A Note on Quasilinear Parabolic Equations on Manifolds

By Carlo Mantegazza, L. Martinazzi. 2010

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We prove short time existence, uniqueness and continuous dependence on the initial data of smooth solutions of quasilinear locally parabolic equations of arbitrary even order on closed manifolds.

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Citations in the corpus

The Łojasiewicz–Simon gradient inequality for open elastic curves (2016, unknown): “L2Eλ(f˜)(t,Φ(t,x)) = −∇L2Eλ(f˜(t,Φ(t,x))) and the initial condition f˜(0,Φ(0,x)) = f˜(0,x) = fˆ 0 ◦Φ 0(x) = f(tk,Ψ−1(tk,Φ 0(x))). Due to the uniqueness of the solution of the elastic ﬂow (recall also [MM12], [KS12]) we ﬁnd f(tk+ t) ≡ f˜(t,Φ ◦(Ψ−1(tk) ◦Φ 0)−1), or equivalently f˜(t,·) = f(tk+ t,Ψ−1(t”
Ricci flow coupled with harmonic map flow (2009, background): “ξiξj · id, where id is the identity on tensors ˜g. Hence this ﬂow equation is strictly parabolic and we get short-time existence from the standard parabolic theory for quasilinear equations, see e.g. [24] for a recent and detailed proof. If we additionally assume that (M,h) = (M,g0) and ψ(0) = idM, ”
Ricci flow on surfaces with conic singularities (2013, background): “ric. Then there exists some T >0 depending on the Dk,δ b norm of g 0, and a unique solution φ∈ D k+δ,(k+δ)/2 b ([0,T] ×Mf) to (3.19) with φ|t=0 = 0. The proof follows arguments similar to those in [30]. We now prove the short-time existence result for the angle-changing ﬂow. Since this is sidenot”