I will show rough path estimates for smooth L^p functions whose derivatives are in L^q. The application part related to (linear or nonlinear) Fourier analysis will be also discussed.
The classical gravity-capillary water-wave problem is the
study of the irrotational flow of a three-dimensional perfect
fluid bounded below by a flat, rigid bottom and above by a
free surface subject to the forces of gravity and surface
tension. In this lecture I will present a survey of currently
available existence theories for travelling-wave solutions of
this problem, that is, waves which move in a specific
direction with constant speed and without change of shape.
The talk will focus upon wave motions which are truly
three-dimensional, so that the free surface of the water
exhibits a two-dimensional pattern, and upon solutions of the
complete hydrodynamic equations for water waves rather than
model equations. Specific examples include (a) doubly
periodic surface waves; (b) wave patterns which have a
single- or multi-pulse profile in one distinguished
horizontal direction and are periodic in another; (c)
so-called 'fully-localised solitary waves' consisting of a
localised trough-like disturbance of the free surface which
decays to zero in all horizontal directions.
I will also sketch the mathematical techniques required to
prove the existence of the above waves. The key is a
formulation of the problem as a Hamiltonian system with
infinitely many degrees of freedom together with an
associated variational principle.
When studying invariant quantities and stability of discretization schemes for time-dependent differential equations(ODEs), Backward error analysis (BEA) has proven itself an invaluable tool. Although the established results give very accurate estimates, the known results are generally given for "worst case" scenarios. By taking into account the structure of the differential equations themselves further improvements on the estimates can be established, and sharper estimates on invariant quantities and stability can be established. In the talk I will give an overview of BEA, and its applications as it stands emphasizing the shortcoming in the estimates. An alternative strategy is then proposed overcoming these shortcomings, resulting in a tool which when used in connection with results from dynamical systems theory gives a very good insight into the dynamics of discretized differential equations.
/notices/events/abstracts/applied-analysis/ht05/gudrun.pdf
This talk attempts to survey key aspects of the mathematics that has been developed in recent years towards an explicit understanding of the structure of exponential functionals of Brownian motion, starting with work of Yor's in the 1990s
It is known that a continuous path of bounded variation
can be reconstructed from a sequence of its iterated integrals (called the signature) in a similar way to a function on the circle being reconstructed from its Fourier coefficients. We study the radius of convergence of the corresponding logarithmic signature for paths in an arbitrary Banach space. This convergence has important consequences for control theory (in particular, it can be used for computing the logarithm of a flow)and the efficiency of numerical approximations to solutions of SDEs. We also discuss the nonlinear structure of the space of logarithmic signatures and the problem of reconstructing a path by its signature.