Events
Department of Mathematics and Statistics
Texas Tech University
Since the emergence of the Calculus of Variations, understanding the shape of equilibria for certain functionals has played a central role in Differential Geometry and Geometric Analysis.
Geometric variational problems are characterized by energies whose Lagrangians depend on geometric invariants. However, there exist as well other functional arising from physical contexts which are closely related to these geometric variational problems.
In this talk, we will examine the history behind several of these pioneering variational problems and their development up to the present, paying special attention to the speaker's own results in the last three years.
Dr. Pampano's Tenure and Promotion Colloquium has been archived in the Mediasite catalog.
Coupled hyperbolic problems are PDE systems that have a hybrid nature. Part of these systems may have a Hilbert space structure; therefore, they are tractable using traditional variational techniques. However, the other portion of these systems usually corresponds to nonlinear wave behavior and cannot be treated with the usual Galerkin methods. In this talk, we provide some background and examples of these systems, discuss what makes them interesting and what makes them relevant. Moving to the central portion of the talk, we discuss a new numerical method in order to solve the compressible magneto-hydrodynamics (MHD) system. In essence, the MHD system is decomposed into two components: a vanishing-viscosity PDE and a purely Hamiltonian PDE. This is a significant point of departure from the dominant body of literature advocating for divergence formulations. Our approach involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom in the choice of Euler's equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. Similarly, if the scheme used to solve Euler's equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. To the best of our knowledge, it is the first scheme in the literature capable of preserving invariant-domain properties, total energy, and involution constraints exactly.
Dr. Tomas' Tenure and Promotion Colloquium has been archived in the Mediasite catalog.
We study systems of nonlinear ordinary differential equations where the dominant term, with respect to large spatial variables, causes blow-ups and is positively homogeneous of a degree $1+\alpha$ for some $\alpha>0$. We prove that the asymptotic behavior of a solution $y(t)$ near a finite blow-up time $T_*$ is $(T_*-t)^{-1/\alpha}\xi_*$ for some nonzero vector $\xi_*$. Specific error estimates for $|(T_*-t)^{1/\alpha}y(t)-\xi_*|$ are provided. In some typical cases, they can be a positive power of $(T_*-t)$ or $1/|\ln(T_*-t)|$. This depends on whether the decaying rate of the lower order term, relative to the size of the dominant term, is of a power or logarithmic form. Similar results are obtained for a class of nonlinear differential inequalities with finite time blow-up solutions. Our results cover larger classes of nonlinear equations, differential inequalities and error estimates than those in the previous work.
The Differential Geometry, Partial Differential Equations and Mathematical Physics seminar group recommends people attend the Colloquium given on the 9th in Experimental Sciences Building 1, room 120.
 | Wednesday
Sep. 10
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| Algebra and Number Theory
No Seminar
Please attend the departmental colloquium
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The Applied Math seminar group recommends people attend the Colloquium given on the 10th in Experimental Sciences Building 1, room 120.
abstract 2 PM CDT (UT-5)
Zoom link available from Dr. Brent Lindquist upon request.