E-grāmata: Infinite Time Blow-Up Solutions to the Energy Critical Wave Maps Equation

"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))"--