Michiel Bertsch

Nonuniqueness for the heat flow of harmonic maps on the disk

Abstract We prove that for suitable initial data the heat flow of harmonic maps, from the disk to the sphere, admits infinitely many solutions, characterised by “backward bubbling” at some arbitrarily large time, all having uniformly bounded energy.

Positivity properties of viscosity solutions of a degenerate parabolic equation

We consider the problem: (I) {u t =uΔu−γ|⊇u| 2 in R N ×R + , u(x, 0)=u 0 (x) in R N , where γ is a nonnegative constant and u 0 is a bounded continuous and nonnegative function on R N

Hyperbolic Phenomena in a Strongly Degenerate Parabolic Equation

We consider the equation ut=(ϕ(u) ψ (ux))x, where ϕ>0 and where ψ is a strictly increasing function with lims→∞ψ=ψ∞<∞. We solve the associated Cauchy problem for an increasing initial function, and discuss to what extent the solution behaves qualitatively like solutions of the first-order conservation law ut=ψ∞(ϕ(u))x. Equations of this type arise, for example, in the theory of phase transitions where the corresponding free-energy functional has a linear growth rate with respect to the gradient.

Discontinuous “viscosity” solutions of a degenerate parabolic equation

We study a nonlinear degenerate parabolic equation of the second order. Regularizing the equation by adding some artificial viscosity, we construct a generalized solution. We show that this solution is not necessarily continuous at all points.

A mathematical model of turbulent heat and mass transfer in stably stratified shear flow

It is commonly assumed that heat flux and temperature diffusivity coefficients obtained in steady-state measurements can be used in the derivation of the heat conduction equation for fluid flows. Meanwhile it is also known that the steady-state heat flux as a function of temperature gradient in stably stratified turbulent shear flow is not monotone: at small values of temperature gradient the flux is increasing, whereas it is decreasing after a certain critical value of the temperature gradient. Therefore the problem of heat conduction for large values of temperature gradient becomes mathematically ill-posed, so that its solution (if it exists) is unstable

Nonuniqueness of solutions of a degenerate parabolic equation

SummaryWe give some results about nonuniqueness of the solutions of the Cauchy problem for a class of nonlinear degenerate parabolic equations arising in several applications in biology and physics. This phenomenon is a truly nonlinear one and occurs because of the degeneracy of the equation at the points where u=0. For a given set of values of the parameter involved, we prove that there exists a one parameter family of weak solutions; moreover, restricting the parameter set, nonuniqueness appears even in the class of classical solutions.

Analysis of oil trapping in porous media flow

We analyze a one-dimensional nonlinear convection-diffusion equation describing the flow of water and oil through a porous medium composed of two types of rock with different permeability. We prove existence, uniqueness, and regularity properties, as well as matching conditions between the two rock types.

Self-similar intermediate asymptotics for a degenerate parabolic filtration-absorption equation

The equation partial differential(t)u = u partial differential(xx)(2)u -(c-1)( partial differential(x)u)(2) is known in literature as a qualitative mathematical model of some biological phenomena. Here this equation is derived as a model of the groundwater flow in a water-absorbing fissurized porous rock; therefore, we refer to this equation as a filtration-absorption equation. A family of self-similar solutions to this equation is constructed. Numerical investigation of the evolution of non-self-similar solutions to the Cauchy problems having compactly supported initial conditions is performed. Numerical experiments indicate that the self-similar solutions obtained represent intermediate asymptotics of a wider class of solutions when the influence of details of the initial conditions disappears but the solution is still far from the ultimate state: identical zero. An open problem caused by the nonuniqueness of the solution of the Cauchy problem is discussed.

Point Singularities and Nonuniqueness for the Heat Flow for Harmonic Maps

Abstract We consider weak solutions of the harmonic map flow between the three-dimensional unit ball B 3and the two-dimensional unit sphere, with as initial and boundary data. In this situation, we show the existence of infinitely many weak solutions. Indeed for any different from the origin we construct a weak solution whose singular set for all positive time is precisely Qand which satisfies a modified energy inequality.

A free boundary problem arising in a simplified tumour growth model of contact inhibition

It is observed in vitro and in vivo that when two populations of different types of cells come near to each other, the rate of proliferation of most cells decreases. This phenomenon is often called contact inhibition of growth between two cells. In this paper, we consider a simplified 1-dimensional PDEmodel for normal and abnormal cells, motivated by the paper by Chaplain, Graziano and Preziosi ([5]). We show that if the two populations are initially segregated, then they remain segregated due to the contact inhibition mechanism. In this case the system of PDE’s can be formulated as a free boundary problem.