Local and Global Solvability for Advection-Diffusion Equation on an Evolving Surface with a Boundary
aa r X i v : . [ m a t h . A P ] A p r LOCAL AND GLOBAL SOLVABILITY FORADVECTION-DIFFUSION EQUATION ON AN EVOLVINGSURFACE WITH A BOUNDARY
HAJIME KOBA
Abstract.
This paper considers the existence of local and global-in-timestrong solutions to the advection-diffusion equation with variable coefficientson an evolving surface with a boundary. We apply both the maximal L p -in-time regularity for Hilbert space-valued functions and the semigroup theory toconstruct local and global-in-time strong solutions to an evolution equation.Using the approach and our function spaces on the evolving surface, we showthe existence of local and global-in-time strong solutions to the advection-diffusion equation. Moreover, we derive the asymptotic stability of the global-in-time strong solution. Introduction
We are interested in the existence of local and global-in-time solutions to theadvection-diffusion equation on an evolving surface with a boundary. An evolvingsurface means that the surface is moving or the shape of the surface is changingalong with the time. This paper has three purposes. The first one is to constructa strong solution of an evolution equation by applying both the maximal L p -in-time regularity for Hilbert space-valued functions and the semigroup theory. Thesecond one is to introduce and study function spaces on evolving surfaces. Thethird one is to apply our approach and function spaces to show the existence oflocal and global-in-time strong solutions to the advection-diffusion equation withvariable coefficients on an evolving surface with a boundary.Let us first introduce basic notations. Let x = t ( x , x , x ) ∈ R , X = t ( X , X ) ∈ R be the spatial variables, and t, τ ≥ ∇ and ∇ X are two gradient operators defined by ∇ = t ( ∂ , ∂ , ∂ ) and ∇ X = t ( ∂ X , ∂ X ),where ∂ j = ∂/∂x j and ∂ X α = ∂/∂X α . Let T ∈ (0 , ∞ ], and let Γ( t )(= { Γ( t ) } ≤ t Key words and phrases. Advection-diffusion equation with variable coefficients, Time-dependent Laplace-Beltrami operator, Function spaces on evolving surfaces, Maximal L p -regularity, Asymptotic stability. where the unknown function u = u ( x, t ) is the concentration of amount of a sub-stance on Γ( t ), while the given function w = w ( x, t ) = t ( w , w , w ) is the motionvelocity of Γ( t ), the given function κ = κ ( x, t ) is the diffusion coefficient , and u = u ( x ) is the given initial datum . Here D wt u := ∂ t u + ( w · ∇ ) u , ∂ t := ∂/∂t ,Γ := Γ(0), Γ T := [ Here λ , λ > 0. More precisely, we use the elliptic regularity of A , i.e., there is C ♯ > f ∈ D ( A )(1.5) k f k L ( U ) + k∇ X f k L ( U ) + k∇ X f k L ( U ) ≤ C ♯ k Af k L ( U ) , and the maximal L -regularity of A , i.e., there is C A > F, V ) ∈ L (0 , T ; L ( U )) × D ( A ) there exists a unique function V satisfying the system: ( ddt V + AV = F on (0 , T ) ,V | t =0 = V , and the estimate:(1.6) Z T (cid:13)(cid:13)(cid:13)(cid:13) dVdt (cid:13)(cid:13)(cid:13)(cid:13) L ( U ) dt ! + Z T k AV k L ( U ) dt ! ≤ √ k A V k L ( U ) + C A Z T k F k L ( U ) dt ! . Remark that the two positive constants C ♯ , C A depend only on ( λ , λ , U ). Remarkalso that we can write C ♯ = C ′ ♯ ( q, U ) λ and C A = C ′ A ( q, U ) λ when λ = qλ for some q > 0. See Section 3 for details.The second one is to consider the following approximate equations: ( ddt v + Av = 0 on (0 , T ) ,v | t =0 = v , ( ddt v m +1 + Av m +1 = − Bv m on (0 , T ) ,v m +1 | t =0 = v . Here B = B ( t ) is the evolution operator defined by B = L − A . Using the approx-imate equations, the maximal L -regularity of A , the semigroup theory, and thenorm k · k Z T : k ϕ k Z T := sup We first introduce the definition of our evolving surface with a boundary andnotations. Then, we state the main results of this paper. Definition 2.1 (Evolving surface with boundary) . Let T ∈ (0 , ∞ ]. For 0 ≤ t < T ,let Γ( t ) ⊂ R be a set. We call Γ( t )(= { Γ( t ) } ≤ t 3, and α, β = 1 , (cid:12)(cid:12)(cid:12)(cid:12) ∂ b x j ∂X α (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂ b x j ∂X α ∂X β (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂ b x j ∂t∂X α (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂ b x j ∂t∂X α ∂X β (cid:12)(cid:12)(cid:12)(cid:12) ≤ λ max < + ∞ . Let us explain the conventions used in this paper. We use the Greek characters α, β, ζ, η as 1 , 2. Moreover, we often use the following Einstein summation conven-tion: c α d αβ = P α =1 c α d αβ and c ζ d ζη = P ζ =1 c ζ d ζη . The symbol d H kx denotes the k -dimensional Hausdorff measure.Next we define some notations. Let T ∈ (0 , ∞ ] and Γ( t )(= { Γ( t ) } ≤ t Fix j ∈ { , , } . For ψ ∈ C ( R ) and φ = t ( φ , φ , φ ) ∈ [ C ( R )] , ∂ Γ j ψ := X i =1 ( δ ij − n i n j ) ∂ i ψ, ∇ Γ := t ( ∂ Γ1 , ∂ Γ2 , ∂ Γ3 ) , grad Γ ψ := ∇ Γ ψ = t ( ∂ Γ1 ψ, ∂ Γ2 ψ, ∂ Γ3 ψ ) , div Γ φ := ∇ Γ · φ = ∂ Γ1 φ + ∂ Γ2 φ + ∂ Γ3 φ . Here δ ij is the Kronecker delta. We write e X = e X ( x, t ) as the inverse mapping of b x = b x ( X, t ), i.e., for each 0 ≤ t < T , e X ( · , t ) : Γ( t ) → U , b x ( e X ( x, t ) , t ) = x for x ∈ Γ( t ) , e X ( b x ( X, t ) , t ) = X for X ∈ U . Note that there exists an inverse mapping e X of b x by Definition 2.1 and an inversefunction theorem. Set w ( x, t ) = b x t ( e X ( x, t ) , t ) . We call w the motion velocity of the evolving surface Γ( t ). We also call w the speedof the evolving surface Γ( t ).We state the definition of a strong solution to system ( 1.1) with u ∈ W , (Γ ). Definition 2.2 (Strong solution to ( 1.1)) . Let u ∈ W , (Γ ) and u ∈ L (0 , T : W , ∩ W , (Γ( · ))) ∩ W , (0 , T ; L (Γ( · ))) . We call u a strong solution of system ( 1.1) with initial datum u if the function u satisfies the following three properties:(i) (Equation) k D wt u + (div Γ w ) u − div Γ ( κ ∇ Γ u ) k L (0 ,T ; L (Γ( · ))) = 0 . (ii) (Boundary condition) k γu k L (0 ,T ; L ( ∂ Γ( · ))) = 0 . Here γ : W , (Γ( t )) → L ( ∂ Γ( t )) is the trace operator defined by Proposition 4.3.(iii) (Initial condition) b u ∈ C ([0 , T ); L ( U )) andlim t → k u ( b x ( · , t ) , t ) − u ( b x ( · , k L ( U ) = 0 . Here b u = b u ( X, t ) := u ( b x ( X, t ) , t ).See Section 4 for function spaces on the evolving surface Γ( t ) and its norms. Seealso Section 5 for the strong solutions of system ( 1.1).Before introducing the main results of this paper, we state the assumption of theevolving surface Γ( t ). For every 0 ≤ t < T and X ∈ U , g = g ( X, t ) := ∂ b x∂X , g = g ( X, t ) := ∂ b x∂X , g αβ := g α · g β , ( g αβ ) × := ( g αβ ) − × = 1 g g − g g (cid:18) g − g − g g (cid:19) , g α := g αβ g β , G = G ( X, t ) := g g − g g . HAJIME KOBA Note that G = | g × g | = g g − g g = λ min > 0. Write b κ = b κ ( X, t ) = κ ( b x ( X, t ) , t ). Throughout this paper, we assume the following restrictions. Assumption 2.3. (i) Assume that κ ∈ C (Γ T ) . Here C (Γ T ) := { ψ ; ψ ( x, t ) = ϕ ( e X ( x, t ) , t ) , ϕ ∈ C ( U × [0 , T )) } . (ii) Assume that there are two positive constants κ min , κ max such that for all ( X, t ) ∈ U × [0 , T ) and α = 1 , , κ min ≤ b κ ≤ κ max and (cid:12)(cid:12)(cid:12)(cid:12) ∂ b κ∂t (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂ b κ∂X α (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂ b κ∂t∂X α (cid:12)(cid:12)(cid:12)(cid:12) < κ max . (iii) Assume that there are two positive constants λ , λ such that for all ( X, t ) ∈ U × [0 , T ) , b κ g > λ and b κ g > λ . Remark that inf g = inf g in general. For 0 ≤ t < T , M ( t ) := (cid:13)(cid:13)(cid:13)(cid:13)b κ g G − λ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( U ) + (cid:13)(cid:13)(cid:13)(cid:13)b κ g G − λ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( U ) + (cid:13)(cid:13)(cid:13)(cid:13)b κ g G (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( U ) , M ( t ) := (cid:13)(cid:13)(cid:13)(cid:13) b κ G (cid:18) ∂ g ∂X − ∂ g ∂X (cid:19) − b κ G (cid:18) g ∂ G ∂X − g ∂ G ∂X (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( U ) , M ( t ) := (cid:13)(cid:13)(cid:13)(cid:13) b κ G (cid:18) ∂ g ∂X − ∂ g ∂X (cid:19) − b κ G (cid:18) g ∂ G ∂X − g ∂ G ∂X (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( U ) , M ( t ) := (cid:13)(cid:13)(cid:13)(cid:13) g G ∂ b κ∂X − g G ∂ b κ∂X (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( U ) + (cid:13)(cid:13)(cid:13)(cid:13) g G ∂ b κ∂X − g G ∂ b κ∂X (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( U ) , M ( t ) := (cid:13)(cid:13)(cid:13)(cid:13) G d G dt (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( U ) , and M j := sup ≤ t Theorem 2.4 (Local existence (I)) . Assume that T < ∞ and that C ♯ M ( C A + 1) ≤ √ . Then for each u ∈ W , (Γ ) , there exists a unique strong solution u in L (0 , T ⋆ ; W , ∩ W , (Γ( · ))) ∩ W , (0 , T ⋆ ; L (Γ( · ))) of system ( 1.1) with initial datum u , where T ⋆ = min (cid:26) T, 12 log (cid:18) C ⋆ ( C A + 1) (cid:19)(cid:27) . Here C ♯ , C A , and C ⋆ are the three positive constants in ( 1.5) , ( 1.6) , and ( 7.2) ,respectively. See Section 4 for function spaces on the evolving surface Γ( t ). Theorem 2.5 (Local existence (II)) . Assume that T < ∞ and that C ♯ ( M + M + M + M )( C A + 1) ≤ √ . DVECTION-DIFFUSION EQUATION ON EVOLVING SURFACE WITH BOUNDARY 7 Then for each u ∈ W , (Γ ) , there exists a unique strong solution u in L (0 , T ∗ ; W , ∩ W , (Γ( · ))) ∩ W , (0 , T ∗ ; L (Γ( · ))) of system ( 1.1) with initial datum u , where T ∗ = min ( T, 12 log C ♯ M ( C A + 1) !) . Here C ♯ , C A are the two positive constants in ( 1.5) and ( 1.6) , respectively. Theorem 2.6 (Global existence and stability) . Assume that T = ∞ and that C ♯ ( M + M + M + M + M )( C A + 1) ≤ √ . Then for each u ∈ W , (Γ ) , there exists a unique strong solution u in L (0 , ∞ ; W , ∩ W , (Γ( · ))) ∩ W , (0 , ∞ ; L (Γ( · ))) of system ( 1.1) with initial datum u . Moreover, the solution u satisfies the fol-lowing three properties: (i) (Energy equality) For all ≤ s ≤ t , k u ( t ) k L (Γ( t )) + Z ts k√ κ ∇ Γ u ( τ ) k L (Γ( τ )) dτ = 12 k u ( s ) k L (Γ( s )) . (ii) (Stability) There is C > such that for all t > k u ( t ) k L (Γ( t )) ≤ Ct − k b u k W , ( U ) . That is, lim t →∞ k u ( t ) k L (Γ( t )) = 0 . (iii) (Regularity) There is C > independent of u such that k D wt u k L (0 ,T ; L (Γ( · ))) + k div Γ ( κ ∇ Γ u ) k L (0 ,T ; L (Γ( · ))) ≤ C k b u k W , ( U ) . Here b u = b u ( X ) = u ( b x ( X, and C ♯ , C A are the two positive constants in ( 1.5) and ( 1.6) . Remark that we prove Theorems 2.4-2.6 in Sections 5-7.3. Preliminaries In this section, we first recall some basic properties of several differential opera-tors on the evolving surface Γ( t ). Then we state the fundamental properties of anelliptic operator.3.1. Differential Operators on Evolving Surface Γ( t ) . We first introduce therepresentation of some differential operators. Then we state the surface divergencetheorem. For ψ = ψ ( x ), Ψ = Ψ( x, t ), b ψ = b ψ ( X, t ) := ψ ( b x ( X, t )) and b Ψ = b Ψ( X, t ) := Ψ( b x ( X, t ) , t ) . From [10, Chapter 3], [7, Appendix], [11, Section 3], and [12, Section 3], we obtainthe following lemma. HAJIME KOBA Lemma 3.1 (Representation formula for differential operators) . (i) For each ψ ∈ C ( R ) , Z Γ( t ) ψ ( x ) d H x = Z U b ψ ( X, t ) p G ( X, t ) dX. (ii) For each j = 1 , , , and ψ ∈ C ( R ) , Z Γ( t ) ∂ Γ j ψ d H x = Z U g αβ ∂ b x j ∂X α ∂ b ψ∂X β √G dX. (iii) For each i, j = 1 , , , and ψ ∈ C ( R ) , Z Γ( t ) ∂ Γ i ∂ Γ j ψ d H x = Z U g ζη ∂ b x i ∂X ζ ∂∂X η g αβ ∂ b x j ∂X α ∂ b ψ∂X β ! √G dX. (iv) For each ψ ∈ C ( R ) , Z Γ( t ) div Γ ( κ ∇ Γ ψ ) d H x = Z U ( √G ∂∂X α b κ √G g αβ ∂ b ψ∂X β !) √G dX. (v) Z Γ( t ) div Γ w d H x = Z U G (cid:18) d G dt (cid:19) √G dX (cid:18) = ddt Z U √G dX (cid:19) . (vi) For every Ψ ∈ C ( R ) , Z Γ( t ) D wt Ψ d H x = Z U (cid:18) ddt b Ψ (cid:19) √G dX. Remark that one can derive system ( 1.2) from system ( 1.1) by applying Lemma3.1. From [16, Chapter 2] and [13, Section 3], we obtain the following surfacedivergence theorem. Lemma 3.2 (Surface divergence theorem) . For every φ = t ( φ , φ , φ ) ∈ [ C ( R )] , Z Γ( t ) div Γ φ d H x = − Z Γ( t ) H Γ ( n · φ ) d H x + Z ∂ Γ( t ) ν · φ d H x , where H Γ = H Γ ( x, t ) denotes the mean curvature in the direction n defined by H Γ = − div Γ n . Here n is the unit outer normal vector to Γ( t ) and ν is the unitouter co-normal vector to ∂ Γ( t ) . Elliptic Operators. Let 1 < p < ∞ . Define(3.1) ( A p f = − (cid:16) λ ∂ ∂X + λ ∂ ∂X (cid:17) f,D ( A p ) = L p ( U ) ∩ W ,p ( U ) ∩ W ,p ( U ) . We call A p the Dirichlet-Laplace operator if λ = λ . In particular, we write A as A . We easily have the following basic properties of the operator A p . Lemma 3.3. (i) (Strictly ellipticity) For every ξ = t ( ξ , ξ ) ∈ R , min { λ , λ }| ξ | ≤ λ ξ + λ ξ ≤ max { λ , λ }| ξ | . (ii) (Fundamental solution) For X = t ( X , X ) ∈ R \ { t (0 , } , E ( X ) := 12 log( λ X + λ X ) . DVECTION-DIFFUSION EQUATION ON EVOLVING SURFACE WITH BOUNDARY 9 Then − λ ∂ E ∂X − λ ∂ E ∂X = 0 in R \ { t (0 , } . (iii) (Formally selfadjoint operator) For all f ♯ , f ♭ ∈ D ( A ) , Z U ( Af ♯ ) f ♭ dX = Z U f ♯ ( Af ♭ ) dX. (iv) (Fractional power of A ) For all f ∈ D ( A ) Z U ( Af ) f dX = λ k ∂ X f k L ( U ) + λ k ∂ X f k L ( U ) . (v) (Heat system) Let φ = φ ( X, t ) be a C -function. Set Φ( X, t ) = φ ( X / p λ , X / p λ , t ) . Assume that φ satisfies ∂ t φ − ∆ X φ = 0 . Then Φ satisfies ∂ t Φ − ( λ ∂ X + λ ∂ X )Φ = 0 . Here ∆ X := ∂ X + ∂ X . From Lemma 3.3, [15, Chapter 7], [9, Chapters 7-9], and [8, Chapter 6], we havethe following lemma. Lemma 3.4. (i) (Elliptic regularity) There is C ♯ = C ♯ ( p, λ , λ , U ) > such thatfor all f ∈ D ( A p )(3.2) k f k L p ( U ) + k∇ X f k L p ( U ) + k∇ X f k L p ( U ) ≤ C ♯ k A p f k L p ( U ) . Moreover, if λ = qλ for some q > , then we write (3.3) C ♯ = C ′ ♯ ( q, p, U ) λ . (ii) (Interpolation inequality) For each ε > there is C = C ( ε, p, λ , λ , U ) > such that for all f ∈ D ( A p )(3.4) k∇ X f k L p ( U ) ≤ ε k∇ X f k L p ( U ) + C k f k L p ( U ) . (iii) The operator − A p generates a bounded analytic semigroup on L p ( U ) . (iv) The operator − A generates a contraction C -semigroup on L ( U ) . (v) The operator A is a non-negative selfadjoint operator in L ( U ) . (vi) D ( A ) = W , ( U ) and for all f ∈ D ( A )(3.5) k A f k L ( U ) = λ k ∂ X f k L ( U ) + λ k ∂ X f k L ( U ) . Proof of Lemma 3.4. We only derive ( 3.3). Assume that λ = qλ for some q > ( A ′ p f = − (cid:16) ∂ ∂X + q ∂ ∂X (cid:17) f,D ( A ′ p ) = L p ( U ) ∩ W ,p ( U ) ∩ W ,p ( U ) . From the elliptic regularity ( 3.2) of A ′ p , there is C ′ ♯ = C ′ ♯ ( q, p, U ) > f ∈ D ( A ′ p ) k f k L p ( U ) + k∇ X f k L p ( U ) + k∇ X f k L p ( U ) ≤ C ′ ♯ k A ′ p f k L p ( U ) . Since A p = λ A ′ p and D ( A p ) = D ( A ′ p ), we have ( 3.3). (cid:3) Since L ( U ) is a Hilbert space and − A generates a bounded analytic semigroupon L ( U ), it follows from [4] to find that the operator A has the maximal L p -regularity (Proposition 8.1). Therefore we have the following lemma. Lemma 3.5 (Maximal L -regularity of A ) . For each T ∈ (0 , ∞ ] and ( F, V ) ∈ L (0 , T ; L ( U )) × D ( A ) , there exists a uniquefunction V satisfying the system: (3.7) ( ddt V + AV = F on (0 , T ) ,V | t =0 = V , and the estimate: (3.8) k dV /dt k L (0 ,T ; H ) + k AV k L (0 ,T ; L ( U )) ≤ √ k A V k L ( U ) + C A k F k L (0 ,T ; L ( U )) . Here the positive constant C A depends only on ( λ , λ , U ) . Moreover, if λ = qλ for some q > , then we write (3.9) C A = C ′ A ( q, U ) λ . Proof of Lemma 3.5. Fix V ∈ D ( A ) and F ∈ L (0 , ∞ ; L ( U )). We first show(3.10) (cid:18)Z ∞ k A e − tA V k L ( U ) dt (cid:19) ≤ √ k A V k L ( U ) . Set V ♮ ( t ) = e − tA A V . Since A is a non-negative selfadjoint operator and e − tA isan analytic semigroup on L ( U ), we check that12 ddt k V ♮ ( t ) k L ( U ) = Z U ( − AV ♮ ) V ♮ dX = −k A V ♮ ( t ) k L ( U ) . Integrating with respect to time, we see that for all t > k V ♮ ( t ) k L ( U ) + Z t k A V ♮ ( t ) k L ( U ) dt = 12 k V ♮ (0) k L ( U ) = 12 k A V k L ( U ) . Since A V ♮ ( t ) = A e − tA A V = A e − tA V , we have ( 3.10).To study system ( 3.7), we now consider the following two systems:(3.11) ( ddt V ♯ + AV ♯ = 0 on (0 , ∞ ) ,V ♯ | t =0 = V , (3.12) ( ddt V ♭ + AV ♭ = F on (0 , ∞ ) ,V ♭ | t =0 = 0 . Since − A generates an analytic C -semigroup on L ( U ), we find that V ♯ = V ♯ ( t ) =e − tA V and V ♯ ∈ C ([0 , ∞ ); L ( U )) ∩ C ((0 , ∞ ); D ( A )) ∩ C ((0 , ∞ ); L ( U )) . DVECTION-DIFFUSION EQUATION ON EVOLVING SURFACE WITH BOUNDARY 11 Since dV ♯ dt = − AV ♯ = − A e − tA V , it follows from ( 3.10) to check that(3.13) k dV ♯ /dt k L (0 , ∞ ; L ( U )) + k AV ♯ k L (0 , ∞ ; L ( U )) ≤ √ k A V k L ( U ) . From the maximal L -regularity of A (Proposition 8.1), there exists a unique func-tion V ♭ satisfying system ( 3.12) and(3.14) k dV ♭ /dt k L (0 , ∞ ; L ( U )) + k AV ♭ k L (0 , ∞ ; L ( U )) ≤ C A k F k L (0 , ∞ ; L ( U )) . Here C A > U, λ , λ . Set V = V ( t ) = V ♮ ( t ) + V ♭ ( t ). It is easy tocheck that V satisfies system ( 3.7). From ( 3.13) and ( 3.14), we have k dV /dt k L (0 , ∞ ; L ( U )) + k AV k L (0 , ∞ ; L ( U )) ≤ √ k A V k L ( U ) + C A k F k L (0 , ∞ ; L ( U )) . Applying the maximal L -regularity of A , we easily deduce the uniqueness of solu-tions to system ( 3.7) with ( V , F ).Finally, we derive ( 3.9). To this end, we consider the following system:(3.15) ( ddt W ♭ + A ′ W ♭ = Fλ on (0 , ∞ ) ,W ♭ | t =0 = 0 . Here A ′ := A ′ is the operator defined by ( 3.6). From the maximal L -regularityof A ′ , there exists a unique function W ♭ satisfying ( 3.15) and(3.16) k dW ♭ /dt k L (0 , ∞ ; L ( U )) + k A ′ W ♭ k L (0 , ∞ ; L ( U )) ≤ C A ′ λ k F k L (0 , ∞ ; L ( U )) . Here C A ′ > q, U . Now we set V ♭♭ = V ♭♭ ( t ) = W ♭ ( λ t ). It is easyto see that V ♭♭ satisfies ( 3.12). From ( 3.16), we have(3.17) (cid:13)(cid:13)(cid:13)(cid:13) dV ♭♭ dt (cid:13)(cid:13)(cid:13)(cid:13) L (0 , ∞ ; L ( U )) + k AV ♭♭ k L (0 , ∞ ; L ( U )) ≤ C A ′ λ k F k L (0 , ∞ ; L ( U )) . Combing ( 3.13) and ( 3.17) gives k dV /dt k L (0 , ∞ ; L ( U )) + k AV k L (0 , ∞ ; L ( U )) ≤ √ k A V k L ( U ) + C A ′ λ k F k L (0 , ∞ ; L ( U )) . Thus, we have ( 3.9). Therefore, the lemma follows. (cid:3) Function Spaces on Evolving Surfaces In this section, we introduce and study function spaces on the evolving surfaceΓ( t ). Let e X = e X ( x, t ) be the inverse mapping of b x = b x ( X, t ). For each 0 ≤ t < T , ψ = ψ ( X ), and ϕ = ϕ ( x, t ), b ψ = b ψ ( X, t ) := ψ ( b x ( X, t )) , e ϕ = e ϕ ( x, t ) := e ϕ ( e X ( x, t )) . Moreover, for ψ = ψ ( x, t ) and ϕ ( X, t ), b ψ = b ψ ( X, t ) := ψ ( b x ( X, t ) , t ) , e ϕ = e ϕ ( x, t ) := ϕ ( e X ( x, t ) , t ) . Function Spaces on Evolving Surface Γ( t ) . Let us define function spaceson the evolving surface Γ( t ). Throughout this subsection, we fix t ∈ [0 , T ).For k = 0 , , 2, and 1 ≤ p < ∞ , C k (Γ( t )) := { ψ : Γ( t ) → R ; ψ = e ϕ, ϕ ∈ C k ( U ) } ,C k (Γ( t )) := { ψ : Γ( t ) → R ; ψ = e ϕ, ϕ ∈ C k ( U ) } ,C k (Γ( t )) := { ψ : Γ( t ) → R ; ψ = e ϕ, ϕ ∈ C k ( U ) } ,L p (Γ( t )) := { ψ : Γ( t ) → R ; ψ = e ϕ, ϕ ∈ L p ( U ) , k ψ k L p (Γ( t )) < ∞} ,L ∞ (Γ( t )) := { ψ : Γ( t ) → R ; ψ = e ϕ, ϕ ∈ L ∞ ( U ) , k ψ k L ∞ (Γ( t )) < ∞} . Here k ψ k L p (Γ( t )) := Z Γ( t ) | e ϕ ( x, t ) | p d H x ! p , k ψ k L ∞ (Γ( t )) := ess.sup x ∈ Γ( t ) | e ϕ ( x, t ) | . Moreover, for ψ ∈ C (Γ( t ))(= C (Γ( t ))), k ψ | ∂ Γ( t ) k L p ( ∂ Γ( t )) := Z ∂ Γ( t ) | e ϕ | ∂ Γ( t ) | p d H x ! p . It is easy to check that k ψ k pL p (Γ( t )) = Z U | be ϕ ( X, t ) | p p G ( X, t ) dX ≤ C ( λ max ) Z U | ϕ | p dX = C k ϕ k pL p ( U ) (4.1)and that k ψ k pL p (Γ( t )) = Z U | ϕ | p p G ( X, t ) dX ≥ λ min k ϕ k pL p ( U ) . We also see that for all ψ ♯ ∈ L p (Γ( t )) and ψ ♭ ∈ L p ′ (Γ( t ))(4.2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Γ( t ) ψ ♯ ψ ♭ d H x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k ψ ♯ k L p (Γ( t )) k ψ ♭ k L p ′ (Γ( t )) , where 1 ≤ p, p ′ ≤ ∞ such that 1 /p + 1 /p ′ = 1.Next we define a weak derivative for functions on the surface Γ( t ). For ψ ∈ C k (Γ( t )) or ψ ∈ C k (Γ( t )), we define the differential operators ∂ Γ j and ∂ Γ i ∂ Γ j as inLemma 3.1. Definition 4.1 (Weak derivatives) . Let 1 ≤ p ≤ ∞ , ψ ∈ L p (Γ( t )), and i, j = 1 , , ∂ Γ j ψ ∈ L p (Γ( t )) if there exists Ψ ∈ L p (Γ( t )) such that for all φ ∈ C (Γ( t )), Z Γ( t ) Ψ φ d H x = − Z Γ( t ) ψ ( ∂ Γ j φ + H Γ n j φ ) d H x . DVECTION-DIFFUSION EQUATION ON EVOLVING SURFACE WITH BOUNDARY 13 In particular, we write ∂ Γ j ψ as Ψ.(ii) We say that ∂ Γ j ∂ Γ i ψ ∈ L p (Γ( t )) if ∂ Γ i ψ ∈ L p (Γ( t )) and there exists Ψ ∈ L p (Γ( t ))such that for all φ ∈ C (Γ( t )), Z Γ( t ) Ψ φ d H x = − Z Γ( t ) ∂ Γ i ψ ( ∂ Γ j φ + H Γ n j φ ) d H x . In particular, we write ∂ Γ j ∂ Γ i ψ as Ψ.We easily see the uniqueness of weak derivatives. See [6] for weak derivatives forfunctions on a closed surface.Now we introduce Sobolev spaces on the surface Γ( t ). For 1 ≤ p < ∞ , W ,p (Γ( t )) := { ψ : Γ( t ) → R ; ψ = e ϕ, ϕ ∈ W ,p ( U ) , k ψ k W ,p (Γ( t )) < ∞} ,W ,p (Γ( t )) := { ψ : Γ( t ) → R ; ψ = e ϕ, ϕ ∈ W ,p ( U ) , k ψ k W ,p (Γ( t )) < ∞} ,W ,p (Γ( t )) := { ψ : Γ( t ) → R ; ψ = e ϕ, ϕ ∈ W ,p ( U ) , k ψ k W ,p (Γ( t )) < ∞} . Here k ψ k W ,p (Γ( t )) := Z Γ( t ) ( | e ϕ ( x, t ) | p + |∇ Γ e ϕ ( x, t ) | p ) d H x ! p , k ψ k W ,p (Γ( t )) := Z Γ( t ) ( | e ϕ ( x, t ) | p + |∇ Γ e ϕ ( x, t ) | p + |∇ e ϕ ( x, t ) | p ) d H x ! p . From Lemma 3.1 we check that(4.3) Z Γ( t ) | ∂ Γ j ψ ( x ) | p d H x = Z U (cid:12)(cid:12)(cid:12)(cid:12) g αβ ∂ b x j ∂X α ∂ϕ∂X β ( X, t ) (cid:12)(cid:12)(cid:12)(cid:12) p p G ( X, t ) dX ≤ C ( λ min , λ max ) Z U |∇ X ϕ | p dX ≤ C k ϕ k pW ,p ( U ) if ψ ∈ C (Γ( t ))and that(4.4) Z Γ( t ) | ∂ Γ i ∂ Γ j ψ ( x ) | p d H x ≤ C ( λ min , λ max ) k ϕ k pW ,p ( U ) if ψ ∈ C (Γ( t )) . From Lemmas 4.2 and 4.4, we see that ( 4.3) holds for all ψ ∈ W ,p (Γ( t )) and that( 4.4) holds for all ψ ∈ W ,p (Γ( t )). Lemma 4.2 (Properties of W ,p (Γ( t )) and W ,p (Γ( t ))) . (i) Let ψ ∈ W ,p (Γ( t )) and ϕ ∈ W ,p ( U ) such that ψ = e ϕ . Then for each j = 1 , , , ∂ Γ j ψ ∈ L p (Γ( t )) and Z Γ( t ) ∂ Γ j ψ d H x = Z U g αβ ∂ b x j ∂X α ∂ϕ∂X β √G dX. (ii) Let ψ ∈ W ,p (Γ( t )) and ϕ ∈ W ,p ( U ) such that ψ = e ϕ . Then for each i, j =1 , , , ∂ Γ j ∂ Γ i ψ ∈ L p (Γ( t )) and Z Γ( t ) ∂ Γ i ∂ Γ j ψ d H x = Z U g ζη ∂ b x i ∂X ζ ∂∂X η (cid:18) g αβ ∂ b x j ∂X α ∂ϕ∂X β (cid:19) √G dX. (iii) (Formula of the integration by parts) Let < p, p ′ < ∞ such that /p +1 /p ′ = 1 .Then for each j = 1 , , , ψ ♯ ∈ W ,p (Γ( t )) , and ψ ♭ ∈ W ,p ′ (Γ( t )) , (4.5) Z Γ( t ) ψ ♯ ( ∂ Γ j ψ ♭ ) d H x = − Z Γ( t ) ( ∂ Γ j ψ ♯ + H Γ n j ψ ♯ ) ψ ♭ d H x . Proof of Lemma 4.2. We only prove (i) since (ii) and (iii) are similar. Let ψ ∈ W ,p (Γ( t )) and ϕ ∈ W ,p ( U ) such that ψ = e ϕ . Fix j = 1 , , 3. Since C ( U ) ∩ W ,p ( U ) is dense in W ,p ( U ), there are ϕ m ∈ C ( U ) ∩ W ,p ( U ) such that(4.6) lim m →∞ k ϕ − ϕ m k W ,p ( U ) = 0 . Set ψ m = f ϕ m . By definition, we find that ψ m ∈ C (Γ( t )). Applying Lemma 3.2,we see that for all φ ∈ C (Γ( t )),(4.7) Z Γ( t ) ( ∂ Γ j ψ m ) φ d H x = − Z Γ( t ) ψ m ( ∂ Γ j φ + H Γ n j φ ) d H x . By ( 4.1), ( 4.3), and ( 4.6), we see that k ψ m − ψ m ′ k W ,p (Γ( t )) ≤ C k ϕ m − ϕ m ′ k W ,p ( U ) → m, m ′ → ∞ ) . Since L p (Γ( t )) is a Banach space, there is Ψ ∈ L p (Γ( t )) such that(4.8) lim m →∞ k ∂ Γ j ψ m − Ψ k L p (Γ( t )) = 0 . Using ( 4.2), ( 4.7), ( 4.8), andlim m →∞ k ψ m − ψ k L p (Γ( t )) = 0 , we check that for all φ ∈ C (Γ( t )) Z Γ( t ) Ψ φ d H x = − Z Γ( t ) ψ ( ∂ Γ j φ + H Γ n j φ ) d H x . This implies that ∂ Γ j ψ ∈ L p (Γ( t )) . From the assertion (ii) of Lemma 3.1, we have Z Γ( t ) ∂ Γ j ψ m d H x = Z U g αβ ∂ b x j ∂X α ∂ϕ m ∂X β √G dX. Using ( 4.2), the H¨older inequality, ( 4.6), and ( 4.8), we see that Z Γ( t ) ∂ Γ j ψ d H x = Z U g αβ ∂ b x j ∂X α ∂ϕ∂X β √G dX. Note that R Γ( t ) d H x + R U dX < + ∞ . Therefore, the lemma follows. (cid:3) Now we study the trace operator γ : W ,p (Γ( t )) → L p ( ∂ Γ( t )). Proposition 4.3. For each ψ ∈ W ,p (Γ( t )), k γψ k L p ( ∂ Γ( t )) = 0 . Here γ : W ,p (Γ( t )) → L p ( ∂ Γ( t )) is the trace operator defined by the proof ofProposition 4.3.To prove Proposition 4.3, we prepare the following lemma. DVECTION-DIFFUSION EQUATION ON EVOLVING SURFACE WITH BOUNDARY 15 Lemma 4.4. Let ≤ p < ∞ . Then (i) C (Γ( t )) is dense in L p (Γ( t )) . (ii) C (Γ( t )) is dense in W ,p (Γ( t )) . (iii) C (Γ( t )) is dense in W ,p (Γ( t )) . (iv) C (Γ( t )) is dense in W ,p (Γ( t )) .Proof of Lemma 4.4. We only prove (iii) since (i), (ii), and (iv) are similar. Fix ψ ∈ W ,p (Γ( t )). By definition, there is ϕ ∈ W ,p ( U ) such that ψ = e ϕ . Since C ( U ) ∩ W ,p ( U ) is dense in W ,p ( U ), there are ϕ m ∈ C ( U ) ∩ W ,p ( U ) such that(4.9) lim m →∞ k ϕ − ϕ m k W ,p ( U ) = 0 . Set ψ m = f ϕ m . By definition, we find that ψ m ∈ C (Γ( t )). By ( 4.1), ( 4.3), and( 4.9), we check that k ψ − ψ m k W ,p (Γ( t )) = k e ϕ − f ϕ m k W ,p (Γ( t )) ≤ C k ϕ − ϕ m k W ,p ( U ) → m → ∞ ) . Therefore, we conclude that C (Γ( t )) is dense in W ,p (Γ( t )). (cid:3) Proof of Proposition 4.3. We first introduce the trace operator γ on W ,p (Γ( t )).Let ψ ∈ W ,p (Γ( t )). By definition, there are ϕ ∈ W ,p ( U ) and ϕ m ∈ C ( U ) suchthat ψ = e ϕ and ( 4.9). From the definition of line integral, we see that Z ∂ Γ( t ) | f ( x ) | p d H x = Z ∂U | f ( b x ( X, t )) | p | n U g − n U g | d H X . Here n U = n U ( X ) = t ( n U , n U ) is the unit outer normal vector at X ∈ ∂U and g α = ∂ b x/∂X α . From ( n U ) + ( n U ) = 1 and Definition 2.1, we find that k f k L p ( ∂ Γ( t )) ≤ C ( λ max ) k b f k L p ( ∂U ) . Since k · k L p ( ∂U ) ≤ C ( U ) k · k W ,p ( U ) , we see that k f ϕ m | ∂ Γ( t ) − g ϕ m ′ | ∂ Γ( t ) k L p ( ∂ Γ( t )) ≤ C k ϕ m | ∂U − ϕ m ′ | ∂U k L p ( ∂U ) ≤ C k ϕ m − ϕ m ′ k W ,p ( U ) → m, m ′ → ∞ ) . Therefore, we set γψ = lim m →∞ e ϕ m | ∂ Γ( t ) in L p ( ∂ Γ( t )) . Since k e ϕ m | ∂ Γ( t ) k L p ( ∂ Γ( t )) ≤ C k ϕ m | ∂U k L p ( ∂U ) , we use ( 4.9) to see that k γψ k L p ( ∂ Γ( t )) ≤ C k b γϕ k L p ( ∂U ) . Here b γ : W ,p ( U ) → L p ( ∂U ) is the trace operator.Now we assume that ψ ∈ W ,p (Γ( t )). By definition, there are ϕ ∈ W ,p ( U ) and ϕ m ∈ C ( U ) such that ψ = e ϕ and ( 4.9). Since k f ϕ m | ∂ Γ( t ) k L p ( ∂ Γ( t )) ≤ C k ϕ m | ∂U k L p ( ∂U ) = 0 , we find that k γψ k L p ( ∂ Γ( t )) = lim m →∞ k e ϕ m | ∂ Γ( t ) k ∂ Γ( t ) = 0 . Therefore we conclude that γψ = 0 if ψ ∈ W ,p (Γ( t )). (cid:3) Function Spaces on Evolving Surface Γ T . In this section we define andstudy function spaces on Γ T . Set L (0 , T ; L (Γ( · ))) = { ψ ; ψ = e ϕ, ϕ ∈ L (0 , T ; L ( U )) , k ψ k L (0 ,T ; L (Γ( · ))) < ∞} ,L (0 , T ; W , (Γ( · )))= { ψ ; ψ = e ϕ, ϕ ∈ L (0 , T ; W , ( U )) , k ψ k L (0 ,T ; W , (Γ( · ))) < ∞} ,L (0 , T ; W , ∩ W , (Γ( · )))= { ψ ; ψ = e ϕ, ϕ ∈ L (0 , T ; W , ∩ W , ( U )) , k ψ k L (0 ,T ; W , (Γ( · ))) < ∞} ,W , (0 , T ; L (Γ( · )))= { ψ ; ψ = e ϕ, ϕ ∈ W , (0 , T ; L ( U )) , k ψ k W , (0 ,T ; L (Γ( · ))) < ∞} . Here k ψ k L (0 ,T ; L (Γ( · ))) := Z T k e ϕ k L (Γ( t )) dt ! , k ψ k L (0 ,T ; W , (Γ( · ))) := Z T k e ϕ k W , (Γ( t )) dt ! , k ψ k L (0 ,T ; W , (Γ( · ))) := Z T k e ϕ k W , (Γ( t )) dt ! , and k ψ k W , (0 ,T ; L (Γ( · ))) := Z T k e ϕ k L (Γ( t )) dt + Z T (cid:13)(cid:13)(cid:13)(cid:13) d e ϕdt (cid:13)(cid:13)(cid:13)(cid:13) L (Γ( t )) dt ! . However, we can not define e ϕ for ϕ ∈ L (0 , T ; L ( U )) in general. Therefore wedefine e ϕ as follows: Let ϕ ∈ L (0 , T ; L ( U )). From the definition of the Bochnerintegral and the Fubini-Tonelli theorem, there exists ϕ ♯ = ϕ ♯ ( X, t ) ∈ L ( U × (0 , T ))such that(4.10) k ϕ − ϕ ♯ k L (0 ,T ; L ( U )) = 0 . Set e ϕ = e ϕ ( x, t ) := ϕ ♯ ( e X ( x, t ) , t ) . It is easy to check that k ψ k L (0 ,T ; L (Γ( · ))) = Z T k e ϕ ( · , t ) k L (Γ( t )) dt = Z T Z U | ϕ ♯ ( X, t ) | p G ( X, t ) dXdt ≤ C k ϕ ♯ k L (0 ,T ; L ( U )) = C k ϕ k L (0 ,T ; L ( U )) . Now we study the case when ϕ ∈ L (0 , T ; W , ∩ W , ( U )). Assume that ϕ ∈ L (0 , T ; W , ∩ W , ( U )). We prove that for a.e. 0 ≤ t < T ,(4.11) ϕ ♯ ( · , t ) ∈ W , ( U ) ∩ W , ( U ) . From ( 4.10), we see that for a.e. 0 ≤ t < T k ϕ ( t ) − ϕ ♯ ( · , t ) k L ( U ) = 0 . DVECTION-DIFFUSION EQUATION ON EVOLVING SURFACE WITH BOUNDARY 17 Fix t . It is easy to check that for all φ ∈ C ∞ ( U ) and α, β = 1 , Z U ϕ ♯ ( X, t ) ∂φ∂X α dX = Z U ϕ ( t ) ∂φ∂X α dX = − Z U ∂ϕ∂X α φ dX, Z U ϕ ♯ ( X, t ) ∂ φ∂X α ∂X β dX = Z U ϕ ( t ) ∂ φ∂X α ∂X β dX = Z U ∂ ϕ∂X α ∂X β φ dX. Since ϕ ( t ) ∈ W , ( U ), it follows from the definition of weak derivative for functionsin U to find that ϕ ♯ ( · , t ) ∈ W , ( U ). This implies that k ϕ ♯ ( · , t ) − ϕ ( t ) k W , ( U ) = 0.Thus, we see ( 4.11). It is easy to check that k ψ k L (0 ,T ; W , (Γ( · ))) ≤ C k ϕ k L (0 ,T ; W , ( U )) . Finally, we define k γψ k L (0 ,T ; L ( ∂ Γ( · ))) for ψ ∈ L (0 , T ; W , (Γ( · ))). Let ψ ∈ L (0 , T ; W , (Γ( · ))). By the previous argument, there are ϕ ∈ L (0 , T ; W , ( U )) and ϕ ♯ ∈ L ( U × (0 , T )) such that ( 4.10) and for a.e. t , ϕ ♯ ( · , t ) ∈ W , ( U ) . Set e ϕ = e ϕ ( x, t ) := ϕ ♯ ( e X ( x, t ) , t ) . We define k γψ k L (0 ,T ; L ( ∂ Γ( · ))) = Z T k γ e ϕ k L ( ∂ Γ( t )) dt ! . Here γ is the trace operator defined by Proposition 4.3. By an argument similar tothat in the proof of Proposition 4.3, we see that k γψ k L (0 ,T ; L ( ∂ Γ( · ))) = Z T k γ e ϕ k L ( ∂ Γ( t )) dt ≤ C Z T k b γϕ ♯ k L ( ∂U ) dt ≤ C ( U ) Z T k ϕ ♯ k W , ( ∂U ) dt = C k ϕ k L (0 ,T ; W , ( U )) . Remark that k ψ k L (0 ,T ; W , (Γ( · ))) ≤ C k ϕ k L (0 ,T ; W , ( U )) if ψ ∈ L (0 , T ; W , (Γ( · ))) , k ψ k W , (0 ,T ; L (Γ( · ))) ≤ C k ϕ k W , (0 ,T ; L ( U )) if ψ ∈ W , (0 , T ; L (Γ( · ))) . On Strong Solutions to Advection-Diffusion Equation In this section we study basic properties of the strong solutions to ( 1.1) with u ∈ W , (Γ ). Let L = L ( t ) be the evolution operator defined by ( 1.3) (seeSection 7 for details). For ψ = ψ ( x, t ) and ϕ ( X, t ), b ψ = b ψ ( X, t ) := ψ ( b x ( X, t ) , t )and e ϕ = e ϕ ( x, t ) := ϕ ( e X ( x, t ) , t ). Lemma 5.1 (Strong solution to ( 1.1)) . Let u ∈ W , (Γ ) and u ∈ L (0 , T : W , ∩ W , (Γ( · ))) ∩ W , (0 , T ; L (Γ( · ))) . Assume that b u ∈ C ([0 , T ); L ( U )) and that the function u satisfies the followingthree properties: (i) (Equation) (cid:13)(cid:13)(cid:13)(cid:13) d b udt + L b u (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; L ( U )) = 0 . (ii) (Boundary condition) k b γ b u k L (0 ,T ; L ( ∂U )) = 0 . Here b γ : W , ( U ) → L ( ∂U ) is the trace operator. (iii) (Initial condition) lim t → k b u ( t ) − u ( b x ( · , k L ( U ) = 0 . Then u is a strong solution to system ( 1.1) with initial datum u .Proof of Lemma 5.1. From Lemma 3.1 and an argument in Section 4, we see that k D wt u + (div Γ w ) u − div Γ ( κ ∇ Γ u ) k L (0 ,T ; L (Γ( · ))) = Z T (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) d b udt + L b u (cid:19) √G (cid:13)(cid:13)(cid:13)(cid:13) L ( U ) dt ≤ C Z T (cid:13)(cid:13)(cid:13)(cid:13) d b udt + L b u (cid:13)(cid:13)(cid:13)(cid:13) L ( U ) dt = 0and that k γu k L (0 ,T ; L ( ∂ Γ( · ))) = Z T k γu k L ( ∂ Γ( t )) dt ≤ C Z T k b γ b u k L ( ∂U ) dt = 0 . Therefore, we find that u satisfies the properties of strong solutions to system ( 1.1)with u ∈ W , (Γ ) as in Definition 2.2. (cid:3) Lemma 5.2 (Sufficient condition for existence of a sol to ( 1.1)) . Let v ∈ W , ( U ) and v ∈ C ([0 , T ); L ( U )) ∩ L (0 , T ; W , ∩ W , ( U )) ∩ W , (0 , T ; L ( U )) . Assume that v satisfies (cid:13)(cid:13)(cid:13)(cid:13) ddt v + Lv (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; L ( U )) = 0 , and lim t → k v ( t ) − v k L ( U ) = 0 . Then there exists a strong solution u of system ( 1.1) with u . Here u = u ( x ) = v ( e X ( x, .Proof of Lemma 5.2. Since v ∈ L (0 , T ; L ( U )), it follows from the definition ofthe Bochner integral and the Fubini-Tonelli theorem, there exists v ♯ = v ♯ ( X, t ) ∈ L ( U × (0 , T )) such that k v − v ♯ k L (0 ,T ; L ( U )) = 0 . By an argument similar to that in Section 4, we see that k v − v ♯ k L (0 ,T ; W , ( U )) = 0 . DVECTION-DIFFUSION EQUATION ON EVOLVING SURFACE WITH BOUNDARY 19 Now we set u = u ( x, t ) = v ♯ ( e X ( x, t ) , t ). It is clear that k b γ b u k L (0 ,T ; L ( ∂U )) = 0 andthat k v − b u k L (0 ,T ; W , ( U )) = 0 . By Lemma 5.1 and the assumptions of Lemma 5.2, the lemma follows. (cid:3) Next we study the basic properties of solutions to system ( 1.1). Lemma 5.3 (Properties of solutions to ( 1.1)) . Let v ∈ W , ( U ) and v, v ♮ ∈ C ([0 , T ); L ( U )) ∩ L (0 , T ; W , ∩ W , ( U )) ∩ W , (0 , T ; L ( U )) . Set u = e v , u ♮ = e v ♮ , and u = u ( x ) = v ( e X ( x, . Assume that u and u ♮ are twostrong solutions of system ( 1.1) with initial datum u . Then (i) (Uniqueness) u = u ♮ on [0 , T ) . (ii) (Energy equality) For all ≤ s ≤ t < T , (5.1) 12 k u ( t ) k L (Γ( t )) + Z ts k√ κ ∇ Γ u ( τ ) k L (Γ( τ )) dτ = 12 k u ( s ) k L (Γ( s )) . Moreover, assume in addition that T = ∞ and that there is C > independent of v such that (5.2) Z ∞ (cid:13)(cid:13)(cid:13)(cid:13) dvdt (cid:13)(cid:13)(cid:13)(cid:13) L ( U ) dt ! + (cid:18)Z ∞ k Av k L ( U ) dt (cid:19) ≤ C k v k W , ( U ) , where A is the operator defined by ( 1.4) . Then, (iii) (Stability) There is C > such that for all t > , k u ( t ) k L (Γ( t )) ≤ Ct − k v k W , ( U ) . (iv) (Regularity) There is C > independent of v such that k D wt u k L (0 , ∞ ; L (Γ( · ))) + k div Γ ( κ ∇ Γ u ) k L (0 , ∞ ; L (Γ( · ))) ≤ C k v k W , ( U ) . Proof of Lemma 5.3. We first show (i) and (ii). Set u ♯ = u − u ♮ . Then D wt u ♯ + (div Γ w ) u ♯ − div Γ ( κ ∇ Γ u ♯ ) = 0 on Γ T ,u ♯ | ∂ Γ T = 0 ,u ♯ | t =0 = 0 . Using Lemma 3.1 and ( 4.5), we see that ddt Z Γ( t ) | u ♯ | d H x = Z Γ( t ) { D wt u ♯ + (div Γ w ) u ♯ } u ♯ d H x = Z Γ( t ) { div Γ ( κ ∇ Γ u ♯ ) } u ♯ d H x = − Z Γ( t ) κ |∇ Γ u ♯ | d H x . Integrating with respect to time, we check that for 0 < t < T ,12 k u ♯ ( t ) k L (Γ( t )) + Z t k√ κ ∇ Γ u ♯ k L (Γ( τ )) dτ = 12 k u ♯ (0) k L (Γ ) = 0 . Since κ ≥ κ min > 0, we see that u ♯ = 0 on [0 , T ). Therefore, we conclude that u = u ♮ on [0 , T ). In the same manner, we have ( 5.1). Next we prove (iii). From ( 5.1) and κ ≥ κ min > 0, we check that for all s < t k u ( t ) k L (Γ( t )) ≤ k u ( s ) k L (Γ( s )) . By the H¨older inequality, we see that k u ( t ) k L (Γ( t )) ≤ t Z t k u ( s ) k L (Γ( s )) ds ≤ t (cid:18)Z t k u ( s ) k L (Γ( s )) ds (cid:19) =: (R.H.S.) . Using ( 4.1), ( 3.2), and ( 5.2), we observe that(R.H.S.) ≤ Ct (cid:18)Z t k v ( s ) k L ( U ) ds (cid:19) ≤ Ct (cid:18)Z ∞ k Av ( s ) k L ( U ) ds (cid:19) ≤ Ct k v k W , ( U ) . Thus, we see (iii).Finally, we prove (iv). Using Lemma 3.1 and ( 3.2), we check that (cid:18)Z ∞ k D wt u k L (Γ( t )) dt (cid:19) + (cid:18)Z ∞ k div Γ ( κ ∇ Γ u ) k L (Γ( t )) dt (cid:19) ≤ C Z ∞ (cid:13)(cid:13)(cid:13)(cid:13) dvdt (cid:13)(cid:13)(cid:13)(cid:13) L ( U ) dt ! + C (cid:18)Z ∞ k Av k L ( U ) dt (cid:19) ≤ C k v k W , ( U ) . Therefore, the lemma follows. (cid:3) Evolution Operator on Hilbert Space In this section we provide the key method for constructing local and global-in-time strong solutions to the evolution equation ( 1.2). To this end, we study anevolution operator on a Hilbert space by applying the maximal L p -in-time regularityfor Hilbert space-valued functions.Let H be a Hilbert space and k · k H its norm. Let T ∈ (0 , ∞ ] and A : D ( A )( ⊂ H ) → H be a linear operator on H . For 0 ≤ t < T , let B ( t ) : D ( B ( t )) → H be alinear operator on H such that D ( A ) ⊂ D ( B ( t )). Define L ( t ) f := A f + B ( t ) f and D ( L ( t )) := D ( A ). We consider the following evolution system:(6.1) ( ddt v + L v = 0 on (0 , T ) ,v | t =0 = v , under the following assumptions: Assumption 6.1. (i) Assume that −A generates a bounded analytic semigroup on H . (ii) Assume that −A generates a contraction C -semigroup on H . (iii) The operator A is a non-negative selfadjoint operator on H . (iv) There are C , C > such that for all f ∈ D ( A ) and ≤ t < T , (6.2) kB ( t ) f k H ≤ C kA f k H + C k f k H . DVECTION-DIFFUSION EQUATION ON EVOLVING SURFACE WITH BOUNDARY 21 (v) There is C > such that for all ϕ ∈ C ((0 , T ); D ( A )) and < s ≤ t < T , (6.3) kB ( t ) ϕ ( t ) − B ( s ) ϕ ( s ) k H ≤ C ( t − s )( kA ϕ ( t ) k H + k ϕ ( t ) k H )+ C ( kA ϕ ( t ) − A ϕ ( s ) k H + k ϕ ( t ) − ϕ ( s ) k H ) . (vi) For each fixed t ∈ [0 , T ) the operator −L ( t ) generates an analytic semigroupon H . Since −A generates an analytic semigroup on H , it follows from the perturbationtheory (Proposition 8.2) to see that (vi) holds if C is sufficiently small. Remarkthat (vi) is not essential but important.In this section we apply the maximal L -regularity of A to construct strongsolutions to system ( 6.1). Lemma 6.2 (Maximal L -regularity of A ) . For each T ∈ (0 , ∞ ] and ( F, V ) ∈ L (0 , T ; H ) × D ( A ) , there exists a uniquefunction V satisfying the following system: ( ddt V + A V = F on (0 , T ) ,V | t =0 = V , and the estimate: (6.4) k dV /dt k L (0 ,T ; H ) + kA V k L (0 ,T ; H ) ≤ √ kA V k H + C A k F k L (0 ,T ; H ) . Here the positive constant C A does not depend on T , F , and V . Since −A generates a bounded analytic semigroup on H and A is a non-negativeselfadjoint operator on H , we use an argument similar to that in the proof of Lemma3.5 to have Lemma 6.2.Before stating the main result of this section, we introduce the two functionspaces and the definition of strong solutions to system ( 6.1). Define Y T = { ϕ ∈ C ((0 , T ); H ); k ϕ k Y T < ∞} with k ϕ k Y T := sup Assume that T < ∞ and that (6.5) C ( C A + 1) < √ . Then for each v ∈ D ( A ) system ( 6.1) admits a unique strong solution v in Y T ∗ ,satisfying k v k Y T ∗ ≤ k v k H + 2 √ kA v k H , and for each < t < T ∗ (6.6) v ( t ) = e − t A v − Z t e − ( t − τ ) A B v ( τ ) dτ. Here T ∗ = min (cid:26) T, 12 log (cid:18) C ( C A + 1) (cid:19)(cid:27) . (ii) Assume that T = ∞ and C = 0 . Suppose that ( 6.5) holds. Then for each v ∈ D ( A ) system ( 6.1) admits a unique strong solution v in Y ∞ , satisfying k v k Y ∞ ≤ k v k H + 2 √ kA V k H , and for each < t < ∞ , ( 6.6) holds.Here C , C , and C A are the two positive constants appearing in ( 6.2) and thepositive constant appearing in ( 6.4) , respectively. To prove Theorem 6.3, we consider the following approximate equations. Proposition 6.4. Let v ∈ D ( A ). For each m ∈ N , set v and v m +1 as follows:(6.7) ( ddt v + A v = 0 on (0 , T ) ,v | t =0 = v , (6.8) ( ddt v m +1 + A v m +1 = −B v m on (0 , T ) ,v m +1 | t =0 = v . Then the following two assertions hold:(i) Assume that T < ∞ . Then for each m ∈ N ,lim t → v m ( t ) = v in H,v m ∈ C ([0 , T ); H ) ∩ C ((0 , T ); D ( A )) ∩ C ((0 , T ); H ) ,v m , A v m ∈ C ( ) m loc ((0 , T ); H ) , k v k Y T ≤ k v k H + √ kA v k H , k v m +1 k Y T ≤ k v k H + √ kA v k H + ( √ C + C p e T − C A + 1) k v m k Y T , k v m +2 − v m +1 k Y T ≤ ( √ C + C p e T − C A + 1) k v m +1 − v m k Y T , and for each 0 < t < T ,(6.9) v m +1 ( t ) = e − t A v − Z t e − ( t − τ ) A B ( τ ) v m ( τ ) dτ. DVECTION-DIFFUSION EQUATION ON EVOLVING SURFACE WITH BOUNDARY 23 (ii) Assume that T = ∞ and that C = 0. Then for each m ∈ N ,lim t → v m ( t ) = v in H,v m ∈ C ([0 , ∞ ); H ) ∩ C ((0 , ∞ ); D ( A )) ∩ C ((0 , ∞ ); H ) ,v m , A v m ∈ C ( ) m loc ((0 , ∞ ); H ) , k v k Y ∞ ≤ k v k H + √ kA v k H , k v m +1 k Y ∞ ≤ k v k H + √ kA v k H + √ C ( C A + 1) k v m k Y ∞ , k v m +2 − v m +1 k Y ∞ ≤ √ C ( C A + 1) k v m +1 − v m k Y ∞ , and for each 0 < t < ∞ , ( 6.9) holds.To attack Proposition 6.4, we prepare the two lemmas. Lemma 6.5. Let V ∈ D ( A ) and F ∈ L (0 , T ; H ) . Let V be the function satisfy-ing system ( 6.2) with ( V , F ) , obtained by Lemma 6.2. Assume that F ∈ C a loc ((0 , T ); H ) for some < a ≤ / . Then lim t → V ( t ) = V in H , (6.10) V ∈ C ([0 , T ); H ) ∩ C ((0 , T ); D ( A )) ∩ C ((0 , T ); H ) , and for ≤ t < T , V ( t ) = e − t A V − Z t e − ( t − τ ) A F ( τ ) dτ. Moreover, (6.11) V, A V ∈ C a / loc ((0 , T ); H ) . Lemma 6.6. (i) Assume T < ∞ . Then for ϕ ∈ C ([0 , T ); H ) ∩ L (0 , T ; D ( A )) , (6.12) kB ϕ k L (0 ,T ; H ) ≤ ( √ C + C p e T − k ϕ k Y T . (ii) Assume T = ∞ and C = 0 . Then for ϕ ∈ C ([0 , ∞ ); H ) ∩ L (0 , ∞ ; D ( A )) , kB ϕ k L (0 , ∞ ; H ) ≤ √ C k ϕ k Y ∞ . Proof of Lemma 6.5. Fix V ∈ D ( A ) and F ∈ L (0 , T ; H ) ∩ C a loc ((0 , T ); H ). Let V be the function satisfying system ( 6.2) with ( V , F ), obtained by Lemma 6.2.Since −A generates an analytic semigroup on H and F ∈ C a loc ((0 , T ); H ), it followsfrom the semigroup theory to see that lim t → V ( t ) = V , ( 6.10), and that for0 < s < t < T , V ( t ) = e − t A V + Z t e − ( t − τ ) A F ( τ ) dτ,V ( s ) = e − s A V + Z s e − ( s − τ ) A F ( τ ) dτ. We now prove ( 6.11). Fix ε, T ′ such that 0 < ε < T ′ < T . Let ε ≤ s ≤ t ≤ T ′ . Wefirst consider the H¨older continuity of V . Since V ( t ) − V ( s ) = (e − ( t − s ) A − − s A V + Z ts e − ( t − τ ) A F ( τ ) dτ + Z s (e − ( t − s ) A − − ( s − τ ) A F ( τ ) dτ, we use Proposition 8.3 and the H¨older inequality to see that k V ( t ) − V ( s ) k H = C ( t − s ) kA e − s A V k H + C ( t − s ) (cid:18)Z ts k F ( τ ) k H dτ (cid:19) + C Z s ( t − s ) ( s − τ ) k F ( τ ) k H dτ ≤ C ( t − s ) kA V k H + { C ( t − s ) + C ( t − s ) s }k F k L (0 ,T ; H ) ≤ C ( T ′ )( t − s ) . Next we consider the H¨older continuity of A V . It is easy to check that A V ( t ) − A V ( s ) = H ( t, s ) + H ( t, s ) + H ( t, s ) + H ( t, s ) . Here H := (e − ( t − s ) A − A e − s A V ,H := A Z ts e − ( t − τ ) A { F ( τ ) − F ( t ) } dτ,H := A Z s { e − ( t − s ) A − } e − ( s − τ ) A { F ( τ ) − F ( s ) } dτ,H := A Z ts e − ( t − τ ) A F ( t ) dτ + A Z s { e − ( t − s ) A − } e − ( s − τ ) A F ( s ) dτ. From Proposition 8.3 and F ∈ C a loc ((0 , T ); H ), we see that k H k H ≤ C ( t − s ) kA e − s A A V k H ≤ C ( t − s ) ε kA V k H and that k H k H ≤ Z ts C ( t − τ ) a t − τ dτ ≤ C ( t − s ) a . We also see that k H k H ≤ ( t − s ) a C (cid:13)(cid:13)(cid:13)(cid:13)Z s A a e − ( s − τ ) A { F ( τ ) − F ( s ) } dτ (cid:13)(cid:13)(cid:13)(cid:13) H ≤ ( t − s ) a C Z s ( s − τ ) a ( s − τ ) a dτ ≤ C ( t − s ) a s a ≤ C ( t − s ) a T ′ a . From ddτ e − τ A f = −A e − τ A f , we find that H = F ( t ) − F ( s ) − e − ( t − s ) A { F ( t ) − F ( s ) } + (e − ( t − s ) A − − s A F ( s ) . Since F ∈ C a loc ((0 , T ); H ) and k (e − ( t − s ) A − − s A F ( s ) k H = C ( t − s ) a kA a e − s A F ( s ) k H ≤ C ( t − s ) a ε a sup ε ≤ s ≤ T ′ k F ( s ) k H , we check that k H k H ≤ C ( ε, T ′ )( t − s ) a . As a result, we see that kA V ( t ) − A V ( s ) k H = C ( ε, T ′ )( t − s ) a . DVECTION-DIFFUSION EQUATION ON EVOLVING SURFACE WITH BOUNDARY 25 Therefore, we conclude that k V ( t ) − V ( s ) k H + kA V ( t ) − A V ( s ) k H = C ( ε, T ′ )( t − s ) a . Since ε, T ′ are arbitrary, we see ( 6.11). Therefore the lemma follows. (cid:3) Proof of Lemma 6.6. We only prove (i). Assume T < ∞ . Let ϕ ∈ C ([0 , T ); H ) ∩ L (0 , T ; D ( A )). By ( 6.2), we observe that Z T kB ϕ k H dt ≤ C Z T kA ϕ k H dt + 2 C Z T e t e − t k ϕ k H dt ≤ C Z T kA ϕ k H dt + C (e T − Let us attack Proposition 6.4. Proof of Proposition 6.4. We only prove (i) since (ii) is similar. Assume that T < ∞ . Fix v ∈ D ( A ) and 0 < ε < T ′ < T . Let s, t > ε ≤ s ≤ t ≤ T ′ .We first consider v , i.e., system ( 6.7). From the maximal L -regularity (Lemma6.2) of A , there is a unique function v satisfying system ( 6.7) and(6.13) k dv /dt k L (0 ,T ; H ) + kA v k L (0 ,T ; H ) ≤ √ kA v k H . Since −A generates an analytic C -semigroup on H , we see thatlim t → v ( t ) = v in H,v ∈ C ([0 , T ); H ) ∩ C ((0 , T ); D ( A )) ∩ C ((0 , T ); H ) , and for each 0 ≤ t < T v ( t ) = e − t A v . Since e − t A is contraction C -semigroup on H , we check that(6.14) sup From ( 6.3) and ( 6.23), we see that kB ( t ) v ( t ) − B ( s ) v ( s ) k H ≤ C ( ε, T ′ )( t − s ) . By ( 6.12), ( 6.15), and ( 6.22), we observe that kB v k L (0 ,T ; H ) ≤ ( √ C + C p e T − k v k Y T < C ( T ) < ∞ . Therefore, we conclude that B v ∈ L (0 , T ; H ) and B v ∈ C loc ((0 , T ); H ) . Now we assume that for each m ∈ N there are C ( m, ε, T ′ ) > C ( m, T ) > k v m ( t ) − v m ( s ) k H ≤ C ( m, ε, T ′ )( t − s )( ) m , (6.24) kA v m ( t ) − A v m ( s ) k H ≤ C ( m, ε, T ′ )( t − s )( ) m , (6.25) k v m k Y T ≤ C ( m, T ) < ∞ , (6.26)and lim t → v m ( t ) = v in H , v m ∈ C ([0 , T ); H ) ∩ C ((0 , T ); D ( A )) ∩ C ((0 , T ); H ) . From ( 6.3), ( 6.24), and ( 6.25), we check that kB ( t ) v m ( t ) − B ( s ) v m ( s ) k H ≤ C ( t − s )( kA v m ( t ) k H + k v m ( t ) k H ) + C ( m, ε, T ′ )( t − s )( ) m ≤ C ( m, ε, T ′ )( t − s )( ) m . From ( 6.12) and ( 6.26), we find that(6.27) kB v m k L (0 ,T ; H ) ≤ ( √ C + C p e T − k v m k Y T < + ∞ . Therefore, we conclude that(6.28) B v m ∈ L (0 , T ; H ) and B v m ∈ C ( ) m loc ((0 , T ); H ) . Now we consider v m +1 , i.e.,(6.29) ( ddt v m +1 + A v m +1 = −B v m on (0 , T ) ,v m +1 | t =0 = v . Since v ∈ D ( A ) and ( 6.28) holds, it follows from Lemma 6.2 to see that thereexists a unique function v m +1 satisfying system ( 6.29) and(6.30) k dv m +1 /dt k L (0 ,T ; H ) + kA v m +1 k L (0 ,T ; H ) ≤ √ kA v k H + C A kB v m k L (0 ,T ; H ) . Since −A generates an analytic semigroup on H , it follows from ( 6.28) to find that v m +1 ∈ C ((0 , T ); H ) ∩ C ((0 , T ); D ( A )) ∩ C ((0 , T ); H )and for each 0 < t < Tv m +1 ( t ) = e − t A v − Z t e − ( t − τ ) A B v m dτ. Since k v m +1 ( t ) k H ≤ k v k H + t (cid:18)Z t kB v m k H dτ (cid:19) from ||| e − t A ||| ≤ Finally, we prove Theorem 6.3. Proof of Theorem 6.3. Fix v ∈ D ( A ). Let v and v m +1 be the solutions tosystem ( 6.7) and ( 6.8), obtained by Proposition 6.4. We first show (i). Assumethat T < ∞ and that C ( C A + 1) < √ . Write T ∗ = min (cid:26) T, 12 log (cid:18) C ( C A + 1) (cid:19)(cid:27) . It is easy to check that √ C ( C A + 1) ≤ , (6.37) C ( C A + 1) p e T ∗ − ≤ . (6.38)From the assertion (i) of Proposition 6.4, ( 6.37), and ( 6.38), we find that k v m +1 k Y T ∗ ≤ k v k H + √ kA v k H + 12 k v m k Y T ∗ , k v m +2 − v m +1 k Y T ∗ ≤ k v m +1 − v m k Y T ∗ . Since k v k Y T ∗ ≤ k v k H + √ kA v k H , we see that for each m ∈ N , k v m k Y T ∗ ≤ k v k H + 2 √ kA v k H , and k v m +2 − v m +1 k Y T ∗ ≤ (cid:18) (cid:19) m k v − v k Y T ∗ ≤ (cid:18) (cid:19) m { k v k H + 3 √ kA v k H } . From a fixed point argument, we have a unique function v in Y T ∗ satisfyinglim m →∞ k v m − v k Y T ∗ = 0 , (6.39) k v k Y T ∗ ≤ k v k H + 2 √ kA v k H . Using ( 6.12), ( 6.37), ( 6.38), and ( 6.39), we see that(6.40) B v ∈ L (0 , T ; H ) . Since v m +1 satisfies the system ( ddt v m +1 + A v m +1 = −B v m on (0 , T ∗ ) ,v m +1 | t =0 = v , and that for 0 < t < T ∗ v m +1 ( t ) = e − t A v − Z t e − ( t − τ ) A B v m dτ, we apply ( 6.39) and ( 6.12) to see that (cid:13)(cid:13)(cid:13)(cid:13) ddt v + A v + B v (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; H ) = 0 , and that for 0 < t < T ∗ v ( t ) = e − t A v − Z t e − ( t − τ ) A B v dτ. Since e − t A is a contraction C -semigroup on H and ( 6.40) holds, we use the H¨olderinequality to check that k v ( t ) − v k H ≤ k e − t A v − v k H + t kB v k L (0 ,T ; H ) → t → . Therefore the assertion (i) of Theorem 6.3 is proved. The assertion (ii) is similar. (cid:3) Existence of Strong Solutions to Advection-Diffusion Equation In this section we prove the existence of local and global-in-time strong solutionto system ( 1.1). From Lemmas 5.1 and 5.2, we only have to consider the existenceof strong solutions to system ( 1.2). To consider ( 1.2), we write system ( 1.2) asfollows:(7.1) ( ddt v + Av = − Bv on (0 , T ) ,v | t =0 = v , where B = B ( t ) = L ( t ) − A . Here L and A (= A ) are the two operators definedby ( 1.3) and ( 3.1), respectively. Set ( Bf = B ( t ) f = ( L − A ) f,D ( B ( t )) = D ( A ) , ( Lf = L ( t ) f,D ( L ( t )) = D ( A ) . In this section, we apply Theorem 6.3 to construct strong solutions to system ( 7.1).Set M j ( t ) and M j as in Section 2 ( j = 1 , , , , DVECTION-DIFFUSION EQUATION ON EVOLVING SURFACE WITH BOUNDARY 31 Time-dependent Laplace-Beltrami Operator. The aim of this subsectionis to prove the following key lemma to apply Theorem 6.3. Lemma 7.1. Let C ♯ be the positive constant appearing in ( 3.2) . Then (i) There is C ⋆ > such that for all f ∈ D ( A ) and ≤ t < T , (7.2) k B ( t ) f k L ( U ) ≤ C ♯ M k Af k L ( U ) + C ⋆ k f k L ( U ) . (ii) For all f ∈ D ( A ) and ≤ t < T , (7.3) k B ( t ) f k L ( U ) ≤ C ♯ X j =1 M j k Af k L ( U ) + M k f k L ( U ) . (iii) For all f ∈ D ( A ) and ≤ t < T , (7.4) k B ( t ) f k L ( U ) ≤ C ♯ ( M + M + M + M + M ) k Af k L ( U ) . (iv) There is C > such that for all ϕ ∈ C ((0 , T ); D ( A )) and < s ≤ t < T , (7.5) k B ( t ) ϕ ( t ) − B ( s ) ϕ ( s ) k L ( U ) ≤ C ( t − s )( k Aϕ ( t ) k L ( U ) + k ϕ ( t ) k L ( U ) )+ C ( k Aϕ ( t ) − Aϕ ( s ) k L ( U ) + k ϕ ( t ) − ϕ ( s ) k L ( U ) ) . Applying Lemmas 3.4, 7.1, and Proposition 8.2, we have the following lemma. Lemma 7.2. Assume that C ♯ M ≤ . Then for each fixed t ∈ [0 , T ) the operator − L ( t ) generates an analytic semigroupon L ( U ) .Proof of Lemma 7.1. Fix f ∈ D ( A ). A direct calculation shows that B ( t ) = L ( t ) f − Af = − √G ∂∂X α (cid:18)b κ √G g αβ ∂f∂X β (cid:19) + 12 G (cid:18) d G dt (cid:19) f + (cid:18) λ ∂ f∂X + λ ∂ f∂X (cid:19) f = B ( t ) f + B ( t ) f + B ( t ) f + B ( t ) f + B ( t ) f. Here B ( t ) f := − ( b κ g − λ ) ∂ f∂X − ( b κ g − λ ) ∂ f∂X − b κ g ∂ f∂X ∂X ,B ( t ) f := − b κ G (cid:18) ∂ G ∂X α (cid:19) g αβ ∂f∂X β ,B ( t ) f := − b κ ∂ g αβ ∂X α ∂f∂X β ,B ( t ) f := − ∂ b κ∂X α g αβ ∂f∂X β ,B ( t ) f := 12 G (cid:18) d G dt (cid:19) f. We first consider B ( t ) f . From ( g αβ ) × = ( g αβ ) − × , we find that B ( t ) f = − (cid:18)b κ g G − λ (cid:19) ∂ f∂X − (cid:18)b κ g G − λ (cid:19) ∂ f∂X + 2 b κ g G ∂ f∂X ∂X . This gives(7.6) k B ( t ) f k L ( U ) ≤ (cid:18)(cid:13)(cid:13)(cid:13)(cid:13)b κ g G − λ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ + (cid:13)(cid:13)(cid:13)(cid:13)b κ g G − λ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ + 2 (cid:13)(cid:13)(cid:13)(cid:13)b κ g G (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (cid:19) k∇ X f k L ( U ) . Next we consider B ( t ) f + B ( t ) f + B ( t ) f . From ( g αβ ) × = ( g αβ ) − × , we findthat B ( t ) f + B ( t ) f + B ( t ) f = (cid:26) b κ G (cid:18) ∂ g ∂X − ∂ g ∂X (cid:19) − b κ G (cid:18) g ∂ G ∂X − g ∂ G ∂X (cid:19) (cid:27) ∂f∂X + (cid:26) b κ G (cid:18) ∂ g ∂X − ∂ g ∂X (cid:19) − b κ G (cid:18) g ∂ G ∂X − g ∂ G ∂X (cid:19) (cid:27) ∂f∂X + (cid:18) g G ∂ b κ∂X − g G ∂ b κ∂X (cid:19) ∂f∂X + (cid:18) g G ∂ b κ∂X − g G ∂ b κ∂X (cid:19) ∂f∂X . Therefore, we see that(7.7) k B ( t ) f + B ( t ) f + B ( t ) k L ( U ) ≤ ( M ( t ) + M ( t ) + M ( t )) k∇ X f k L . It is easy to check that(7.8) k B ( t ) f k L ( U ) ≤ M ( t ) k f k L ( U ) . Now we derive ( 7.2). Using the interpolation inequality ( 3.4), we find thatthere is C ⋆⋆ > f such that for all 0 ≤ t < T ,(7.9) k B ( t ) f + B ( t ) f + B ( t ) f k L ( U ) ≤ (cid:18)(cid:13)(cid:13)(cid:13)(cid:13)b κ g G − λ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ + (cid:13)(cid:13)(cid:13)(cid:13)b κ g G − λ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (cid:19) k∇ X f k L ( U ) + C ⋆⋆ k f k L ( U ) . By ( 3.4), ( 7.6), ( 7.8), and ( 7.9), we have k B ( t ) f k L ( U ) ≤ C ♯ M k Af k L ( U ) + ( C ⋆⋆ + M ) k f k L ( U ) , which is ( 7.2). Similarly, we use ( 3.4), ( 7.6), ( 7.7), and ( 7.8) to have ( 7.3) and( 7.4).Finally, we derive ( 7.5). Fix ϕ ∈ C ((0 , T ); D ( A )) and 0 < s ≤ t < T . Since b x = b x ( X, t ) is a C -function (Definition 2.1), b κ = b κ ( X, t ) is a C -function (Assumption2.3), and B ( t ) ϕ ( t ) − B ( s ) ϕ ( s ) = { B ( t ) − B ( s ) } ϕ ( t ) + B ( s ) { ϕ ( t ) − ϕ ( s ) } , we use ( 3.2) and the mean-value theorem to have ( 7.5). Therefore, Lemma 7.1 isproved. (cid:3) Existence of Strong Solutions to Advection-Diffusion Equation. Letus show the existence of strong solutions to the advection-diffusion equation ( 1.1)by Theorem 6.3 and Lemmas 5.1-5.3, 7.1, and 7.2. Let C ♯ , C A , and C ⋆ be the threepositive constants in ( 3.2), ( 3.8), and ( 7.2). Define Z T = { ϕ ∈ C ((0 , T ); L ( U )); k ϕ k Z T < ∞} with k ϕ k Z T := sup Assume that(7.10) 2 C ♯ M ≤ . From Lemmas 3.4, 7.1 and 7.2, we see that the three operators A , B and L satisfythe properties as in Assumption 6.1. Therefore, we apply Theorem 6.3 and Lemma7.1 to have the following three propositions. Proposition 7.3. Assume that T < ∞ and that(7.11) 2 C ♯ M ( C A + 1) < √ . Then for each v ∈ D ( A ) system ( 7.1) admits a unique strong solution v in Z T ⋆ ,satisfying k v k Z T⋆ ≤ k v k L ( U ) + 2 √ k A v k L ( U ) . Here T ⋆ = min (cid:26) T, 12 log (cid:18) C ⋆ ( C A + 1) (cid:19)(cid:27) . Proposition 7.4. Assume that T < ∞ and that(7.12) 2 C ♯ ( M + M + M + M ) ( C A + 1) < √ . Then for each v ∈ D ( A ) system ( 7.1) admits a unique strong solution v in Z T ∗ ,satisfying k v k Z T ∗ ≤ k v k L ( U ) + 2 √ k A v k L ( U ) . Here T ∗ = min (cid:26) T, 12 log (cid:18) M ( C A + 1) (cid:19)(cid:27) . Proposition 7.5. Assume that T = ∞ and that(7.13) 2 C ♯ ( M + M + M + M + M )( C A + 1) < √ . Then for each v ∈ D ( A ) system ( 7.1) admits a unique strong solution v in Y ∞ ,satisfying k v k Z ∞ ≤ k v k L ( U ) + 2 √ k A V k L ( U ) . Remark that ( 7.10) holds if either ( 7.11), ( 7.12), or ( 7.13) holds. Applying ( 3.5),Lemmas 5.1-5.3, Propositions 7.3-7.5, we have Theorems 2.4-2.6.8. Appendix: Maximal L p -Regularity and Semigroup Theory In this section we introduce the maximal L p -in-time regularity for Hilbert space-valued functions and the basic semigroup theory. Let H be a (complex) Hilbertspace and k · k H its norm. Let A : D ( A )( ⊂ H ) → H and B : D ( B ) → H betwo linear operators on H such that D ( A ) ⊂ D ( B ). Define L f := A f + B f and D ( L ) := D ( A ).We first state the maximal L p -regularity of the generator of a bounded analyticsemigroup on H . From [4], we have the following proposition. Proposition 8.1 (Maximal L p -regularity) . Let 1 < p < ∞ . Assume that −A generates a bounded analytic semigroup on H . Then A has the maximal L p -regularity, i.e., there is C A ( p ) > T ∈ (0 , ∞ ] and F ∈ L p (0 , T ; H )there exists a unique function V satisfying the system: ( ddt V + A V = F on (0 , T ) ,V | t =0 = 0 , and the estimates: k dV /dt k L p (0 ,T ; H ) + kA V k L p (0 ,T ; H ) ≤ C A ( p ) k F k L p (0 ,T ; H ) . See also [3] and [14] for maximal L p -regularity.Next we introduce basic semigroup properties. From [15, Sections 2 and 3] and[17, Section 2], we have the following two propositions. Proposition 8.2 (Perturbation theory) . Assume that −A generates an analytic semigroup on H . Suppose that there are C , C > f ∈ D ( A ) kB f k H ≤ C kA f k H + C k f k H . Then the following two assertions hold;(i) There is δ A > C ≤ δ A then the operator −L generates an analyticsemigroup on H .(ii) Assume in addition that −A generates a contraction C -semigroup on H . 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