Exercise · ZW-DIFFEOMORPHIC-FLOW-004 · ★☆☆ Basic
Which hypothesis on \(v_t\) helps the flow equation have a smooth, well-determined trajectory?
Choose one option.
Show answer
- B. Enough regularity of the field to guarantee existence and uniqueness
Explanation
With \(v_t\) continuous and locally Lipschitz in \(x\), Picard-Lindelöf guarantees existence and uniqueness of the trajectory. Without that regularity the flow may fail to be well defined.
Review the concept: Diffeomorphic flow
A diffeomorphic flow \(\varphi_t\) is a family of deformations parametrized by time \(t\), where at each instant the transformation remains invertible and smooth.
\[\{\varphi_t\}_{t\in[0,1]},\quad \varphi_0=\operatorname{id}\]
More practice problems: diffeomorphic flow
- ZW-DIFFEOMORPHIC-FLOW-001 For a point x, the flow equation is read as φ_t(x)=v_t(φ_t(x)). What does it describe?
- ZW-DIFFEOMORPHIC-FLOW-002 What role does φ_0=id play in a flow?
- ZW-DIFFEOMORPHIC-FLOW-003 If two flows have the same final map but travel different paths, what can vary?
- ZW-DIFFEOMORPHIC-FLOW-005 Which description is compatible with a diffeomorphic flow \φ_t\?
- ZW-DIFFEOMORPHIC-FLOW-006 What does it mean for φ_t to be a diffeomorphism?
- ZW-DIFFEOMORPHIC-FLOW-007 What is the field v_t in LDDMM?
- ZW-DIFFEOMORPHIC-FLOW-008 What does it mean for the flow to be parametrized by t∈[0,1]?
- ZW-DIFFEOMORPHIC-FLOW-009 What does the final map φ_1 do in LDDMM?