On the principle of linearized stability in interpolation spaces for quasilinear evolution equations

Abstract

Well-posedness in time-weighted spaces for quasilinear (and semilinear) parabolic evolution equations u ′ = A(u)u + f (u) is established in a certain critical case of strict inclusion dom(f ) dom(A) for the domains of the (superlinear) function u → f (u) and the quasilinear part u → A(u). Based upon regularizing effects of parabolic equations, it is proven that the solution map generates a semiflow in a critical intermediate space.The applicability of the abstract results is demonstrated by several examples including a model for atmospheric flows and semilinear and quasilinear evolution equations with scaling invariance for which well-posedness in the critical scaling invariant intermediate spaces is shown.

Well-posedness and stability results for a quasilinear periodic Muskat problem

“…Besides, this paper provides the full picture of the set of equilibrium solutions to (1.1) -which are described by either flat of finger-shaped interfaces (similarly as in the bounded periodic case [31]) -and the stability properties of the flat equilibria and of small finger-shaped equilibria are studied in the phase space H r (S). For the latter purpose we use a quasilinear principle of linearized stability derived recently in [50].…”

“…This property needs to be verified before applying the abstract quasilinear parabolic theory outlined in [1][2][3][4][5] (see also [50]) in the particular context of (4.1). We begin by solving the equation (1.1a) 2 for ω.…”

“…. , 2 p+1 } and τ ∈ [0, 1] operators A c j,τ ∈ L(H 3 (S), L 2 (S)) (A c j,τ is the complexification of A j,τ defined in (4.10)) such that We now come to the proof our first main result which uses on the one hand the abstract theory for quasilinear parabolic problems outlined in [1][2][3][4][5] (see also [50,Theorem 1.1]), and on the other hand a parameter trick which has been employed in various versions in [8,35,[47][48][49]55] in the context of improving the regularity of solutions to certain parabolic evolution equations. We point out that the parameter trick can only be used because the uniqueness claim of Theorem 1.…”

Well‐posedness and stability for a mixed order system arising in thin film equations with surfactant

“…(C2) in Section 2.1), we affirm that the mixed‐order matrix A with fixed coefficients in a certain regularity space generates an analytic semigroup. The well‐posedness result then follows from a classical result on abstract parabolic equations by Amann in [3, Section 12] and [15, Theorem 1.1].…”

“…Eventually, we recall that the one and only equilibrium of the system (1.1) is given by the flat state, which is uniquely determined by the initial data. We apply a recently established result in [15] to prove that the flat equilibrium is asymptotically stable in interpolation spaces. This improves the existing stability result in [4, Theorem 4.5].…”