TY - THES
AU - Roxanas, Dimitrios
PY - 2017
TI - Long-time dynamics for the energy-critical harmonic map heat flow and nonlinear heat equation
KW - Thesis/Dissertation
LA - eng
M3 - Text
AB - The main focus of this thesis is on critical parabolic problems, in particular, the
harmonic map heat from the plane to S2, and nonlinear focusing heat equations
with an algebraic nonlinearity.
The focus of this work has been on long-time dynamics, stability and singularity
formation, and the investigation of the role of special, soliton-like, solutions to the
asymptotic behaviour of solutions.
Harmonic Map Heat Flow: Flow: we consider m-corotational solutions to the
harmonic map heat flow from R2 to S2. We first work in a class of maps with
trivial topology and energy of the initial data below two times the energy of
the stationary harmonic map solutions. We give a new proof of global existence
and decay. The proof is based on the "concentration-compactness plus rigidity"
approach of Kenig and Merle and relies on the dissipation of the energy and a
profile decomposition. We also treat m-corotational maps (m greater than 3) with
non-trivial topology and energy of the initial data less than three times the energy
of the stationary harmonic map solutions. Through a new stability argument we
rule out finite-time blow-up and show that the global solution asymptotically
converges to a harmonic map.
Nonlinear Heat Equation: we also study solutions of the focusing energy-critical
nonlinear heat equation. We show that solutions emanating from initial data
with energy and kinetic energy below those of the stationary solutions
are global and decay to zero. To prove that global solutions dissipate to zero
we rely on a refined small data theory, L2-dissipation and an
approximation argument. We then follow the "concentration-compactness
plus rigidity" roadmap of Kenig and Merle (and in particular the approach taken
by Kenig and Koch for Navier-Stokes) to exclude finite-time blow-up.
N2 - The main focus of this thesis is on critical parabolic problems, in particular, the
harmonic map heat from the plane to S2, and nonlinear focusing heat equations
with an algebraic nonlinearity.
The focus of this work has been on long-time dynamics, stability and singularity
formation, and the investigation of the role of special, soliton-like, solutions to the
asymptotic behaviour of solutions.
Harmonic Map Heat Flow: Flow: we consider m-corotational solutions to the
harmonic map heat flow from R2 to S2. We first work in a class of maps with
trivial topology and energy of the initial data below two times the energy of
the stationary harmonic map solutions. We give a new proof of global existence
and decay. The proof is based on the "concentration-compactness plus rigidity"
approach of Kenig and Merle and relies on the dissipation of the energy and a
profile decomposition. We also treat m-corotational maps (m greater than 3) with
non-trivial topology and energy of the initial data less than three times the energy
of the stationary harmonic map solutions. Through a new stability argument we
rule out finite-time blow-up and show that the global solution asymptotically
converges to a harmonic map.
Nonlinear Heat Equation: we also study solutions of the focusing energy-critical
nonlinear heat equation. We show that solutions emanating from initial data
with energy and kinetic energy below those of the stationary solutions
are global and decay to zero. To prove that global solutions dissipate to zero
we rely on a refined small data theory, L2-dissipation and an
approximation argument. We then follow the "concentration-compactness
plus rigidity" roadmap of Kenig and Merle (and in particular the approach taken
by Kenig and Koch for Navier-Stokes) to exclude finite-time blow-up.
UR - https://open.library.ubc.ca/collections/24/items/1.0347430
ER - End of Reference