By Robert Denk
In this article, a thought for common linear parabolic partial differential equations is demonstrated which covers equations with inhomogeneous image constitution in addition to mixed-order platforms. commonplace purposes contain numerous variations of the Stokes approach and unfastened boundary worth difficulties. We express well-posedness in Lp-Lq-Sobolev areas in time and area for the linear difficulties (i.e., maximal regularity) that's the main step for the therapy of nonlinear difficulties. the idea is predicated at the proposal of the Newton polygon and will disguise equations which aren't obtainable by way of average equipment as, e.g., semigroup thought. effects are received in numerous kinds of non-integer Lp-Sobolev areas as Besov areas, Bessel strength areas, and Triebel–Lizorkin areas. The last-mentioned classification seems to be in a typical method as lines of Lp-Lq-Sobolev areas. We additionally current a variety of purposes within the entire area and on half-spaces. between others, we turn out well-posedness of the linearizations of the generalized thermoelastic plate equation, the two-phase Navier–Stokes equations with Boussinesq–Scriven floor, and the Lp-Lq two-phase Stefan challenge with Gibbs–Thomson correction.
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Extra info for General Parabolic Mixed Order Systems in Lp and Applications
Example text
T❤❡ r❡❛❧ ✐♥t❡r♣♦❧❛t✐♦♥ s♣❛❝❡ (X0 , X1 )θ,p ✐s ❞❡✜♥❡❞ ❛s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ u ∈ X0 + X1 s❛t✐s❢②✐♥❣ (t → t−θ K(t, u)) ∈ Lp (R+ , dt t ). ❚❤❡ ♥♦r♠ ✐♥ (X0 , X1 )θ,p ✐s ❞❡✜♥❡❞ ❜② ∞ u θ,p t−θ K(t, u) := 0 p dt t 1/p . ❉❡✜♥✐t✐♦♥ ✶✳✸✽ ✭❈♦♠♣❧❡① ✐♥t❡r♣♦❧❛t✐♦♥ ❢✉♥❝t♦r✮✳ ▲❡t θ ∈ (0, 1)✱ ❛♥❞ s❡t S := (0, 1) + iR ⊆ C✳ ❚❤❡♥ F (X0 , X1 ) ✐s ❞❡✜♥❡❞ ❛s t❤❡ s❡t ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s f : S → X0 + X1 s❛t✐s❢②✐♥❣ ✭✐✮ f ✐s (X0 + X1 )✲❝♦♥t✐♥✉♦✉s ✐♥ S ✱ ✭✐✐✮ f ✐s (X0 + X1 )✲❤♦❧♦♠♦r♣❤✐❝ ✐♥ S ✱ ✭✐✐✐✮ f (k + it) ∈ Xk (t ∈ R) ❛♥❞ (t → f (k + it)) ∈ C(R, Xk ) ❢♦r k = 0, 1✱ ✭✐✈✮ lim f (k + it) |t|→∞ Xk = 0 ❢♦r k = 0, 1✳ ❲❡ ❡♥❞♦✇ F (X0 , X1 ) ✇✐t❤ t❤❡ ♥♦r♠ f F (X0 ,X1 ) := max sup f (it) t∈R X0 , sup f (1 + it) t∈R X1 .
ZN ) ❢♦r ❛❧❧ f ∈ HP (Ω)✳ Pr♦♦❢✳ ✭■✮ ❋✐rst✱ ✇❡ ♣r♦✈❡ t❤❡ ❛ss❡rt✐♦♥ ❢♦r f ∈ H0∞ (Ω)✳ ❋♦r ❛❧❧ z ∈ Ω := N k=2 Ωk ✇❡ ❞❡✜♥❡ g(z ) := f (σ + T1 , z )✳ ❚❤❡♥ ▲❡♠♠❛ ✶✳✸✸ ❛♥❞ ▲❡♠♠❛ ✶✳✷✾ ②✐❡❧❞ g(z ) = fσ (T1 , z ) ❛♥❞ f (S) = g(T ) = fσ (T) ✇❤❡r❡ T := (T2 , . . , TN )✳ ❖❜✈✐♦✉s❧②✱ f (S) L(X) ≤ C fσ ∞ ≤ C f ❛❧❧ f ∈ H0∞ (Ω) ❛♥❞ t❤✉s S ❛❧s♦ ❛❞♠✐ts ❛ ❜♦✉♥❞❡❞ H ∞ (Ω)✲❝❛❧❝✉❧✉s✳ ∞ ❢♦r ✭■■✮ ▲❡t f ∈ HP (Ω)✳ ❚❤❡♥ ✇❡ ❣❡t f (S) = ψ(S)−m (ψ m f )(S) = ψσ (T)−m (ψσm fσ )(T) ⊇ ψσ (T)−m ψσ (T)m fσ (T) = fσ (T) ❞✉❡ t♦ ❚❤❡♦r❡♠ ✶✳✷✻ ❛♥❞ t❤❡ r❡s✉❧t ♦❢ ♣❛rt ✭■✮✳ ❚❤❡ ❝♦♥✈❡rs❡ ✐♥❝❧✉s✐♦♥ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛② ❜② fσ (T) = ψ(T)−m (ψ m fσ )(T) = ψ(T)−m ⊇ ψ(T)−m = ψ(T)−m ψm ψσm ψm ψσm ψm ψσm · (ψσm fσ ) (T) (T) (ψσm fσ ) (T) (T) (ψ m f ) (S) ⊇ ψ(T)−m ψ(T)m ψσ−m (T) (ψ m f ) (S) = ψ(S)−m (ψ m f ) (S) = f (S).
N ❜❡ s❡❝t♦r✐❛❧ ♦r ❜✐s❡❝t♦r✐❛❧ ♦♣❡r❛t♦rs ✭♦❢ t❤❡ s❛♠❡ t②♣❡✮ ✇✐t❤ D(Sk ) ⊆ D(Tk ) ✭✇✐t❤ r❡s♣❡❝t t♦ Y → X ✮ s✉❝❤ t❤❛t t❤❡ t✉♣❧❡s T := (T1 , . . , TN ) ❛♥❞ S := (S1 , . . , SN ) ❈❤❛♣t❡r ✶✳ ❚❤❡ ❥♦✐♥t t✐♠❡✲s♣❛❝❡ H ∞ ✲❝❛❧❝✉❧✉s ✷✻ ❛r❡ ❛❞♠✐ss✐❜❧❡ ❛♥❞ Tk y = Sk y ❢♦r ❛❧❧ y ∈ D(Sk ) ❛♥❞ k ∈ {1, . . , N }✳ ❚❤❡♥ ✇❡ ♦❜t❛✐♥ f (S) ⊆ f (T) ❢♦r ❛❧❧ f ∈ HP (Ω) ✇❤❡r❡ Ω ✐s ❛❞♠✐ss✐❜❧❡ ❢♦r T ❛♥❞ S✳ Pr♦♦❢✳ ✭■✮ ▲❡t f ∈ H0∞ (Ω) ❛♥❞ y ∈ Y ✳ ❚❤❡♥ ✇❡ ❤❛✈❡ 1 f (S)y = (2πi)N = = = 1 (2πi)N 1 (2πi)N 1 (2πi)N N (zk − Sk )−1 dz y f (z) Γ k=1 N (zk − Sk )−1 y dz f (z) Γ k=1 N (zk − Tk )−1 y dz f (z) Γ k=1 N f (z) Γ (zk − Tk )−1 dz y k=1 = f (T)y ❜❡❝❛✉s❡ ♦❢ (zk −Tk )−1 |Y = (zk −Sk )−1 ❢♦r ❛❧❧ zk ∈ ρ(Tk )∩ρ(Sk ) ❛♥❞ Y → X ✳ ✭■■✮ ▲❡t f ∈ HP (Ω) ❛♥❞ m ∈ N0 ✇✐t❤ (ψ m f ) ∈ H0∞ (Ω)✳ ❯s✐♥❣ ✭■✮ ✇❡ ♦❜t❛✐♥ D(f (S)) = {y ∈ Y : (ψ m f )(S)y ∈ R(ψ(S)m )} = {y ∈ Y : (ψ m f )(T)y ∈ R(ψ(S)m )} ⊆ {y ∈ Y : (ψ m f )(T)y ∈ R(ψ(T)m )} = Y ∩ D(f (T)).