We study travelling wave solutions of a one-dimensional two-phase Free Boundary Problem, which models premixed flames propagating in a gaseous mixture with dust. The model combines diffusion of mass and temperature with reaction at the flame front, the reaction rate being temperature dependent. The radiative effects due to the presence of dust account for the divergence of the radiative flux entering the equation for temperature. This flux is modelled by the Eddington equation. In an appropriate limit the divergence of the flux takes the form of a nonlinear heat loss term. The resulting reduced model is able to capture a hysteresis effect that appears if the amount of fuel in front of the flame, or equivalently, the adiabatic temperature is taken as a control parameter.

In this work we describe some aspects of the dynamics of the Cahn–Hilliard equation. In particular, we consider the dynamics of ‘bubble’ solutions that is spherical interfaces which move superslowly towards the boundary without changing their shape. We show for the Cahn–Hilliard that the bubble drifts towards the closest point on the boundary provided it is sufficiently small. This is contrasted with the related mass conserving Allen–Cahn equation where size is not an issue.

In this paper we study a nonlocal parabolic/elliptic system which models thermistor behaviour in cases where heat losses to the surrounding gas play a significant role. The existence of time periodic solutions for the system is established through Faedo-Galerkin approximations and the Leray–Schauder degree theory. We show that for the small gas pressure case, the temperature of the time periodic solutions is positive. Moreover we consider the long time behaviour of the system and prove the existence of a uniform attractor. Finally, the finite dimensionality of the attractor is discussed.

A vortex sheet is a surface across which the velocity field of incompressible and inviscid flows has a jump discontinuity. Mathematical and numerical studies reveal that a two-dimensional vortex sheet, which is governed by the Birkhoff–Rott equation, acquires a singularity in finite time without forming rolling-up spiral. On the other hand, numerical computation of a regularized Birkhoff–Rott equation shows that the vortex sheet evolves into a rolling-up doubly branched spiral. Because of the finite-time singularity, it is impossible to regard the rolling-up spiral as a solution of the Birkhoff–Rott equation as long as time is real. However, it may be possible to analytically continue the equation to the spiral along a path to get around the singularity in complex-time plane. In the present article, we consider singularities in complex-time plane for the regularized Birkhoff–Rott equation by numerical means. Distribution of the complex singularities and their limiting behaviour indicate that it is absolutely impossible to perform analytic continuation in complex-time domain to the spiral solution. Furthermore, we propose a simple model of a doubly branched spiral and investigate it mathematically. The model is successful in approximating the rolling-up motion of the vortex sheet. Comparing the vortex-sheet motion with the model indicate that the doubly branched spiral with infinite windings at the centre could be a solution of the Birkhoff–Rott equation beyond the singularity time.

‘Parametric dependence of exponents and eigenvalues in focusing porous media flows’ by D. G. Aronson, J. B. Van den Berg and J. Hulshof, European Journal of Applied Mathematics, Vol. 14 No. 4 August 2003, pp. 485–512

Recent numerical evidence [8, 28, 33] suggests that in the Hele–Shaw suction problem with vanishingly small surface tension $\gamma$, the free boundary generically approaches the sink in a wedge-like configuration, blow-up occurring when the wedge apex reaches the sink. Sometimes two or more such wedges approach the sink simultaneously [33]. We construct a family of solutions to the zero-surface tension (ZST) problem in which fluid is injected at the (coincident) apices of an arbitrary number $N$ of identical infinite wedges, of arbitrary angle. The time reversed suction problem then models what is observed numerically with non-zero surface tension. We conjecture that (for a given value of $N$) a particular member of this family of ZST solutions, with special complex plane singularity structure, is selected in the limit $\gamma\,{\to}\,0$.

In this paper, we define and study the Weil–Petersson geometry. Under the framework of the Weil–Petersson geometry, we study the Weil–Petersson metric and the Hodge metric. Among the other results, we represent the Hodge metric in terms of the Weil–Petersson metric and the Ricci curvature of the Weil–Petersson metric for Calabi–Yau fourfold moduli. We also prove that the Hodge volume of the moduli space is finite. Finally, we proved that the curvature of the Hodge metric is bounded if the Hodge metric is complete and the dimension of the moduli space is 1.

AMS 2000 Mathematics subject classification: Primary 53A30. Secondary 32C16

We consider the Cauchy problem for a class of scalar conservation laws with flux having a single inflection point. We prove existence of global weak solutions satisfying a single entropy inequality together with a kinetic relation, in a class of bounded variation functions. The kinetic relation is obtained by the travelling-wave criterion for a regularization consisting of balanced diffusive and dispersive terms. The result is applied to the one-dimensional Buckley-Leverett equation.

We study the long-time behaviour of solutions of autonomous and non-autonomous reaction-diffusion equations in unbounded domains of R3. It is shown that, under appropriate assumptions on the nonlinear interaction function and on the external forces, these equations possess compact global (uniform) attractors in the corresponding phase space. Estimates for Kolmogorov's ε-entropy of these attractors in terms of Kolmogorov's entropy of the external forces are given. Moreover, (infinite-dimensional) exponential attractors with the same entropy estimate as that of the corresponding global (uniform) attractor are also constructed.

We prove the existence of a non-trivial solution for the nonlinear elliptic problem −Δu + V(x)u = a(x)g(u) in RN, where g is superlinear near zero and near infinity, a(x) changes sign and V ∈ C(RN) is positive at infinity. For g odd, we prove the existence of an infinite number of solutions.

In this paper we analyse a singular perturbation problem for linear wave equations with interior and boundary damping. We show how the solutions converge to the formal parabolic limit problem with dynamic boundary conditions. Conditions are given for uniform convergence in the energy space.

We give an example of an indefinite weight Sturm-Liouville problem whose eigenfunctions form a Riesz basis under Dirichlet boundary conditions but not under anti-periodic boundary conditions.

We construct a relative de Rham–Witt complex $W\varOmega^{\cdot}_{X/S}$ for a scheme $X$ over a base scheme $S$. It coincides with the complex defined by Illusie (Annls Sci. Ec. Norm. Super.12 (1979), 501–661) if $S$ is a perfect scheme of characteristic $p>0$. The hypercohomology of $W\varOmega^{\cdot}_{X/S}$ is compared to the crystalline cohomology if $X$ is smooth over $S$ and $p$ is nilpotent on $S$. We obtain the structure of a $3n$-display on the first crystalline cohomology group if $X$ is proper and smooth over $S$.

AMS 2000 Mathematics subject classification: Primary 14F30; 14F40

An initial boundary-value problem for the modified Korteweg–de Vries equation on the half-line, $0<x<\infty$, $t>0$, is analysed by expressing the solution $q(x,t)$ in terms of the solution of a matrix Riemann–Hilbert (RH) problem in the complex $k$-plane. This RH problem has explicit $(x,t)$ dependence and it involves certain functions of $k$ referred to as the spectral functions. Some of these functions are defined in terms of the initial condition $q(x,0)=q_0(x)$, while the remaining spectral functions are defined in terms of the boundary values $q(0,t)=g_0(t)$, $q_x(0,t)=g_1(t)$, and $q_{xx}(0,t)=g_2(t)$. The spectral functions satisfy an algebraic global relation which characterizes, say, $g_2(t)$ in terms of $\{q_0(x),g_0(t),g_1(t)\}$. It is shown that for a particular class of boundary conditions, the linearizable boundary conditions, all the spectral functions can be computed from the given initial data by using algebraic manipulations of the global relation; thus, in this case, the problem on the half-line can be solved as efficiently as the problem on the whole line.

AMS 2000 Mathematics subject classification: Primary 37K15; 35Q53. Secondary 35Q15; 34A55

We study some properties of space-like submanifolds in Minkowski n-space, whose points are all umbilic with respect to some normal field. As a consequence of these and some results contained in a paper by Asperti and Dajczer, we obtain that being ν-umbilic with respect to a parallel light-like normal field implies conformal flatness for submanifolds of dimension n − 2 ≥ 3. In the case of surfaces, we relate the umbilicity condition to that of total semi-umbilicity (degeneracy of the curvature ellipse at every point). Moreover, if the considered normal field is parallel, we show that it is everywhere time-like, space-like or light-like if and only if the surface is included in a hyperbolic 3-space, a de Sitter 3-space or a three-dimensional light cone, respectively. We also give characterizations of total semi-umbilicity for surfaces contained in hyperbolic 4-space, de Sitter 4-space and four-dimensional light cone.

In this paper we study the convexity of the level sets of solutions of the problemwhere f is a suitable function with subcritical or critical growth. Under some assumptions on the Gauss curvature of ∂Ω, we prove that the level sets of the solution of (0.1) are strictly convex.

For a wide class of nonlinearities f(u) satisfyingbut not necessarily Lipschitz continuous, we study the quasi-linear equationwhere T = {x = (x1, x2, …, xN) ∈ RN: x1 > 0} with N ≥ 2. By using a new approach based on the weak maximum principle, we show that any positive solution on T must be a function of x1 only. Under our assumptions, the strong maximum principle does not hold in general and the solution may develop a flat core; our symmetry result allows an easy and precise determination of the flat core.

This paper deals with the existence and the behaviour of global connected branches of positive solutions of the problemThe function f is allowed to change sign and has an asymptotically linear or a superlinear behaviour.

Given a lattice in an isocrystal, Mazur’s inequality states that the Newton point of the isocrystal is less than or equal to the invariant measuring the relative position of the lattice and its transform under Frobenius. Conversely, it is known that any potential invariant allowed by Mazur’s inequality actually arises from some lattice. These can be regarded as statements about the group $GL_n$. This paper proves an analogous converse theorem for all split classical groups.

AMS 2000 Mathematics subject classification: Primary 14L05. Secondary 11S25; 14F30; 20G25