SCIENTIFIC PROGRAMS AND ACTIVITIES
|September 1, 2014|
7. Christian Klingenberg
8. Petronela Radu
10. Elizabeth Thoren
Inelastic Boltzmann Equation: Existence and uniqueness theorem for granular and dilute materials.
The University of Texas at Austin, 1 University Station, C1200, Austin, Texas 78712.
The Cauchy problem for the inelastic Boltzmann equation is studied
for small data. Existence and uniqueness of mild and weak solutions
is obtained for sufficiently small data that lies in the space of
functions bounded by Maxwellians. The technique used to derive the
result is the well known iteration process of Kaniel & Shinbrot.
We consider finite energy solutions of a wave equation with supercritical
nonlinear sources and nonlinear damping. A distinct feature of the
model under consideration is the presence of the double interaction
of source and damping, both in the interior of the domain and on
the boundary. Moreover, we consider nonlinear sources on the boundary
driven by Neumann boundary conditions. Since Lopatinski condition
fails to hold (unless the dim( W) = 1), the analysis of the nonlinearities
supported on the boundary, within the framework of weak solutions,
is a rather subtle issue and involves strong interaction between
the source and the damping. We provide positive answers to the questions
of local existence and uniqueness of weak solutions and moreover
we give complete and sharp description of parameters corresponding
to global existence and blow-up of solutions in finite time.
Multiscale dynamics of 2D rotational compressible
Euler equations: an analytical approach.
We study the 2D rotational compressible Euler equations with two independent parameters: the Rossby number t for rotational forcing and the Froude/Mach number s for pressure forcing. The competition of these two forces leads to a newly found parameter d = ts-2 that serves as a characteristic scale separating two major dynamics regimes: d << 1 for the strong rotation regime () and d >> 1 () for the mid/weak rotation regime. Our results reveal, in an analytic level, the stabilizing effect of rotation and the dispersive effect of pressure when these singular forces interact with the inherent nonlinearity of Euler dynamics. The understanding of such interaction is essential to the analysis/simulation of rotating dynamics, primarily to geophysical flows. Our results are consistent with geophysical observations of e.g. Near Inertial Oscillation and nonlinear Rossby adjustment.
The analytical novelty relies on several approximation and associated error estimates. Differing from existing literature, our approach imposes algebraic constraint not on individual parameters t and s, but on their relative strength d. In the d << 1 regime, we utilize the method of iterative approximation, starting with the pressureless rotational Euler equations (). The resulting approximation yields a periodic-in-time, fast rotating flow that reflects the domination of rotation in a nonlinear fashion. On the other hand, for d >> 1, we combine an invariant-based nonlinear wave analysis with Strichartz type estimates to reveal an approximate incompressible flow. This approach, free of Fourier analysis, has the potential to be extended to e.g. domains with nontrivial geometry.
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Various authors such as J. Lidsey, T. Christodoulakis, T. Grammenos,
C. Helias, P. Kevrekidis, G. Papadopoulos and F. Williams are known
to have formulated equivalent versions of the 3+1-dimensional Einstein's
field equations in terms of a so-called generalized Ermakov-Milne-Pinney
(EMP) differential equation. This reformulation provides an alternate
method for acquiring exact solutions to the field equations, and
has been accomplished within the frameworks of FRLW and some Bianchi
universe models. Further inspired by an EMP-Schrödinger correspondence
as noted by J. Lidsey, the author has recently published a linear
Schrödinger version of the Bianchi I scalar field cosmology.
This model has now been extended to an arbitrary number of dimensions,
and will be presented here.
I will discuss the stability of traveling waves for a model that
describes combustion of high density liquid fuels. The stability
analysis is performed for a parameter regime when the spectral information
is not definitive. It is shown that the wave is orbitally stable
with respect to a carefully chosen exponentially weighted norm.
Scattering For the Focusing 2D Quintic Nonlinear
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We have a new numerical method to compute turbulence flows arising
in astrophysical applications. The idea is to combine subgrid scale
modeling with adaptive mesh refinement. This has been implemented
into the cosmological code called ENZO.
Wave equations with variable coefficients
and space dependent damping
Damped wave equations with variable coefficients can be seen as
models of either hyperbolic diffusion or wave propagation under
the action of friction forces in a heterogeneous medium. We establish
decay rates for the energy and the L2 norm of the solution by employing
a strengthened multiplier method. The central piece in the proof
is an approximating profile constructed from a special subsolution
of a related elliptic problem. Decay rates for higher energies are
obtained by following an approach due to Nakao.
We propose an explicit construction of the fundamental solutions
to the one-dimensional Schrodinger equation with a particular linear
time-dependent Hamiltonian such that the sum of the order of derivative
and the degree of polynomial in the respective coefficient equals
two. For some special choice of coefficients of the Hamiltonian
this system can be integrated and therefore the fundamental solution
has an explicit form. Applications to physics are outlined.
One criteria for linear instability of a steady flow of an ideal incompressible fluid involves computing the essential spectral radius of the associated evolution operator for the linear perturbation about the steady equilibrium. This quantity is known to be equal to a Lyapunov type exponent associated with the equilibrium flow. In this work, the essential spectral radius of the linear evolution operator is investigated in the invariant subspace corresponding to the perturbations preserving the topology of the vortex lines and the associated factor space.
We address the problem of analyticity and Gevrey-regularity of smooth solutions u of the incompressible Euler equations. If the initial datum is real-analytic, the solution remains real-analytic as long as ?0t ??u(·, s)?L8 ds < 8 (cf. Bardos and Benachour). In the periodic case, using a Fourier method, we obtain a lower bound on the uniform radius of space analyticity which depends algebraically on exp?0t ??u(·, s)?L8ds. In particular, we positively answer a question posed by Levermore and Oliver.