Infinite Time Blow-Up Solutions to the Energy Critical Wave Maps Equation / Mohandas Pillai.

Format:

Book

Author/Creator:

Pillai, Mohandas, author.

Series:

Memoirs of the American Mathematical Society ; Volume 284.

Memoirs of the American Mathematical Society Series ; Volume 284

Language:

English

Subjects (All):

Wave equation.

Solitons.

Harmonic maps.

Blowing up (Algebraic geometry).

Physical Description:

1 online resource (254 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"We consider the wave maps problem with domain R2 + 1 and target S2 in the 1- equivariant, topological degree one setting. In this setting, we recall that the soliton is a harmonic map from R2 to S2, with polar angle equal to Q1(r) = 2arctan(r). By applying the scaling symmetry of the equation, Q[lambda](r) = Q1(r[lambda]) is also a harmonic map, and the family of all such Q[lambda] are the unique minimizers of the harmonic map energy among finite energy, 1-equivariant, topological degree one maps. In this work, we construct infinite time blowup solutions along the Q[lambda] family. More precisely, for b < 0, and for all [lambda]0,0,b [element of] C[superscript infinity]([100,[infinity])) satisfying, for some Cl, Cm,k > 0, Cl logb(t) [less than or equal to] [lamda]0,0,b(t) [less than or equal to] Cm logb(t) , |[lambda](k) 0,0,b(t)| [less than or equal to] Cm,k tk logb+1(t) , k [greater than or equal to] 1 t [greater than or equal to] 100 there exists a wave map with the following properties. If ub denotes the polar angle of the wave map into S2, we have ub(t, r) = Q 1 [lambda]b(t) (r) + v2(t, r) + ve(t, r), t [greater than or equal to] T0 where - [partial derivative]ttv2 + [partial derivative]rrv2 + 1 r [partial derivative]rv2 - v2 r2 = 0 [parallel][partial derivative]t(Q 1 [lambda]b(t) + ve)[parallel]2 L2(rdr) + [parallel]ve r [parallel]2 L2(rdr) + [parallel][partial derivative]rve[parallel]2 L2(rdr) [less than or equal to] C t2 log2b(t) , t [greater than or equal to] T0 and [lambda]b(t) = [lambda]0,0,b(t) [plus] O [x in a square] 1 logb(t) [x in a square] log(log(t))"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

Notation

Acknowledgments

Chapter 2. Overview of the proof

Chapter 3. Construction of the ansatz

Chapter 4. Solving the final equation

Chapter 5. The energy of the solution, and its decomposition as in Theorem 1.1

Appendix A. Proof of Theorem 1.2

Bibliography

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

Other Format:

Print version: Pillai, Mohandas Infinite Time Blow-Up Solutions to the Energy Critical Wave Maps Equation

ISBN:

9781470474454