Global Well Posedness Of High Dimensional Maxwell Dirac For Small Critical Data
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| Author | : Cristian Gavrus |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 106 |
| Release | : 2020-05-13 |
| Genre | : Education |
| ISBN | : 147044111X |
In this paper, the authors prove global well-posedness of the massless Maxwell–Dirac equation in the Coulomb gauge on R1+d(d≥4) for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of the authors' proof are A) uncovering null structure of Maxwell–Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell–Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru, which says that the most difficult part of Maxwell–Dirac takes essentially the same form as Maxwell-Klein-Gordon.
| Author | : Cristian Dan Gavrus |
| Publisher | : |
| Total Pages | : 94 |
| Release | : 2020 |
| Genre | : Differential equations, Partial |
| ISBN | : 9781470458089 |
In this paper, the authors prove global well-posedness of the massless Maxwell-Dirac equation in the Coulomb gauge on \mathbb{R}^{1+d} (d\geq 4) for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of the authors' proof are A) uncovering null structure of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Kri.
| Author | : Chao Wang |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 119 |
| Release | : 2021-07-21 |
| Genre | : Education |
| ISBN | : 1470446898 |
In this paper, we prove the local well-posedness of the free boundary problem for the incompressible Euler equations in low regularity Sobolev spaces, in which the velocity is a Lipschitz function and the free surface belongs to C 3 2 +ε. Moreover, we also present a Beale-Kato-Majda type break-down criterion of smooth solution in terms of the mean curvature of the free surface, the gradient of the velocity and Taylor sign condition.
| Author | : Benjamin Jaye |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 97 |
| Release | : 2020-09-28 |
| Genre | : Mathematics |
| ISBN | : 1470442132 |
Fix $dgeq 2$, and $sin (d-1,d)$. The authors characterize the non-negative locally finite non-atomic Borel measures $mu $ in $mathbb R^d$ for which the associated $s$-Riesz transform is bounded in $L^2(mu )$ in terms of the Wolff energy. This extends the range of $s$ in which the Mateu-Prat-Verdera characterization of measures with bounded $s$-Riesz transform is known. As an application, the authors give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-Delta )^alpha /2$, $alpha in (1,2)$, in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.
| Author | : Angel Castro |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 89 |
| Release | : 2020-09-28 |
| Genre | : Mathematics |
| ISBN | : 1470442140 |
In this paper, the authors show the existence of the first non trivial family of classical global solutions of the inviscid surface quasi-geostrophic equation.
| Author | : Lisa Berger |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 131 |
| Release | : 2020-09-28 |
| Genre | : Mathematics |
| ISBN | : 1470442191 |
The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $mathbb F_p(t)$, when $p$ is prime and $rge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $mathbb F_q(t^1/d)$.
| Author | : Zhi Qi |
| Publisher | : American Mathematical Society |
| Total Pages | : 123 |
| Release | : 2021-02-10 |
| Genre | : Mathematics |
| ISBN | : 1470443252 |
In this article, the author studies fundamental Bessel functions for $mathrm{GL}_n(mathbb F)$ arising from the Voronoí summation formula for any rank $n$ and field $mathbb F = mathbb R$ or $mathbb C$, with focus on developing their analytic and asymptotic theory. The main implements and subjects of this study of fundamental Bessel functions are their formal integral representations and Bessel differential equations. The author proves the asymptotic formulae for fundamental Bessel functions and explicit connection formulae for the Bessel differential equations.
| Author | : Ulrich Bunke |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 177 |
| Release | : 2021-06-21 |
| Genre | : Education |
| ISBN | : 1470446855 |
We develop differential algebraic K-theory for rings of integers in number fields and we construct a cycle map from geometrized bundles of modules over such a ring to the differential algebraic K-theory. We also treat some of the foundational aspects of differential cohomology, including differential function spectra and the differential Becker-Gottlieb transfer. We then state a transfer index conjecture about the equality of the Becker-Gottlieb transfer and the analytic transfer defined by Lott. In support of this conjecture, we derive some non-trivial consequences which are provable by independent means.
| Author | : Hiroshi Iritani |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 92 |
| Release | : 2021-06-21 |
| Genre | : Education |
| ISBN | : 1470443635 |
Gromov-Witten theory started as an attempt to provide a rigorous mathematical foundation for the so-called A-model topological string theory of Calabi-Yau varieties. Even though it can be defined for all the Kähler/symplectic manifolds, the theory on Calabi-Yau varieties remains the most difficult one. In fact, a great deal of techniques were developed for non-Calabi-Yau varieties during the last twenty years. These techniques have only limited bearing on the Calabi-Yau cases. In a certain sense, Calabi-Yau cases are very special too. There are two outstanding problems for the Gromov-Witten theory of Calabi-Yau varieties and they are the focus of our investigation.
| Author | : Pierre Albin |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 126 |
| Release | : 2021-06-21 |
| Genre | : Education |
| ISBN | : 1470444224 |
Manifolds with fibered cusps are a class of complete non-compact Riemannian manifolds including many examples of locally symmetric spaces of rank one. We study the spectrum of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold undergoing degeneration to a manifold with fibered cusps. We obtain precise asymptotics for the resolvent, the heat kernel, and the determinant of the Laplacian. Using these asymptotics we obtain a topological description of the analytic torsion on a manifold with fibered cusps in terms of the R-torsion of the underlying manifold with boundary.
