Turbulence: intermittency in the 2D & 3D-Navier-Stokes equations & regularity
issues; (potential) singularities in the 3D-Euler equations.
Clustering in complex systems. Fluid convection.
One of the most interesting & unexplained phenomenon in 3D incompressible
Navier-Stokes isotropic turbulence is the problem of regularity of solutions
for arbitrarily long times. As one of the major unsolved problems of modern
applied mathematics it is listed as on of the 7 AMS-Clay million-dollar
Prize problems. This issue is intimately related to the problem of dissipation-range
intermittency in solutions of the Navier-Stokes equations at high Reynolds numbers.
Vorticity has the tendency to accumulate on "thin sets" in the form of quasi
one-dimensional tubes & quasi two-dimensional sheets; these structures dominate
the local vortical topology. This phenomenon is closely related to the regularity
issue because the direction of vorticity is one of the factors that determines the
size of the vortex stretching term. It is our inability to bound this term in a
satisfactory fashion that leads to our failure to prove that the 3D Navier-Stokes
equations are regular (see Doering & Gibbon Applied Analysis of the Navier-Stokes
equations CUP '95). Some of those in the list of newer papers above together
with some older ones below address these issues.
- C. R. Doering and J. D. Gibbon: 2001 Bounds on moments of the energy spectrum for
weak solutions of the 3D-Navier-Stokes equations, (Physica D, vol 165,
pp163-175, 2002) (pdf)
- J-L. Thiffeault, C. R. Doering & J. D. Gibbon, A bound on mixing efficiency for
the advection–diffusion equation, (J. Fluid Mechanics, vol 521, 105-114, 2004).
(pdf)
Many active quantities cluster into concentrated sets, like bubbles in a Swiss
cheese. Vorticity in a fluid is just one example; the number of examples in
physics and biology is manifold. The paper below, co-authored with Edriss Titi
of Israel's Weizmann Institute, shows that competition between members of a
hierarchy of length scales in complex multi-scale systems gives generically induces
clustering that dominates the intermittent structure. The clusters have
halo-like surfaces that have scaling exponents lower than that of their kernels,
which can be as high as the domain dimension.
- J. D Gibbon and E. S. Titi, Cluster formation in complex multi-scale systems,
(Proc Royal Soc., vol 461, 3089--3097, 2005. DOI ref: dx.doi.org/10.1098/rspa.2005.1548)
(pdf)
Work on Quaternions and particle dynamics in the Euler fluid equations
below hints that the three-dimensional Euler equations has a geometric structure.
Some of those in the list of newer papers above together with some older ones below
address these issues.
- J. D. Gibbon and D. D. Holm; Lagrangian particle paths and
orthonormal quaternion frames (see above)
- J. D. Gibbon, D. D. Holm, R. M. Kerr & I. Roulstone; Quaternions and particle
dynamics in the Euler fluid equations (Nonlinearity, vol 19, 1969-1983, 2006)
(pdf)
- J. D. Gibbon: A quaternionic structure in the three-dimensional Euler
and equations for ideal MHD (Physica D, vol 166, pp17-28, 2002)(pdf)
- B. Galanti, J. D. Gibbon & M. Heritage: Vorticity alignment results for the 3D-Euler
and Navier-Stokes equations (Nonlinearity, vol 10, pp1675-1695, 1997)(pdf)
Another interesting problem is that of the behaviour of solutions of the 3D incompressible
Euler equations (and related equations) where there is evidence to suggest that finite energy
solutions may become singular in a finite time (at least with well-prepared initial data),
but no proof is known and little is known about the process. The following papers discuss
the infinite energy blow-up process. The original paper in this series is the last.
- J. D. Gibbon, D. R. Moore & J. T. Stuart: Exact, infinite energy, blow-up solutions
of the three-dimensional Euler equations (Nonlinearity, vol 16, pp1823-1831, 2004)
(pdf)
- J. D. Gibbon and K. Ohkitani: Singularity formation in a class of
stretched solutions of the equations for ideal MHD
(Nonlinearity, vol 14, 1239-1264, 2001) (pdf)
- K. Ohkitani & J. D. Gibbon: Numerical study of singularity formation in a class
of Euler and Navier-Stokes flows, (Physics of Fluids, vol 12, pp3181-3194, 2000)
(pdf)
- J. D. Gibbon, A. Fokas & C. R. Doering: Dynamically stretched vortices as solutions
of the 3D Navier-Stokes equations, (Physica D, vol 132, 497-510, 1999)
(pdf)
Teaching:
Mathematics for 2nd Year Electrical Engineering.
- 1) Lecture notes on Vector Calculus: (pdf).
- 2) Lecture notes on Complex variables: (pdf).
- 3) Lecture notes on Transforms: (pdf).
- 4) Lecture notes on Linear Algebra: (pdf).
- 5) The syllabus content: (pdf).
- The handouts available are:
- 1) The first handout on "Things you need to recall about Vector Algebra"
(pdf).
- 2) The second handout on "The role of grad, div and curl in vector calculus"
(pdf).
- 3) The third handout on "Changing the order in double integration"
(pdf).
- 4) The fourth handout on "Green's, Divergence & Stokes' Theorems plus
Maxwell's Equations" (pdf).
- 5) The fifth handout on "Jordan's Lemma" in complex integration
(pdf).
- 6) The sixth handout on "Fourier Transforms" (pdf).
- 7) The seventh handout on "Laplace Transforms" (pdf).
- 8) The Mathematics Department Public Formula Sheet: (pdf)
and another with the stats-exam-formulae (pdf).
- Problem sheets available are:
- Problem Sheet 1 (pdf).
- Solutions to Sheet 1 (pdf).
- Problem Sheet 2 (pdf).
- Solutions to Sheet 2 (pdf).
- Problem Sheet 3 (pdf).
- Solutions to Sheet 3 (pdf).
- Problem Sheet 4 (pdf).
- Solutions to Sheet 4 (pdf).
- Problem Sheet 5 (pdf).
- Solutions to Sheet 5 (pdf).
- Problem Sheet 6 (pdf).
- Solutions to Sheet 6 (pdf).
- Problem Sheet 7 (pdf).
- Solutions to Sheet 7 (pdf).
- Problem Sheet 8 (pdf).
- Solutions to Sheet 8 (pdf).
Mathematics for 2nd Year Aeronautical Engineering.
- The syllabus content: (pdf).
- 1) Lecture notes on Fourier Series: (pdf).
- 2) Lecture notes on PDEs: (pdf).
- The handouts available are:
- Handout on the formulae used for Characteristics and how
they are used to solve 1st and 2nd order PDEs (pdf).
- Handout on the Chain Rule in partial differentiation.
(pdf).
- Handout on the solution of the Wave, Diffusion & Laplace's
equation using separation of variables: (pdf).
See also the supplementary handout on hyperbolic functions
(pdf).
- Handout on similarity solutions of the Diffusion Equation
(pdf).
- Handout on the role of mappings in complex variable theory
(pdf).
- The Mathematics Department Public Formula Sheet: (pdf)
and another with Statistical Tables (pdf).
- Problem sheets now available are:
- Problem Sheet 1 (pdf).
- Solutions to Sheet 1 (pdf).
- Problem Sheet 2 (pdf).
- Solutions to Sheet 2 .
- Problem Sheet 3 (pdf).
- Solutions to Sheet 3
- Problem Sheet 4 (pdf).
- Solutions to Sheet 4
- Problem Sheet 5 (pdf).
- Solutions to Sheet 5
- Problem Sheet 6 (pdf).
- Solutions to Sheet 6
2nd year Applied Mathematics M2A1:
-
The syllabus content & lecture list: (pdf). Office Hour: Tuesday 1-2 in
my office 6M26.
- The available handouts are:
- The first handout on "Classification of critical points" (pdf).
- The second handout on "Double Taylor Series & the Jacobian Matrix" (pdf).
- The third handout on "A nonlinear example" (pdf).
- The fourth handout on "Prey-predator picture" (pdf).
- The fifth handout on "Guerrillas versus a conventional army" (pdf).
- The sixth handout on "Competition of political parties" (pdf).
- The seventh handout on "Reduced form of the Euler-Lagrange Equations" (pdf)
- The eighth handout on "The Kinematic Wave Equation" (pdf)
- Problem sheets available so far are:
- Problem Sheet 1 (pdf).
- Solutions to Sheet 1 (pdf).
- Problem Sheet 2 (pdf).
- Solutions to Sheet 2 (pdf).
- 1st Assessed Coursework: (pdf). Deadline Monday 30th Oct 06.
- Solutions to ACW1: (pdf).
- Problem Sheet 3 (pdf).
- Solutions to Sheet 3 (pdf).
- Problem Sheet 4 (pdf).
- 2nd Assessed Coursework: (pdf). Deadline Friday 24th Nov 2pm.
- Solutions to ACW2 (pdf).
- Solutions to Sheet 4 (pdf).
- Problem Sheet 5 (pdf).
- Solutions to Sheet 5 (pdf).
- 3rd Assessed Coursework: (pdf). Deadline Tuesday 9th Jan 2pm.
- Solutions to ACW3 (pdf).
- Problem Sheet 6 (pdf).
- Solutions to Sheet 6 (pdf).
