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

2020-05-13 Education
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Release : 2020-05-13
Genre : Education
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Book Synopsis Global Well-Posedness of High Dimensional Maxwell–Dirac for Small Critical Data by : Cristian Gavrus

Download or read book Global Well-Posedness of High Dimensional Maxwell–Dirac for Small Critical Data written by Cristian Gavrus. This book was released on 2020-05-13. Available in PDF, EPUB and Kindle. Book excerpt: 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

2020 Differential equations, Partial
Download Global Well-Posedness of High Dimensional Maxwell-Dirac for Small Critical Data PDF Online Free

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Release : 2020
Genre : Differential equations, Partial
Kind : eBook
Book Rating : 089/5 ( reviews)

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Book Synopsis Global Well-Posedness of High Dimensional Maxwell-Dirac for Small Critical Data by : Cristian Dan Gavrus

Download or read book Global Well-Posedness of High Dimensional Maxwell-Dirac for Small Critical Data written by Cristian Dan Gavrus. This book was released on 2020. Available in PDF, EPUB and Kindle. Book excerpt: 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

2021-07-21 Education
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Release : 2021-07-21
Genre : Education
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Book Rating : 898/5 ( reviews)

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Book Synopsis Local Well-Posedness and Break-Down Criterion of the Incompressible Euler Equations with Free Boundary by : Chao Wang

Download or read book Local Well-Posedness and Break-Down Criterion of the Incompressible Euler Equations with Free Boundary written by Chao Wang. This book was released on 2021-07-21. Available in PDF, EPUB and Kindle. Book excerpt: 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

2020-09-28 Mathematics
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Release : 2020-09-28
Genre : Mathematics
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Book Rating : 132/5 ( reviews)

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Book Synopsis The Riesz Transform of Codimension Smaller Than One and the Wolff Energy by : Benjamin Jaye

Download or read book The Riesz Transform of Codimension Smaller Than One and the Wolff Energy written by Benjamin Jaye. This book was released on 2020-09-28. Available in PDF, EPUB and Kindle. Book excerpt: 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

2020-09-28 Mathematics
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Release : 2020-09-28
Genre : Mathematics
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Book Rating : 140/5 ( reviews)

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Book Synopsis Global Smooth Solutions for the Inviscid SQG Equation by : Angel Castro

Download or read book Global Smooth Solutions for the Inviscid SQG Equation written by Angel Castro. This book was released on 2020-09-28. Available in PDF, EPUB and Kindle. Book excerpt: 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.

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