Global Well-Posedness of High Dimensional Maxwell–Dirac for Small Critical Data

Global Well-Posedness of High Dimensional Maxwell–Dirac for Small Critical Data PDF Author: Cristian Gavrus
Publisher: American Mathematical Soc.
ISBN: 147044111X
Category : Education
Languages : en
Pages : 106

Get Book Here

Book Description
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.

Global Well-Posedness of High Dimensional Maxwell–Dirac for Small Critical Data

Global Well-Posedness of High Dimensional Maxwell–Dirac for Small Critical Data PDF Author: Cristian Gavrus
Publisher: American Mathematical Soc.
ISBN: 147044111X
Category : Education
Languages : en
Pages : 106

Get Book Here

Book Description
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.

Global Well-Posedness of High Dimensional Maxwell-Dirac for Small Critical Data

Global Well-Posedness of High Dimensional Maxwell-Dirac for Small Critical Data PDF Author: Cristian Dan Gavrus
Publisher:
ISBN: 9781470458089
Category : Differential equations, Partial
Languages : en
Pages : 94

Get Book Here

Book Description
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.

Local Well-Posedness and Break-Down Criterion of the Incompressible Euler Equations with Free Boundary

Local Well-Posedness and Break-Down Criterion of the Incompressible Euler Equations with Free Boundary PDF Author: Chao Wang
Publisher: American Mathematical Soc.
ISBN: 1470446898
Category : Education
Languages : en
Pages : 119

Get Book Here

Book Description
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.

The Riesz Transform of Codimension Smaller Than One and the Wolff Energy

The Riesz Transform of Codimension Smaller Than One and the Wolff Energy PDF Author: Benjamin Jaye
Publisher: American Mathematical Soc.
ISBN: 1470442132
Category : Mathematics
Languages : en
Pages : 97

Get Book Here

Book Description
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.

Global Smooth Solutions for the Inviscid SQG Equation

Global Smooth Solutions for the Inviscid SQG Equation PDF Author: Angel Castro
Publisher: American Mathematical Soc.
ISBN: 1470442140
Category : Mathematics
Languages : en
Pages : 89

Get Book Here

Book Description
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.

Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields

Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields PDF Author: Lisa Berger
Publisher: American Mathematical Soc.
ISBN: 1470442191
Category : Mathematics
Languages : en
Pages : 131

Get Book Here

Book Description
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 $mathbb F_q(t^1/d)$.

Theory of Fundamental Bessel Functions of High Rank

Theory of Fundamental Bessel Functions of High Rank PDF Author: Zhi Qi
Publisher: American Mathematical Society
ISBN: 1470443252
Category : Mathematics
Languages : en
Pages : 123

Get Book Here

Book Description
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.

Differential Function Spectra, the Differential Becker-Gottlieb Transfer, and Applications to Differential Algebraic K-Theory

Differential Function Spectra, the Differential Becker-Gottlieb Transfer, and Applications to Differential Algebraic K-Theory PDF Author: Ulrich Bunke
Publisher: American Mathematical Soc.
ISBN: 1470446855
Category : Education
Languages : en
Pages : 177

Get Book Here

Book Description
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.

Gromov-Witten Theory of Quotients of Fermat Calabi-Yau Varieties

Gromov-Witten Theory of Quotients of Fermat Calabi-Yau Varieties PDF Author: Hiroshi Iritani
Publisher: American Mathematical Soc.
ISBN: 1470443635
Category : Education
Languages : en
Pages : 92

Get Book Here

Book Description
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.

Resolvent, Heat Kernel, and Torsion under Degeneration to Fibered Cusps

Resolvent, Heat Kernel, and Torsion under Degeneration to Fibered Cusps PDF Author: Pierre Albin
Publisher: American Mathematical Soc.
ISBN: 1470444224
Category : Education
Languages : en
Pages : 126

Get Book Here

Book Description
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.