:2035–2048, 1997). Finally, an application to the forced Sine-Gordon equation is presented to illustrate the utility of this technique.     

Nonlinear stability theory of channel flow with heat transfer – an asymptotic approach

A fully developed laminar Poiseuille flow subject to constant heat flux across the wall is analysed with respect to its stability behavior by applying a weakly nonlinear stability theory. It is based on an expansion of the disturbance control equations with respect to a perturbation parameter ε. This parameter is the small initial amplitude of the fundamental wave. This fundamental wave which is the solution of the linear (Orr-Sommerfeld) first order equation triggers all higher order effects with respect to ε. Heat transfer is accounted for asymptotically through an expansion with respect to a small heat transfer parameter ε _T . Both perturbation parameters, ε and ε _T , are linked by the assumption ε _T =O(ε²) by which a certain distinguished limit is assumed. The results for a fluid with temperature dependent viscosity show that heat transfer effects in the nonlinear range continue to act in the same way as in the initial linear range. Received on 11 August 1997     

Time-Periodic Solutions of the Burgers Equation

We investigate the time periodic solutions to the viscous Burgers equation u_t − μu_xx + uu_x = f for irregular forcing terms. We prove that the corresponding Burgers operator is a diffeomorphism between appropriate function spaces.      

Generalized Solution for 1-D Non-Newtonian Flow in a Porous Domain due to an Instantaneous Mass Injection

Non-Newtonian fluid flow through porous media is of considerable interest in several fields, ranging from environmental sciences to chemical and petroleum engineering. In this article, we consider an infinite porous domain of uniform permeability k and porosity f{\phi} , saturated by a weakly compressible non-Newtonian fluid, and analyze the dynamics of the pressure variation generated within the domain by an instantaneous mass injection in its origin. The pressure is taken initially to be constant in the porous domain. The fluid is described by a rheological power-law model of given consistency index H and flow behavior index n; n, < 1 describes shear-thinning behavior, n > 1 shear-thickening behavior; for n = 1, the Newtonian case is recovered. The law of motion for the fluid is a modified Darcy’s law based on the effective viscosity μ _ef , in turn a function of f, H, n{\phi, H, n} . Coupling the flow law with the mass balance equation yields the nonlinear partial differential equation governing the pressure field; an analytical solution is then derived as a function of a self-similar variable η = rt ^β (the exponent β being a suitable function of n), combining spatial coordinate r and time t. We revisit and expand the work in previous papers by providing a dimensionless general formulation and solution to the problem depending on a geometrical parameter d, valid for plane (d = 1), cylindrical (d = 2), and semi-spherical (d = 3) geometry. When a shear-thinning fluid is considered, the analytical solution exhibits traveling wave characteristics, in variance with Newtonian fluids; the front velocity is proportional to t ^(n-2)/2 in plane geometry, t ^{(2n-3)/(3−n)} in cylindrical geometry, and t ^{(3n-4)/[2(2−n)]} in semi-spherical geometry. To reflect the uncertainty inherent in the value of the problem parameters, we consider selected properties of fluid and matrix as independent random variables with an associated probability distribution. The influence of the uncertain parameters on the front position and the pressure field is investigated via a global sensitivity analysis evaluating the associated Sobol’ indices. The analysis reveals that compressibility coefficient and flow behavior index are the most influential variables affecting the front position; when the excess pressure is considered, compressibility and permeability coefficients contribute most to the total response variance. For both output variables the influence of the uncertainty in the porosity is decidedly lower.     

Spinodal Decomposition for the¶Cahn-Hilliard Equation in Higher Dimensions:¶Nonlinear Dynamics

This paper addresses the phenomenon of spinodal decomposition for the Cahn-Hilliard equation

Asymptotics of the Fast Diffusion Equation via Entropy Estimates

We consider non-negative solutions of the fast diffusion equation u _t  = Δ u ^m with m ∈ (0, 1) in the Euclidean space , d ≧ 3, and study the asymptotic behavior of a natural class of solutions in the limit corresponding to t → ∞ for m ≧ m _c  = (d − 2)/d, or as t approaches the extinction time when m < m _c . For a class of initial data, we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if m ≧ m _c , or close enough to the extinction time if m < m _c . Such results are new in the range m ≦ m _c where previous approaches fail. In the range m _c  < m < 1, we improve on known results.     

Radial Symmetry of Overdetermined Boundary-Value Problems in Exterior Domains

In this paper, we extend a classical result by J. Serrin [15] to exterior domains , where Ω is a bounded domain. We prove, under some hypotheses on f, that if there exists a solution of satisfying the overdetermined boundary conditions that and u are constant on , and such that , then the domain Ω is a ball. Under different assumptions on f, this result has been obtained by W. Reichel in [13]. The main result here covers new cases like with . When Ω is a ball, almost the same proof allows us to derive the symmetry of positive bounded solutions satisfying only the Dirichlet condition that u is constant on . Our method relies on Kelvin transforms, various forms of the maximum principle and the device of moving planes up to a critical position. (Accepted May 30, 1997)     

Combined heat and mass transfer along a vertical moving cylinder with a free stream

The mixed convection flow over a continuous moving vertical slender cylinder under the combined buoyancy effect of thermal and mass diffusion has been studied. Both uniform wall temperature (concentration) and uniform heat (mass) flux cases are included in the analysis. The problem is formulated in such a manner that when the ratio λ(= u _w/(u _w + u _∞), where u _w and u _∞ are the wall and free stream velocities, is zero, the problem reduces to the flow over a stationary cylinder, and when λ = 1 it reduces to the flow over a moving cylinder in an ambient fluid. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. We have also obtained the solution using a perturbation technique with Shanks transformation. This transformation has been used to increase the range of the validity of the solution. For some particular cases closed form solutions are obtained. The surface skin friction, heat transfer and mass transfer increase with the buoyancy forces. The buoyancy forces cause considerable overshoot in the velocity profiles. The Prandtl number and the Schmidt number strongly affect the surface heat transfer and the mass transfer, respectively. The surface skin friction decreases as the relative velocity between the surface and free stream decreases. Received on 17 May 1999     

Global Semigroup of Conservative Solutions of the Nonlinear Variational Wave Equation

We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u _tt − c(u)(c(u)u _x ) _x  = 0. We allow for initial data u| _{t = 0} and u _t | _t=0 that contain measures. We assume that 0 < k^-1 \leqq c(u) \leqq k{0 < \kappa^{-1} \leqq c(u) \leqq \kappa}. Solutions of this equation may experience concentration of the energy density (u_t²+c(u)²u_x²)dx{(u_t^2+c(u)^2u_x^2){\rm d}x} into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby singularities in the energy density become manageable. Furthermore, we prove that the energy may focus only on a set of times of zero measure or at points where c′(u) vanishes. A new numerical method for constructing conservative solutions is provided and illustrated with examples.     

A natural neighbour method based on Fraeijs de Veubeke variational principle for materially non-linear problems

Abstract The natural neighbour method can be considered as one of many variants of the meshless methods. In the present paper, a new approach based on the Fraeijs de Veubeke （FdV） functional, which is initially developed for linear elasticity, is extended to the case of geometrically linear but materially non-linear solids. The new approach provides an original treatment to two classical problems： the numerical evaluation of the integrals over the domain A and the enforcement of boundary conditions of the type ui = hi on Su. In the absence of body forces （Fi = 0）, it will be shown that the calculation of integrals of the type fA .dA can be avoided and that boundary conditions of the type ui = hi on Su can be imposed in the average sense in general and exactly if hi is linear between two contour nodes, which is obviously the case for tTi = O.     

BV Solutions of the Semidiscrete Upwind Scheme

  We consider the semidiscrete upwind scheme

Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold

We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation u_t=D(u^m/m) = div (u^m-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for x ? \mathbb R^d{x\in\mathbb R^d}, t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold (\mathbb R^d,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard \mathbb R^d{\mathbb R^d} metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of m.     

Isolated Singularities of the 1D Complex Viscous Burgers Equation

The Cauchy problem for the 1D real-valued viscous Burgers equation u _t +uu _x  = u _xx is globally well posed (Hopf in Commun Pure Appl Math 3:201–230, 1950). For complex-valued solutions finite time blow-up is possible from smooth compactly supported initial data, see Poláčik and Šverák (J Reine Angew Math 616:205–217, 2008). It is also proved in Poláčik and Šverák (J Reine Angew Math 616:205–217, 2008) that the singularities for the complex-valued solutions are isolated if they are not present in the initial data. In this paper we study the singularities in more detail. In particular, we classify the possible blow-up rates and blow-up profiles. It turns out that all singularities are of type II and that the blow-up profiles are regular steady state solutions of the equation.     

Symmetry of Mountain Pass Solutions of Some Vector Field Equations

We prove radial symmetry (or axial symmetry) of the mountain pass solution of variational elliptic systems − AΔu(x) + ∇ F(u(x)) = 0 (or − ∇.(A(r) ∇ u(x)) + ∇ F(r,u(x)) = 0,) u(x) = (u ₁(x),...,u _N (x)), where A (or A(r)) is a symmetric positive definite matrix. The solutions are defined in a domain Ω which can be , a ball, an annulus or the exterior of a ball. The boundary conditions are either Dirichlet or Neumann (or any one which is invariant under rotation). The mountain pass solutions studied here are given by constrained minimization on the Nehari manifold. We prove symmetry using the reflection method introduced in Lopes [(1996), J. Diff. Eq. 124, 378–388; (1996), Eletron. J. Diff. Eq. 3, 1–14].     

A triangular quasi-conforming finite element for transient dynamic analysis

A type of 3 node triangular element is constructed by the Quasi-conforming method, which may be used to solve the equation of a type of inverse problem of wave propagation after Laplace transformation Δu−A ² u=0. The strains in the element are approximated by an exponential function and the string-net function between neighbouring elements is approximated by one dimensional general solution of the equation. Furthermore the strain field satisfies the equation, and therefore in the derivation of the element formulation, no shape function is needed. In this sense, it is a kind of hybrid element. Compared with the ordinary linear triangular element, the new one features higher precision with coarse meshes. Some numerical tests are presented. The project is supported by the National Natural Science Foundation of China.     

Asymptotic Variational Wave Equations

We investigate the equation (u _t +(f(u)) _x ) _x =f ^{′ ′}(u) (u _x )²/2 where f(u) is a given smooth function. Typically f(u)=u ²/2 or u ³/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation u _tt − c(u) (c(u)u _x ) _x =0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the Euler–Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.We establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function f has a Lipschitz continuous second-order derivative. In the case where f is convex, the Cauchy problem is well posed also within the class of dissipative solutions. However, when f is not convex, we show that the dissipative solutions do not depend continuously on the initial data.     

An Efficient Derivation of the Aronsson Equation

For 1<p<∞, the equation which characterizes minima of the functional u↦∫ _U |Du| ^p ,dx subject to fixed values of u on ∂U is −Δ _p u=0. Here −Δ _p is the well-known ``p-Laplacian'. When p=∞ the corresponding functional is u↦|| |Du|<sup>2</sup>|| <sub>L∞(U)</sub> . A new feature arises in that minima are no longer unique unless U is allowed to vary, leading to the idea of ``absolute minimizers'. Aronsson showed that then the appropriate equation is −Δ_∞ u=0, that is, u is ``infinity harmonic' as explained below. Jensen showed that infinity harmonic functions, understood in the viscosity sense, are precisely the absolute minimizers. Here we advance results of Barron, Jensen and Wang concerning more general functionals u↦||f(x,u,Du)|| _L∞(U) by giving a simplified derivation of the corresponding necessary condition under weaker hypotheses. (Accepted September 6, 2002) Published online April 14, 2003 Communicated by S. Muller     

The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation D² u=\fracl(1-u)²{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball B ì \mathbbR^N{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?_n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u _λ with 0 < u _λ < 1 exists for l ? (0,l^*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution u_l^*{u_{\lambda^*}} is regular (sup_B u_l^* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided N \leqq 8{N \leqq 8} while u_l^*{u_{\lambda^*}} is singular (sup_B u_l^* = 1){({\rm sup}_B u_{\lambda^*} =1)} for N \geqq 9{N \geqq 9}, in which case 1-C₀|x|^4/3 \leqq u_l^* (x) \leqq 1-|x|^4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where C₀:=(\fracl^*[`(l)])<sup>\frac</sup>13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and [`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}.