arXiv subjects

Evan Miller

Publications and source records attributed to Evan Miller.

23 records · Page 2Linked to original sources

arXiv2020

A locally anisotropic regularity criterion for the Navier--Stokes equation in terms of vorticity

In this paper, we will prove a regularity criterion that guarantees solutions of the Navier--Stokes equation must remain smooth so long as the the vorticity restricted to a plane remains bounded in the scale critical space $L^4_t L^2_x$, where the plane may vary in space and time as long as the gradient of the vector orthogonal to the plane remains bounded. This extends previous work by Chae and Choe that guaranteed that solutions of the Navier--Stokes equation must remain smooth as long as the vorticity restricted to a fixed plane remains bounded in family of scale critical mixed Lebesgue spaces. This regularity criterion also can be seen as interpolating between Chae and Choe's regularity criterion in terms of two vorticity components and Beirão da Veiga and Berselli's regularity criterion in terms of the gradient of vorticity direction.

math.AP↗

arXiv2020

Bergman Kernels of Elementary Reinhardt Domains

We study the Bergman kernel of certain domains in $\mathbb{C}^n$, called elementary Reinhardt domains, generalizing the classical Hartogs triangle. For some elementary Reinhardt domains, we explicitly compute the kernel, which is a rational function of the coordinates. For some other such domains, we show that the kernel is not a rational function. For a general elementary Reinhardt domain, we obtain a representation of the kernel as an infinite series.

math.CV↗

arXiv2019

A regularity criterion for the Navier-Stokes equation involving only the middle eigenvalue of the strain tensor

This manuscript derives an evolution equation for the symmetric part of the gradient of the velocity (the strain tensor) in the incompressible Navier-Stokes equation on $\mathbb{R}^3$, and proves the existence of $L^2$ mild solutions to this equation. We use this equation to obtain a simplified identity for the growth of enstrophy for mild solutions that depends only on the strain tensor, not on the nonlocal interaction of the strain tensor with the vorticity. The resulting identity allows us to prove a new family of scale-critical, necessary and sufficient conditions for the blow-up of a solution at some finite time $T_{max}<+\infty$, which depend only on the history of the positive part of the second eigenvalue of the strain matrix. Since this matrix is trace-free, this severely restricts the geometry of any finite-time blow-up. This regularity criterion provides analytic evidence of the numerically observed tendency of the vorticity to align with the eigenvector corresponding to the middle eigenvalue of the strain matrix. This regularity criterion also allows us to prove as a corollary a new scale critical, one component type, regularity criterion for a range of exponents for which there were previously no known critical, one component type regularity criteria. Furthermore, our analysis permits us to extend the known time of existence of smooth solutions with fixed initial enstrophy $E_0=\frac{1}{2}\left\|\nabla \otimes u^0\right \|_{L^2}^2$ by a factor of $4{,}920{.}75$---although the previous constant in the literature was not expected to be close to optimal, so this improvement is less drastic than it sounds, especially compared with numerical results. Finally, we will prove the existence and stability of blow-up for a toy model ODE for the strain equation.

math.AP↗

arXiv2017

Learning Abduction under Partial Observability

Juba recently proposed a formulation of learning abductive reasoning from examples, in which both the relative plausibility of various explanations, as well as which explanations are valid, are learned directly from data. The main shortcoming of this formulation of the task is that it assumes access to full-information (i.e., fully specified) examples; relatedly, it offers no role for declarative background knowledge, as such knowledge is rendered redundant in the abduction task by complete information. In this work, we extend the formulation to utilize such partially specified examples, along with declarative background knowledge about the missing data. We show that it is possible to use implicitly learned rules together with the explicitly given declarative knowledge to support hypotheses in the course of abduction. We observe that when a small explanation exists, it is possible to obtain a much-improved guarantee in the challenging exception-tolerant setting. Such small, human-understandable explanations are of particular interest for potential applications of the task.

cs.AI↗

arXiv2015

Maximum principles for the relativistic heat equation

The classical heat equation is incompatible with relativity, since the strong maximum principle allows for disturbances to propagate instantaneously. Some authors have proposed limiting the propagation speed by adding a linear hyperbolic correction term, but then even a weak maximum principle fails to hold. We study a more recently introduced relativistic heat equation, which replaces the Laplace operator by a quasilinear elliptic operator, and show that strong and weak maximum principles hold for stationary and time-varying solutions, respectively, as well as for sub- and supersolutions. Moreover, by transforming the equation into an equivalent form, related to the mean curvature operator, we prove even stronger tangency and comparison principles.

math.AP↗