Infinite time blow-up solutions to the energy critical wave maps equation / Mohandas Pillai.
| Author/creator | Pillai, Mohandas |
| Format | Electronic |
| Publication Info | Providence, Rhode Island : American Mathematical Society, 2023. |
| Description | v, 242 pages ; 26 cm. |
| Supplemental Content | Full text available from Ebook Central - Academic Complete |
| Subjects |
| Series | Memoirs of the American Mathematical Society, 0065-9266 ; Volume 284, Number 1407 |
| Contents | Overview of the proof -- Construction of the ansatz -- Solving the final equation -- The energy of the solution, and its decomposition as in Theorem 1.1. |
| Abstract | "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. |
| Bibliography note | Includes bibliographical references (pages 241-242). |
| Access restriction | Available only to authorized users. |
| Technical details | Mode of access: World Wide Web |
| Genre/form | Electronic books. |
| LCCN | 2023015046 |
| ISBN | 9781470459932 (paperback) |
| ISBN | (pdf) |
Availability
| Library | Location | Call Number | Status | Item Actions |
|---|---|---|---|---|
| Electronic Resources | Access Content Online | ✔ Available |