Philip T. Gressman
A non-local inequality and global existence.
Tue, 02/21/2012 - 15:01 — SomNambulAuthors: Joachim Krieger, Philip T. Gressman, Robert M. Strain
Subjects: Analysis of PDEs
AbstractIn this article we prove a collection of new non-linear and non-local
integral inequalities. As an example for $u\ge 0$ and $p\in (0,\infty)$ we
obtain $$ \int_{\threed} dx ~ u^{p+1}(x) \le (\frac{p+1}{p})^2 \int_{\threed}
dx ~ \{(-\triangle)^{-1} u(x) \} \nsm \nabla u^{\frac{p}{2}}(x)\nsm^2. $$ We
use these inequalities to deduce global existence of solutions to a non-local
heat equation with a quadratic non-linearity for large radial monotonic
positive initial conditions. Specifically, we improve \cite{ksLM} to include
all $\alpha\in (0, 74/75)$.Global Classical Solutions of the Boltzmann Equation with Long-Range Interactions and Soft Potentials.
Mon, 02/22/2010 - 16:01 — SomNambulAuthors: Philip T. Gressman, Robert M. Strain
Subjects: Analysis of PDEs
AbstractIn this work we prove global stability for the Boltzmann equation (1872) with
the physical collision kernels derived by Maxwell in 1866 for the full range of
inverse power intermolecular potentials, $r^{-(p-1)}$ with $p>2$. This
completes the work which we began in (arXiv:0912.0888v1). We more generally
cover collision kernels with parameters $s\in (0,1)$ and $\gamma$ satisfying
$\gamma > -(n-2)-2s$ in arbitrary dimensions $\mathbb{T}^n \times \mathbb{R}^n$
with $n\ge 2$. Moreover, we prove rapid convergence as predicted by the
Boltzmann H-Theorem.On multilinear determinant functionals.
Mon, 11/09/2009 - 16:01 — SomNambulAuthors: Philip T. Gressman
Subjects: Classical Analysis and ODEs
AbstractThis paper considers the problem of $L^p$-estimates for a certain multilinear
functional involving integration against a kernel with the structure of a
determinant. Examples of such objects are ubiquitous in the study of Fourier
restriction and geometric averaging operators. It is shown that, under very
general circumstances, the boundedness of such functionals is equivalent to a
geometric inequality for measures which has recently appeared in work by D.
Oberlin (Math Proc. Cambridge. Philos. Soc., 129, 2000) and Bak, Oberlin, and
Seeger (J. Aust. Math. Soc., 85, 2008).Uniform geometric estimates for sublevel sets.
Mon, 09/07/2009 - 10:33 — SomNambulAuthors: Philip T. Gressman
Subjects: Classical Analysis and ODEs
AbstractThis paper reconsiders the uniform sublevel set estimates of Carbery, Christ,
and Wright (1999) and Phong, Stein, and Sturm (2001) from a geometric
perspective. This perspective leads one to consider a natural collection of
homogeneous, nonlinear differential operators which generalize mixed
derivatives in $\R^d$. As a consequence, it is shown that, in the case of both
of these previous works, improved uniform decay rates are possible in many
situations.Uniform geometric estimates for sublevel sets.
Mon, 09/07/2009 - 10:33 — SomNambulAuthors: Philip T. Gressman
Subjects: Classical Analysis and ODEs
AbstractThis paper reconsiders the uniform sublevel set estimates of Carbery, Christ,
and Wright (1999) and Phong, Stein, and Sturm (2001) from a geometric
perspective. This perspective leads one to consider a natural collection of
homogeneous, nonlinear differential operators which generalize mixed
derivatives in $\R^d$. As a consequence, it is shown that, in the case of both
of these previous works, improved uniform decay rates are possible in many
situations.
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