A Blow-up Criterion for the Curve Diffusion Flow with a Contact Angle

Short time existence for the curve diffusion flow with a contact angle

“…The key ingredient for solving the linear problem in our situation is a result by Meyries and Schnaubelt on maximal L p -regularity with temporal weights, [17]: We deduce that for admissible initial data in W 4(µ− 1 /2) 2 (I), µ ∈ ( 7 8 , 1], there exists a unique solution of the linear problem in the temporal weighted parabolic space W 1 2,µ ((0, T ); L 2 (I) ∩ L 2,µ ((0, T ); W 4 2 (I)) for any T > 0, see Section 2 for the definitions of the spaces. In order to show that the nonlinear terms are contractive for small times, it is crucial to study the structure of the non-linearities and to establish suitable product estimates in time weighted anisotropic L 2 -Sobolev spaces of low regularity.…”

“…Note that in this work, we establish short time existence for curves which can be described as a graph over the reference curve with a height function, which is small in some sense. The corresponding result which allows for starting the flow for a fixed initial curve can be found in Theorem 4.1.3 in [3] and [1], where it is also used to establish a blow-up criterion for the geometric evolution equation (1.1) -(1.4), cf. Theorem 4.1.4 in [3].…”