SCIENTIFIC PROGRAMS AND ACTIVITIES

August 20, 2014

September 18-20, 2008
Conference on Non-linear Phenomena in Mathematical Physics:
Dedicated to Cathleen Synge Morawetz
on her 85th birthday

Co-sponsored by:

Poster Abstracts

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1. Ricardo Alonso
Inelastic Boltzmann Equation: Existence and uniqueness theorem for granular and dilute materials.

2. Lorena Bociu

Existence, Uniqueness and Blow-Up of Solutions to Wave Equations with Supercritical Boundary/Interior Sources and Damping

3. Bin Cheng

Multiscale dynamics of 2D rotational compressible Euler equations: an analytical approach.

4. Jennie D'Ambroise

A Linear Schrodinger Formulation of (d+1)-Dimensional Bianchi I Scalar Field Cosmology

5. Anna Ghazaryan

Existence and stability of waves in combustion of high density liquid fuels

6. Cristi Darley Guevara

Scattering For the Focusing 2D Quintic Nonlinear Schrödinger Equation

7. Christian Klingenberg
Computing turbulence flows using subgrid scale modeling

8. Petronela Radu
Wave equations with variable coefficients and space dependent damping

9. Erwin Suazo
Evolution operator for a one-dimensional Schrodinger equation with a time dependent Hamiltonian.

10. Elizabeth Thoren
Linear instability criteria for Euler's equation: two classes of perturbations

11. Vlad Vicol
The Radius of Analyticity of Solutions to the Three-Dimensional Euler Equations



Inelastic Boltzmann Equation: Existence and uniqueness theorem for granular and dilute materials.
by
Ricardo Alonso
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.
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Existence, Uniqueness and Blow-Up of Solutions to Wave Equations with Supercritical Boundary/Interior Sources and Damping
by
Lorena Bociu
University of Nebraska-Lincoln
Coauthors: Irena Lasiecka

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.
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Multiscale dynamics of 2D rotational compressible Euler equations: an analytical approach.
by
Bin Cheng
Department of Mathematics, University of Michigan, USA
Coauthors: Eitan Tadmor (University of Maryland, USA)

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 ([1]) and d >> 1 ([2]) 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 ([3]). 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.

References
[1] B. Cheng, E. Tadmor, Long time existence of smooth solutions for the rapidly rotating shallow-water and Euler equations. SIAM J. Math. Anal. 39(5) (2008) 1668-1685.
[2] B. Cheng, Effects of scales on 2D rotational compressible Euler equations. (to appear) Proceedings of the 12th International Conference on Hyperbolic Problems. (2008)
[3] H. Liu and E. Tadmor Rotation prevents finite-time breakdown. Phys. D 188 (2004), no. 3-4, 262-276.

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A Linear Schrodinger Formulation of (d+1)-Dimensional Bianchi I Scalar Field Cosmology
by
Jennie D'Ambroise
University of Massachusetts Amherst

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.
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Existence and stability of waves in combustion of high density liquid fuels
by
Anna Ghazaryan
UNC-Chapel Hill & University of Kansas-Lawrence

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.
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Scattering For the Focusing 2D Quintic Nonlinear Schrödinger Equation
by
Cristi Darley Guevara
Arizona State University
Coauthors: Fernando Carreon (Arizona State University), Svetlana Roudenko(Arizona State University)

Recent developments for the energy critical nonlinear Schrödinger equation (NLS) in 3d, and nonlinear wave equation (NLW) by Carlos Kenig and Frank Merle have attracted attention from Harmonic Analysis and PDE audience. Their approach is based on concentration-compactness method which dates back to works of P.-L. Lions and the localized virial argument. It gives a sharp threshold for the scattering and finite time blow up of solutions at least in the case of radial data, and in many problems can be extended to nonradial data as well. These methods have been recently applied to the focusing cubic NLS in 3d by Holmer-Roudenko and Duyckaerts-Holmer-Roudenko as well as to the mass critical (both focusing and defocusing) NLS in 2 and higher dimensions by Killip-Tao-Visan, Killip-Visan-Zhang. Using the above techniques, we characterize the behavior of H1 solutions to the focusing quintic NLS in R2, namely,
i ?t u+\triangle u+|u|4u=0,
(x, t) ? R2×R.

We obtain scattering for globally existing solutions (under an a priori mass-energy threshold) and mention how this extends to a general mass supercritical and energy subcritical NLS with H1 data.

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Computing turbulence flows using subgrid scale modeling
by
Christian Klingenberg
Math Dept., Wurzburg University, Germany
Coauthors: Wolfram Schmidt, Jens, Niemeyer

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.
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Wave equations with variable coefficients and space dependent damping
by
Petronela Radu
University of Nebraska-Lincoln, Department of Mathematics, Lincoln NE 68588
Coauthors: Grozdena Todorova (University of Tennessee, Knoxville) Boris Yordanov (University of Tennessee, Knoxville)

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.
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Evolution operator for a one-dimensional Schrodinger equation with a time dependent Hamiltonian.
by
Erwin Suazo
Arizona State University
Coauthors: Sergei Suslov, Raquel Lopez, Ricardo Cordero-Soto.

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.
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Linear instability criteria for Euler's equation: two classes of perturbations
by
Elizabeth Thoren
University of Texas at Austin

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.

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The Radius of Analyticity of Solutions to the Three-Dimensional Euler Equations
by
Vlad Vicol
University of Southern California
Coauthors: Igor Kukavica (University of Southern California)

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.

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