Postgraduate research supervision

© 2014 Society for Industrial and Applied Mathematics.This article addresses a fundamental concern regarding the incompressible approximation of fluid motions, one of the most widely used approximations in fluid mechanics. Common belief is that its accuracy is O(µ), where µ denotes the Mach number. In this article, however, we prove an O(µ2) accuracy for the incompressible approximation of the isentropic, compressible Euler equations thanks to several decoupling properties. At the initial time, the velocity field and its first time derivative are of O(1) size, but the boundary conditions can be as stringent as the solid-wall type. The fast acoustic waves are still O(µ) in magnitude, since the O(µ2) error is measured in the sense of Leray projection and more physically, in time-averages. We also show when a passive scalar is transported by the flow, it is O(µ2) accurate pointwise in time to use incompressible approximation for the velocity field in the transport equation.

We prove the global-in-time existence of large-data nite-energy weak solutions to an incompressible hybrid Vlasov-magnetohydrodynamic model in three space dimensions. The model couples three essential ingredients of magnetized plasmas: a transport equation for the probability density function, which models energetic rare ed particles of one species; the incompressible Navier{Stokes system for the bulk uid; and a parabolic evolution equation, involving magnetic di usivity, for the magnetic eld. The physical derivation of our model is given. It is also shown that the weak solution, whose existence is established, has nonincreasing total energy, and that it satis es a number of physically relevant properties, including conservation of the total momentum, conservation of the total mass, and nonnegativity of the probability density function for the energetic particles. The proof is based on a one-level approximation scheme, which is carefully devised to avoid increase of the total energy for the sequence of approximating solutions, in conjunction with a weak compactness argument for the sequence of approximating solutions. The key technical challenges in the analysis of the mathematical model are the nondissipative nature of the Vlasov-type particle equation and passage to the weak limits in the multilinear coupling terms.

An inertial manifold (IM) is one of the key objects in the modern theory of dissipative systems generated by partial differential equations (PDEs) since it allows us to describe the limit dynamics of the considered system by the reduced finite-dimensional system of ordinary differential equations (ODEs). It is well known that the existence of an IM is guaranteed when the so called spectral gap conditions are satisfied, whereas their violation leads to the possibility of an infinite-dimensional limit dynamics, at least on the level of an abstract parabolic equation. However, these conditions restrict greatly the class of possible applications and are usually satisfied in the case of one spatial dimension only.

Despite many efforts in this direction, the IMs in the case when the spectral gap conditions are violated remain a mystery especially in the case of parabolic PDEs. On the one hand, there is a number of interesting classes of such equations where the existence of IMs is established without the validity of the spectral gap conditions and, on the other hand there were no examples of dissipative parabolic PDEs where the non-existence of an IM is rigorously proved.

The main aim of this thesis is to bring some light on this mystery by the comprehensive study of three model examples of parabolic PDEs where the spectral gap conditions are not satisfied, namely, 1D reaction-diffusion-advection (RDA) systems (see Chapter 3), the 3D Cahn-Hilliard equation on a torus (see Chapter 4) and the modified 3D Navier-Stokes equations (see Chapter 5). For all these examples the existence or non-existence of IM was an open problem.

As shown in Chapter 3, the existence or non-existence of an IM for RDA systems strongly depends on the boundary conditions. In the case of Dirichlet or Neumann boundary conditions, we have proved the existence of an IM using a specially designed non-local in space diffeomorphism which transforms the equations to the new ones for which the spectral gap conditions are satisfied. In contrast to this, in the case of periodic boundary conditions, we construct a natural example of a RDA system which does not possess an IM.

In Chapters 4 and 5 we develop an extension of the so-called spatial averaging principle (SAP) (which has been suggested by Sell and Mallet-Paret in order to treat scalar reaction-diffusion equation on a 3D torus) to the case of 4th order equations where the nonlinearity loses smoothness (the Cahn-Hilliard equation) as well as

This paper presents a single-column model of moist atmospheric convection. The
problem is formulated in terms of conservation laws for mass, moist potential
temperature and specific humidity of air parcels. A numerical adjustment algorithm
is devised to model the convective adjustment of the column to a statically stable
equilibrium state for a number of test cases. The algorithm is shown to converge to
a weak solution with saturated and unsaturated parcels interleaved in the column as
the vertical spatial grid size decreases. Such weak solutions would not be obtainable via
discrete PDE methods, such as finite differences or finite volumes, from the governing
Eulerian PDEs. An equivalent variational formulation of the problem is presented and
numerical results show equivalence with those of the adjustment algorithm. Results are
also presented for numerical simulations of an ascending atmospheric column as a series
of steady states. The adjustment algorithm developed in this paper is advantageous
over similar algorithms because first it includes the latent heating of parcels due to
the condensation of water vapour, and secondly it is computationally efficient making it
implementable into current weather and climate models.

We study a single column model of moist convection in the atmosphere.
We state the conditions for it to represent a stable steady state. We then evolve
the column by subjecting it to an upward displacement which can release instability,
leading to a time dependent sequence of stable steady states. We propose a definition
of measure valued solution to describe the time dependence and prove its existence.

Singular limits of a class of evolutionary systems of partial differential
equations having two small parameters and hence three time scales are considered.
Under appropriate conditions solutions are shown to exist and remain uniformly
bounded for a fixed time as the two parameters tend to zero at different rates. A
simple example shows the necessity of those conditions in order for uniform bounds
to hold. Under further conditions the solutions of the original system tend to solutions
of a limit equation as the parameters tend to zero.

We study classical solutions of one dimensional rotating shallow water system which
plays an important role in geophysical fluid dynamics. The main results contain two contrasting
aspects. First, when the solution crosses certain thresholds, we prove finite-time singularity
formation for the classical solutions by studying the weighted gradients of Riemann invariants
and utilizing conservation of physical energy. In fact, the singularity formation will take place for
a large class of
C
1
initial data whose gradients and physical energy can be arbitrarily small and
higher order derivatives should be large. Second, when the initial data have constant potential
vorticity, global existence of small classical solutions is established via studying an equivalent
form of a quasilinear Klein-Gordon equation satisfying certain null conditions. In this global
existence result, the smallness condition is in terms of the higher order Sobolev norms of the
initial data.

Time averages are common observables in analysis of experimental data and numerical simulations of physical systems. We
will investigate, from the angle of partial differential equation
analysis, some oscillatory geophysical fluid dynamics in three
dimensions: Navier?Stokes equations in a fast-rotating, spherical shell, and magnetohydrodynamics subject to strong Coriolis
and Lorentz forces. Upon averaging their oscillatory solutions
in time, interesting patterns such as zonal flows can emerge.
More rigorously, we will prove that, when the restoring forces
are strong enough, time-averaged solutions stay close to the null
spaces of the wave operators, whereas the solutions themselves
can be arbitrarily far away from these subspaces.