Well-posedness of fully nonlinear and nonlocal critical parabolic equations
aa r X i v : . [ m a t h . A P ] J a n WELL-POSEDNESS OF FULLY NONLINEAR AND NONLOCAL CRITICALPARABOLIC EQUATIONS ∗ XICHENG ZHANGA bstract . In this paper we prove the existence of smooth solutions to fully nonlinear and non-local parabolic equations with critical index. The proof relies on the apriori H¨older estimate foradvection fractional-di ff usion equation established by Silvestre [11].
1. I ntroduction and main result
In this paper we are interested in solving the following fully nonlinear and nonlocal parabolicequation: ∂ t u = F ( t , x , u , ∇ u , − ( − ∆ ) α u ) , u (0) = ϕ, α ∈ (0 , , (1.1)where F ( t , x , u , w , q ) : [0 , × R d × R × R d × R → R is a measurable function, and ( − ∆ ) α is theusual fractional Laplacian defined by( − ∆ ) α u = F − ( | · | α F u ) , u ∈ S ( R d ) , where F denotes the Fourier’s transform, S ( R d ) is the Schwartz class of smooth real-valuedrapidly decreasing functions.Recently, in the sense of viscosity solutions, fully nonlinear and nonlocal elliptic and par-abolic equations have been extensively studied (cf. [4, 10, 3, 9], etc.). In [4], Ca ff arelli andSilvestre studied the following type of nonlocal equation: I α u ( x ) : = sup i inf j c i j + b i j · ∇ u ( x ) + Z R d [ u ( x + y ) − u ( x )] a i j ( y ) | y | − d − α d y ! = , where α ∈ (0 , i , j ranges in arbitrary sets, c i j ∈ R and b i j ∈ R d , the kernel a i j ( y ) satisfies a i j ( y ) = a i j ( − y ) , a a i j ( y ) a . This type of equation appears in the stochastic control problems. In [4], the extremal Puccioperators are used to characterize the ellipticity, and the ABP estimate, Harnack inequality andinterior C ,β -regularity were obtained. In [11], Silvestre studied the following nonlocal parabolicequation with critical index α = ∂ t u = I u , u (0) = ϕ, and established C ,β -regularity of viscosity solutions. In particular, the following first orderHamilton-Jacobi equation is covered by the above equation when H is Lipschitz continuous: ∂ t u + H ( ∇ u ) + ( − ∆ ) u = . In [9], Lara and Davila extended Silvestre’s result to the more general case, and in particular,focused on the uniformity of regularity as α → F does not depend on u . Taking the gradient with respect to x for equation (1.1), we have ∂ t ∇ u = − ( ∂ q F )( − ∆ ) α ∇ u + ( ∇ w F ) ∇ u + ∇ x F . ∗ This work is supported by NSFs of China (No. 10971076). e make the following observation: − ( − ∆ ) α u = ( − ∆ ) α − div ∇ u = R α · ∇ u , where R α : = ( − ∆ ) α − div is a bounded linear operator from Bessel potential space H α − , p to L p provided p >
1. If we set w : = ∇ u , then w satisfies the following quasi-linear parabolic system: ∂ t w = − ( ∂ q F )( w , R α w )( − ∆ ) α w + ( ∇ w F )( w , R α w ) ∇ w + ( ∇ x F )( w , R α w ) . (1.2)It is noticed that the classical quasi-geostrophic equation takes the same form (cf. [6, 5, 8], etc.): ∂ t θ + ( − ∆ ) α θ + R θ · ∇ θ = , R : = ∇ ⊥ ( − ∆ ) − . Assume now that one can solve equation (1.2), then it is natural to define u ( t , x ) : = ϕ ( x ) + Z t F ( s , x , w ( s , x ) , R α w ( s , x ))d s . Thus, if one can show ∇ u = w , (1.3)then it follows that u ( t , x ) = ϕ ( x ) + Z t F ( s , x , ∇ u ( s , x ) , − ( − ∆ ) α u ( s , x ))d s . For solving equation (1.2), we shall use Silvestre’s H¨older estimate [11] about the followinglinear parabolic equation: ∂ t u = − a ( − ∆ ) α u + b · ∇ u + f . (1.4)For proving (1.3), we need to solve a linear equation like ∂ t u = a ( − ∆ ) α − (cid:3) u + b · ( ∇ u − ( ∇ u ) t ) , (1.5)where (cid:3) : = div ∇ − ∇ div is a symmetric operator on L ( R d ; R d ) and h (cid:3) u , u i = −k∇ u k + k div u k . Notice that in one dimensional case, (cid:3) = α = Theorem 1.1.
Assume that ∂ q F > a > and for some κ > , | F ( t , x , u , , | κ ( | u | + and for any R > , F ∈ L ∞ ([0 , C ∞ b ( R d × B R × B dR × B R )) , (1.7) ∂ q F , ∇ w F ∈ L ∞ ([0 , C b ( R d × B R × B dR × R )) , (1.8) ∂ u F ∈ L ∞ ([0 , × R d × B R × R d × R ) , (1.9) where B dR denotes the open ball in R d with radius R and center ; and for any j ∈ N andR > , there exist C R , j > , γ R , j > and h R , j ∈ ( L ∩ L ∞ )( R d ) such that for all ( t , x , u , w , q ) ∈ [0 , × R d × B R × R d × R , |∇ jx F ( t , x , u , w , q ) | C R , j | w | ( | w | γ R , j + + h R , j ( x ) , (1.10) where γ R , = . Then for any initial value ϕ ∈ U ∞ : = ∩ k , p U k , p , where U k , p is defined by (2.4)below, there exists a unique u ∈ C ([0 , U ∞ ) solving equation (1.1) with α = . Moreover, sup t ∈ [0 , k u ( t ) k ∞ e κ ( k ϕ k ∞ + κ ) . emark 1.2. Let A ( q ) ∈ C ∞ ( R ) have bounded derivatives of first and second orders and ∂ q Abe bounded below by a > . Let H ∈ C ∞ ( R d ) and f ∈ C ∞ ( R ) satisfy | f ( u ) | κ ( | u | + . ThenF ( t , x , u , w , q ) : = A ( q ) + H ( w ) + f ( u ) satisfies all the conditions (1.6)-(1.10). In the subcritical case α ∈ (1 , ffi culties occurring: on one hand, we need toprove a stronger apriori H¨older estimate for equation (1.4) (see Theorem 2.4 below)sup t ∈ [0 , sup x , y | u ( t , x ) − u ( t , y ) || x − y | β C , ∃ β ∈ ( α − , , where C only depends on the bounds of a , b , f and u (0); on the other hand, for α ∈ (1 , α ∈ (0 , (cid:3) u = α ∈ (0 , ∂ t u = F ( t , x , u , − ( − ∆ ) α u ) , u (0) = ϕ. The paper is organized as follows: In Section 2, we prepare some notations and recall somewell-known facts for later use. In Section 3, we solve the linear equation in Sobolev spaces. InSection 4, we prove the existence of smooth solutions for the quasi-linear nonlocal parabolicsystem. In Section 5, we give the proof of Theorem 1.1.2. P reliminaries
Let N : = N ∪ { } . For p ∈ (1 , ∞ ) and β ∈ N , let W β, p be the completion of S ( R d ) withrespect to the norm k f k β, p : = β X k = k∇ k f k p , where ∇ k denotes the k -order gradient; and for 0 < β , integer, let W β, p be the completion of S ( R d ) with respect to the norm k f k β, p : = k f k p + [ β ] X k = " R d × R d |∇ k f ( x ) − ∇ k f ( y ) | p | x − y | d + { β } p d x d y ! p , (2.1)where for a number β >
0, [ β ] denotes the integer part of β and { β } = β − [ β ]. It is well-knownthat for k m , θ ∈ (0 ,
1) with (1 − θ ) k + m θ < N (cf. [14, p.185, (2)]),( W k , p , W m , p ) θ, p = W (1 − θ ) k + m θ, p , (2.2)where ( · , · ) θ, p stands for the real interpolation space. For t ∈ [0 , Y k , pt : = L p ([0 , t ]; W k , p )with the norm k u k Y k , pt : = Z t k u ( s ) k pk , p d s ! p , and let X k , pt be the completion of all functions u ∈ C ∞ ([0 , t ]; S ( R d )) with respect to the norm k u k X k , pt : = sup s ∈ [0 , t ] k u ( s ) k k − , p + k u k Y k , pt + k ∂ t u k Y k − , pt . It is well-known that (cf. [1, p.180, Theorem III 4.10.2]) X k , pt ֒ → C ([0 , t ]; W k − p , p ) . (2.3) et U k , p be the Banach space of the completion of C ∞ b ( R d ) with respect to the norm: k f k U k , p : = k f k ∞ + k∇ f k k , p . (2.4)For simplicity of notation, we also write X k , p : = X k , p , Y k , p : = Y k , p and W ∞ : = ∩ k , p W k , p , Y ∞ : = ∩ k , p Y k , p , X ∞ : = ∩ k , p X k , p , U ∞ : = ∩ k , p U k , p . Let Ω be an open domain of R d . For k ∈ N ∪ {∞} , we use C kb = C kb ( Ω ) to denote the space ofall bounded and k -order continuous di ff erentiable functions with all bounded derivatives up to k -order. For β ∈ (0 , C β be the completion of S ( R d ) with respect to the norm k f k C β : = k f k ∞ + | f | C β , where k · k ∞ is the sup-norm and | f | C β : = sup x , y | f ( x ) − f ( y ) || x − y | β . (2.5)Notice that C ∞ b ( R d ) C β . By the Sobolev embedding theorem, one has W , p ⊂ C − dp , p > d . Let R be the class of all linear operators R : W ∞ → W ∞ satisfying that for each β > p > R : W β, p → W β, p is a bounded linear operator , and for each β ∈ (0 , |R f | C β C d ,β | f | C β , ∀ f ∈ C β . (2.6)A typical example of such an operator is the Riesz transform: R j f : = ( − ∆ ) − ∂ j f = lim ε → Z | y | >ε f ( x − y ) y j | y | d + d y . Indeed, it holds that for any p > k∇ f k p ≃ k ( − ∆ ) f k p . (2.7)Recalling that for any f ∈ S ( R d ),( − ∆ ) f ( x ) = c d Z R d [ f ( x ) − f ( x + y )] | y | − d − d y , (2.8)where c d > − ∆ ) ( f g ) = g ( − ∆ ) f + f ( − ∆ ) g − E ( f , g ) , (2.9)where E ( f , g )( x ) : = c d Z R d ( f ( x ) − f ( x + y ))( g ( x ) − g ( x + y )) | y | − d − d y . (2.10)From this formula, it is easy to derive that (see [16]), Lemma 2.1.
Let ζ ∈ S ( R d ) and set ζ z ( x ) : = ζ ( x − z ) for z ∈ R d . Then for any p ∈ [1 , ∞ ) , thereexists a constant C = C ( p , d ) > such that for all f ∈ W , p , Z R d k ( − ∆ ) ( f ζ z ) − ( − ∆ ) f ζ z k pp d z C k ζ k p , p k f k p / p k f k p / , p . (2.11) or given λ > f ∈ L ∞ ([0 , W ∞ ) and ϕ ∈ W ∞ , let us consider the following heatequation: ∂ t u + λ ( − ∆ ) u = f , u (0) = ϕ. It is well-known that the unique solution can be represented by u ( t , x ) = P λ t ϕ ( x ) + Z t P λ t − s f ( s , x )d s , where ( P λ t ) t > is the Cauchy semigroup associated with λ ( − ∆ ) and given by P λ t ϕ ( x ) : = c d t Z R d ϕ ( λ y + x )( | y | + t ) ( d + / d y , where c d > C > λ , d , p such that for any f ∈ L p ([0 , × R d ) (cf. [12, 16]), Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∇ Z t P λ t − s f ( s )d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pp d s C Z k f ( s ) k pp d s . (2.12)We now use the probabilistic technique to extend the above estimate to the more generalcase. Let ( L t ) t > be a d -dimensional Cauchy process with L´evy measure ν (d x ) = d x / | x | d + . It iswell-known that (cf. [2]) P λ t ϕ ( x ) = E ϕ ( x + λ L t ) . Let ϑ : [0 , → R d and λ : [0 , → [0 , ∞ ) be two bounded measurable functions. Define T λ,ϑ t , s ϕ ( x ) : = E ϕ x − Z ts ϑ ( r )d r + Z ts λ ( r )d L r ! . (2.13)By the theory of stochastic di ff erential equation (cf. [2, p.402, Theorem 6.7.4]), one knows that ∂ t T λ,ϑ t , s ϕ + λ ( − ∆ ) T λ,ϑ t , s ϕ + ϑ · ∇T λ,ϑ t , s ϕ = . Now if we define u ( t , x ) : = Z t T λ,ϑ t , s f ( s , x )d s , then it is easy to see that ∂ t u + λ ( − ∆ ) u + ϑ · ∇ u = f , u (0) = . We have
Theorem 2.2.
Let ϑ : [0 , → R d and λ : [0 , → [ λ , ∞ ) be two bounded measurablefunctions, where λ > . For any p ∈ (1 , ∞ ) , there exists a constant C depending only on λ , d , p such that for all f ∈ L p ([0 , × R d ) , Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∇ Z t T λ,ϑ t , s f ( s )d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pp d s C Z k f ( s ) k pp d s . Proof.
Let ( L ( i ) t ) t > , i = , L t ) t > . By the theoryof stochastic di ff erential equation (cf. [2, 15]), one can write T λ,ϑ t , s ϕ ( x ) = E ϕ x − Z ts ϑ ( r )d r + Z ts ( λ ( r ) − λ )d L (1) r + λ ( L (2) t − L (2) s ) ! = E P λ t − s ϕ ( x − X t + X s ) , (2.14)where P λ t ϕ ( x ) : = E ϕ ( x + λ L (2) t ) is the semigroup associated with λ ( − ∆ ) , and X t : = Z t ϑ ( r )d r − Z t ( λ ( r ) − λ )d L (1) r . efine u ( t , x ) : = Z t P λ t − s f ( s , x + X s ) d s . By (2.14) one has Z t T λ,ϑ t , s f ( s , x )d s = E u ( t , x − X t ) . Hence, by H¨older’s inequality and Fubini’s theorem, Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∇ Z t T λ,ϑ t , s f ( s , x )d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pp d t = Z (cid:13)(cid:13)(cid:13) E ∇ u ( t , · − X t ) (cid:13)(cid:13)(cid:13) pp d t E Z (cid:13)(cid:13)(cid:13) ∇ u ( t , · − X t ) (cid:13)(cid:13)(cid:13) pp d t = E Z k u ( t ) k pp d t = E Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∇ Z t P λ t − s f ( s , · + X s ) d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pp d t ( . ) C E Z (cid:13)(cid:13)(cid:13) f ( s , · + X s ) (cid:13)(cid:13)(cid:13) pp d s = C Z k f ( s ) k pp d s . The proof is finished. (cid:3)
Below we prove a maximum principle for the fully nonlinear equation (1.1).
Theorem 2.3. (Maximum principle) Let F ( t , x , w , q ) : [0 , × R d × R d × R → R be a measurablefunction. Assume that for any R > and all ( t , x , w , q ) ∈ [0 , × R d × R d × R with | w | , | q | R, ∂ q F ( t , x , w , q ) a R , , |∇ w F | ( t , x , w , q ) a R , , (2.15) where a R , , a R , > . Let u ∈ C ([0 , C b ( R d )) satisfyu ( t , x ) = u (0 , x ) + Z t F ( s , x , ∇ u ( s , x ) , − ( − ∆ ) u ( s , x ))d s . If F ( s , x , , , then for all t ∈ [0 , , sup x ∈ R d u ( t , x ) sup x ∈ R d u (0 , x ) . (2.16) In particular, k u ( t ) k ∞ k u (0) k ∞ + Z t k F ( s , · , , k ∞ d s . (2.17) Proof.
First of all, we assume F ( s , x , , δ < . (2.18)Suppose that (2.16) does not hold, then there exists a time t ∈ (0 ,
1] such thatsup x ∈ R d u ( t , x ) = sup t ∈ [0 , sup x ∈ R d u ( t , x ) . Let x n ∈ R d be such thatlim n →∞ u ( t , x n ) = sup x ∈ R d u ( t , x ) > u ( t , x ) , ∀ ( t , x ) ∈ [0 , × R d . We have for any ε ∈ (0 , t ),0 ε (cid:20) lim n →∞ u ( t , x n ) − lim n →∞ u ( t − ε, x n ) (cid:21) (2.19) ε lim n →∞ ( u ( t , x n ) − u ( t − ε, x n )) = ε lim n →∞ Z t t − ε F ( s , x n , ∇ u ( s , x n ) , − ( − ∆ ) u ( s , x n ))d s , (2.20) nd for any h ∈ R d , 0 lim ε ↓ ε (cid:18) lim n →∞ u ( t , x n ) − lim n →∞ u ( t , x n − ε h ) (cid:19) lim ε ↓ ε lim n →∞ ( u ( t , x n ) − u ( t , x n − ε h )) = lim ε ↓ lim n →∞ Z h · ∇ u ( t , x n − ε sh )d s = lim n →∞ ( h · ∇ u ( t , x n )) . In particular, by the arbitrariness of h , we getlim n →∞ ∇ u ( t , x n ) = . (2.21)On the other hand, since for any y ∈ R d , u ( t , x n + y ) − u ( t , x n ) sup x ∈ R d u ( t , x ) − u ( t , x n ) → , by (2.8) we havelim n →∞ − ( − ∆ ) u ( t , x n ) c d Z R d lim n →∞ [ u ( t , x n + y ) − u ( t , x n )] | y | − d − d y . (2.22)Moreover, since by u ∈ C ([0 , C b ( R d )),lim s → t k∇ ( u ( s ) − u ( t )) k ∞ = s → t k ( − ∆ ) ( u ( s ) − u ( t )) k ∞ = , we have by (2.15),lim ε → ε Z t t − ε k F ( s , ∇ u ( s ) , − ( − ∆ ) u ( s )) − F ( s , ∇ u ( t ) , − ( − ∆ ) u ( t )) k ∞ d s = . Hence, by (2.20), (2.21) and (2.18),0 lim ε → ε lim n →∞ Z t t − ε F ( s , x n , ∇ u ( t , x n ) , − ( − ∆ ) u ( t , x n ))d s = lim ε → ε lim n →∞ Z t t − ε F ( s , x n , , − ( − ∆ ) u ( t , x n ))d s lim ε → ε lim n →∞ Z t t − ε [ F ( s , x n , , − ( − ∆ ) u ( t , x n )) − F ( s , x n , , s + δ = lim ε → lim n →∞ h a n ,ε · (cid:16) − ( − ∆ ) u ( t , x n ) (cid:17)i + δ, (2.23)where a n ,ε : = ε Z t t − ε Z ( ∂ q F )( s , x n , , − r ( − ∆ ) u ( t , x n ))d r d s . Let R : = k ( − ∆ ) u ( t ) k ∞ . Noticing that 0 a n ,ε a R , , by (2.22), (2.23) and δ <
0, we obtain a contradiction.We now drop assumption (2.18). For δ <
0, set u δ ( t , x ) = u ( t , x ) + δ t . hen u δ ( t , x ) = u δ (0 , x ) + Z t h δ + F ( s , x , ∇ u δ ( s , x ) , − ( − ∆ ) u δ ( s , x )) i d s . So, for all t ∈ [0 , x ∈ R d u ( t , x ) sup x ∈ R d u δ ( t , x ) − δ t sup x ∈ R d u (0 , x ) − δ t . Letting δ ↑
0, we conclude the proof of (2.16).As for (2.17), by considering˜ u ( t , x ) = u ( t , x ) − Z t k F ( s , · , , k ∞ d s and using (2.16) for ˜ u ( t , x ) and − ˜ u ( t , x ) respectively, we immediately obtain (2.17). (cid:3) Next we recall Silvestre’s H¨older estimate about the linear advection fractional-di ff usionequation. The following result is taken from [16, Corollary 6.2]. Although the proofs givenin [11] and [16] are only for constant di ff usion coe ffi cient a ( t , x ), by slight modifications, theyare also adapted to the general bounded measurable function a ( t , x ). Theorem 2.4. (Silvestre’s H¨older estimate) Let a : [0 , × R d → R and b : [0 , × R d → R d be two bounded measurable functions. Let u ∈ C ([0 , C b ( R d )) satisfyu ( t , x ) = u (0 , x ) − Z t ( a ( − ∆ ) u )( s , x )d s + Z t ( b · ∇ u )( s , x )d s + Z t f ( s , x )d s . If a ( t , x ) > a > , then for any γ ∈ (0 , , there exist β ∈ (0 , and C > depending only ond , a , γ and k a k ∞ , k b k ∞ such that sup t ∈ [0 , | u ( t ) | C β C ( k u k ∞ + k f k ∞ + | u (0) | C γ ) , (2.24) where | · | C β is defined by (2.5).
3. L inear nonlocal parabolic equation
In this section, we consider the following linear scalar nonlocal equation: ∂ t u + a ( − ∆ ) u + b · ∇ u = f , u (0) = ϕ, (3.1)where a : [0 , × R d → R and b : [0 , × R d → R d are two bounded measurable functions.An increasing function ω : R + → R + is called a modulus function if lim s ↓ ω ( s ) =
0. Wemake the following assumptions on a and b : (H a , bk ) Let k ∈ N , a , b ∈ L ∞ ([0 , C kb ), and there are two modulus functions ω a and ω b suchthat for all t ∈ [0 ,
1] and x , y ∈ R d , | a ( t , x ) − a ( t , y ) | ω a ( | x − y | ) , | b ( t , x ) − b ( t , y ) | ω b ( | x − y | ) . (3.2)Moreover, for some a , a > t , x ) ∈ [0 , × R d , a a ( t , x ) a . We first prove the following important apriori estimate.
Lemma 3.1.
For given p ∈ (1 , ∞ ) and k ∈ N , let f ∈ Y k − , p and u ∈ X k , p satisfy that for almostall ( t , x ) ∈ [0 , × R d , ∂ t u ( t , x ) + a ( t , x )( − ∆ ) u ( t , x ) + b ( t , x ) · ∇ u ( t , x ) = f ( t , x ) . (3.3) Then under (H a , bk − ) , there exists a constant C k , p > such that for all t ∈ [0 , , k u k X k , pt C k , p (cid:18) k u (0) k k − p , p + k f k Y k − , pt (cid:19) , (3.4) here C , p depends only on a , a , k b k ∞ , d , p and ω a , ω b . In particular, equation (3.3) admits atmost one solution in X k , p .Proof. Let ( ρ ε ) ε ∈ (0 , be a family of mollifiers in R d , i.e., ρ ε ( x ) = ε − d ρ ( ε − x ), where ρ ∈ C ∞ ( R d )is nonnegative and has support in B and R ρ =
1. Define u ε ( t ) : = u ( t ) ∗ ρ ε , a ε ( t ) : = a ( t ) ∗ ρ ε , b ε ( t ) : = b ( t ) ∗ ρ ε , f ε ( t ) : = f ( t ) ∗ ρ ε . Taking convolutions for both sides of (3.3), we have ∂ t u ε ( t , x ) + a ε ( t , x )( − ∆ ) u ε ( t , x ) + b ε ( t , x ) · ∇ u ε ( t , x ) = h ε ( t , x ) , (3.5)where h ε ( t , x ) : = f ε ( t , x ) + a ε ( t , x )( − ∆ ) u ε ( t , x ) − [( a ( t )( − ∆ ) u ( t )) ∗ ρ ε ]( x ) + b ε ( t , x ) · ∇ u ε ( t , x ) − [( b ( t ) · ∇ u ( t )) ∗ ρ ε ]( x ) . By (3.2), it is easy to see that for all ε ∈ (0 ,
1) and t ∈ [0 ,
1] and x , y ∈ R d , | a ε ( t , x ) − a ε ( t , y ) | ω a ( | x − y | ) , | b ε ( t , x ) − b ε ( t , y ) | ω b ( | x − y | ) , (3.6)and | a ε ( t , x ) − a ( t , x ) | ω a ( ε ) , | b ε ( t , x ) − b ( t , x ) | ω b ( ε ) . Moreover, by the property of convolutions, we also havelim ε → Z k h ε ( t ) − f ( t ) k pp d t = . Below, we use the method of freezing the coe ffi cients to prove that for all t ∈ [0 , k u ε k X , pt C (cid:18) k u ε (0) k − p , p + k h ε k Y , pt (cid:19) , (3.7)where the constant C is independent of ε . After proving this estimate, (3.4) with k = ε below. Fix δ > ζ be a smooth function with support in B δ and k ζ k p =
1. For z ∈ R d , set ζ z ( x ) : = ζ ( x − z ) , λ az : = a ( t , z ) , ϑ bz ( t ) : = b ( t , z ) . Multiplying both sides of (3.5) by ζ z , we have ∂ t ( u ζ z ) + λ az ( − ∆ ) ( u ζ z ) + ϑ bz · ∇ ( u ζ z ) = g ζ z , where g ζ z : = ( λ az − a )( − ∆ ) u ζ z + λ az (( − ∆ ) ( u ζ z ) − ( − ∆ ) u ζ z ) + ( ϑ bz − b ) · ∇ ( u ζ z ) + ub · ∇ ζ z + h ζ z . Let T λ az ,ϑ bz t , s be defined by (2.13). Then u ζ z can be written as u ζ z ( t , x ) = T λ az ,ϑ bz t , ( u (0) ζ z )( x ) + Z t T λ az ,ϑ bz t , s g ζ z ( s , x )d s , and so that for any T ∈ [0 , Z T k∇ ( u ζ z )( t ) k pp d t p − Z T k∇T λ az ,ϑ bz t , ( u (0) ζ z ) k pp d t + Z T (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∇ Z t T λ az ,ϑ bz t , s g ζ z ( s )d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pp d t = : 2 p − ( I ( T , z ) + I ( T , z )) . or I ( T , z ), by (2.14) and (2.7), we have Z T k∇T λ az ,ϑ bz t , ( u (0) ζ z ) k pp d t = Z T (cid:13)(cid:13)(cid:13) ∇P a t ( u (0) ζ z ) (cid:13)(cid:13)(cid:13) pp d t C Z T (cid:13)(cid:13)(cid:13)(cid:13) ( − ∆ ) P a t ( u (0) ζ z ) (cid:13)(cid:13)(cid:13)(cid:13) pp d t C k u (0) ζ z k p − p , p , (3.8)where the last step is due to [14, p.96 Theorem 1.14.5] and (2.2). Thus, by definition (2.1), it iseasy to see that Z R d I ( T , z )d z C Z R d k u (0) ζ z k p − p , p d z C (cid:16) k u (0) k p − p , p k ζ k pp + k u (0) k pp k ζ k p − p , p (cid:17) . For I ( T , z ), by Theorem 2.2, we have I ( T , z ) C Z T k g ζ z ( s ) k pp d s C Z T k (( λ az − a )(( − ∆ ) u ζ z ))( s ) k pp d s + C Z T k λ az (( − ∆ ) u ζ z − ( − ∆ ) ( u ζ z ))( s ) k pp d s + C Z T k (( ϑ bz − b ) · ∇ ( u ζ z ))( s ) k pp d s + C Z T k ( ub · ∇ ζ z )( s ) k pp d s + C Z T k h ζ z ( s ) k pp d s = : I ( T , z ) + I ( T , z ) + I ( T , z ) + I ( T , z ) + I ( T , z ) . For I ( T , z ), by (3.6) and k ζ k p =
1, we have Z R d I ( T , z )d z C ω pa ( δ ) Z R d Z T k (( − ∆ ) u ζ z )( s ) k pp d s d z = C ω pa ( δ ) Z T k ( − ∆ ) u ( s ) k pp d s ( . ) C ω pa ( δ ) Z T k∇ u ( s ) k pp d s . For I ( T , z ), by (2.11) and Young’s inequality, we have Z R d I ( T , z )d z Ca Z T Z R d k (( − ∆ ) u ζ z − ( − ∆ ) ( u ζ z ))( s ) k pp d z d s C Z T k u ( s ) k pp d s + C Z T k u ( s ) k p / p k∇ u ( s ) k p / p d s C Z T k u ( s ) k pp d s + p Z T k∇ u ( s ) k pp d s . For I ( T , z ), as above we have Z R d I ( T , z )d z C ω pb ( δ ) Z T k∇ u ( s ) k pp d s + k∇ ζ k pp Z T k u ( s ) k pp d s ! . Moreover, it is easy to see that Z R d I ( T , z )d z C k b k p ∞ k∇ ζ k pp Z T k u ( s ) k pp d s , Z R d I ( T , z )d z C Z T k h ( s ) k pp d s . ombining the above calculations, we get Z T k∇ u ( s ) k pp d s = Z T Z R d k∇ u ( s ) · ζ z k pp d z d s p − Z T Z R d k∇ ( u ζ z )( s ) k pp d z d s + p − k∇ ζ k pp Z T k u ( s ) k pp d s C k u (0) k p − p , p + (cid:16) + C ( ω pa ( δ ) + ω pb ( δ )) (cid:17) Z T k∇ u ( s ) k pp d s + C Z T k u ( s ) k pp d s + C Z T k h ( s ) k pp d s . Choosing δ > C ( ω pa ( δ ) + ω pb ( δ )) , we obtain that for all T ∈ [0 , Z T k∇ u ( s ) k pp d s C k u (0) k p − p , p + C Z T k u ( s ) k pp d s + C Z T k h ( s ) k pp d s . (3.9)On the other hand, by (3.5), it is easy to see that for all t ∈ [0 , k u ( t ) k pp C k u (0) k pp + C Z t k∇ u ( s ) k pp d s + C Z t k h ( s ) k pp d s , which together with (3.9) and Gronwall’s inequality yields that for all t ∈ [0 , s ∈ [0 , t ] k u ( s ) k pp + Z t k∇ u ( s ) k pp d s C k u (0) k p − p , p + Z t k h ( s ) k pp d s ! . (3.10)From equation (3.3), by (2.7) we also have Z t k ∂ s u ( s ) k pp d s C k a k p ∞ Z t k ( − ∆ ) u ( s ) k pp d s + k b k p ∞ Z t k∇ u ( s ) k pp d s + Z t k h ( s ) k pp d s ! C ( k a k p ∞ + k b k p ∞ ) Z t k∇ u ( s ) k pp d s + Z t k h ( s ) k pp d s ! , which together with (3.10) gives (3.7), and therefore (3.4) with k = n = , , · · · , k , let w ( n ) ( t , x ) : = ∇ n u ( t , x ) . By the chain rule, we have ∂ t w ( n ) + a ( − ∆ ) w ( n ) + b · ∇ w ( n ) = h ( n ) , where h ( n ) : = ∇ n f − n X j = n !( n − j )! j ! (cid:16) ∇ j a · ∇ n − j ( − ∆ ) u + ∇ j b · ∇ n − j + u (cid:17) . Thus, by (3.4) with k = k∇ n u k X , pt = k w ( n ) k X , pt C (cid:18) k w ( n ) (0) k − p , p + k h ( n ) k Y , pt (cid:19) C k∇ n u (0) k − p , p + n X j = k∇ n − j + u k Y , pt + k∇ n f k Y , pt , hich implies that k u k X n + , pt C (cid:18) k u (0) k n + − p , p + k u k Y n , pt + k f k Y n , pt (cid:19) . By induction method, one obtains (3.7). (cid:3)
Now we prove the existence of solutions to equation (3.1).
Theorem 3.2.
Let p > and k ∈ N . Under (H a , bk − ) , for any ϕ ∈ W k − p , p and f ∈ Y k − , p , thereexists a unique u ∈ X k , p with u (0) = ϕ solving equation (3.1).Proof. We use the continuity method. For λ ∈ [0 , U λ : = ∂ t + λ a ( − ∆ ) + λ b · ∇ + (1 − λ )( − ∆ ) . By (2.7), it is easy to see that U λ : X k , p → Y k − , p . (3.11)For given ϕ ∈ W k − p , p , let X k , p ϕ be the space of all functions u ∈ X k , p with u (0) = ϕ . It is clearthat X k , p ϕ is a complete metric space with respect to the metric k · k X k , p . For λ = f ∈ Y k − , p ,it is well-known that there is a unique u ∈ X k , p ϕ such that U u = ∂ t u + ( − ∆ ) u = f . In fact, by Duhamel’s formula, the unique solution can be represented by u ( t , x ) = P t ϕ ( x ) + Z t P t − s f ( s , x )d s . Suppose now that for some λ ∈ [0 , f ∈ Y k − , p , the equation U λ u = f admits a unique solution u ∈ X k , p ϕ . Then, for fixed f ∈ Y k − , p and λ ∈ [ λ , u ∈ X k , p ϕ , by (3.11), the equation U λ w = f + ( U λ − U λ ) u (3.12)admits a unique solution w ∈ X k , p ϕ . Introduce an operator w = Q λ u . We want to use Lemma 3.1 to show that there exists an ε > λ such that forall λ ∈ [ λ , λ + ε ], Q λ : X k , p ϕ → X k , p ϕ is a contraction operator.Let u , u ∈ X k , p ϕ and w i = Q λ u i , i = ,
2. By equation (3.12), we have U λ ( w − w ) = ( U λ − U λ )( u − u ) = ( λ − λ )(( a − − ∆ ) + b · ∇ )( u − u ) . By (3.4) and (2.7), one sees that k Q λ u − Q λ u k X k , p C k , p | λ − λ | · k (( a − − ∆ ) + b · ∇ )( u − u ) k Y k − , p C | λ − λ | · k u − u k Y k , p C | λ − λ | · k u − u k X k , p , where C is independent of λ, λ and u , u . Taking ε = / (2 C ), one sees that Q λ : X k , p ϕ → X k , p ϕ is a 1 / λ ∈ [ λ , λ + ε ], there existsa unique u ∈ X k , p ϕ such that Q λ u = u , hich means that U λ u = f . Now starting from λ =
0, after repeating the above construction [ ε ] + f ∈ Y k − , p , U u = f admits a unique solution u ∈ X k , p ϕ . (cid:3)
4. Q uasi - linear nonlocal parabolic system Consider the following quasi-linear nonlocal parabolic system: ∂ t u + a ( u , R a u )( − ∆ ) u + b ( u , R b u ) · ∇ u = f ( u , R f u ) , (4.1)where u = ( u , · · · , u m ) and a ( t , x , u , r ) : [0 , × R d × R m × R k → R , b ( t , x , u , r ) : [0 , × R d × R m × R k → R d , f ( t , x , u , r ) : [0 , × R d × R m × R k → R m , are measurable functions, and R a = ( R i ja ) , R b = ( R i jb ) , R f = ( R i jf ) ∈ R k × m . Here we have used that R a u = P mj = R i ja u j , similarly for R b u and R f u .The main result of this section is: Theorem 4.1.
Suppose that a ( t , x , u , r ) > a > , and for any R > ,a , b , f ∈ L ∞ ([0 , C ∞ b ( R d × B mR × B kR )) , (4.2) a , b ∈ L ∞ ([0 , C b ( R d × B mR × R k )) , (4.3) where B mR denotes the ball in R m with radius R, and for each j ∈ N , there exist C f , j , γ j > andh j ∈ ( L ∩ L ∞ )( R d ) such that |∇ jx f ( t , x , u , r ) | C f , j | u | ( | u | γ j + + h j ( x ) , (4.4) and for some C f > , h u , f ( t , x , u , r ) i R m C f ( | u | + . (4.5) Then for any ϕ ∈ W ∞ , there exists a unique u ∈ X ∞ solving equation (4.1). Moreover, sup t ∈ [0 , k u ( t ) k ∞ e C f ( k ϕ k ∞ + C f ) . Proof.
First of all, for any
R ∈ R and u ∈ X k , p , by the boundedness of R in Sobolev space W k , p ,one has ( t , x )
7→ R u ( t , x ) ∈ X k , p . Thus, by (4.2) and the chain rules, one sees that for any u ∈ X ∞ ,( t , x ) a ( t , x , u ( t , x ) , R a u ( t , x )) ∈ L ∞ ([0 , C ∞ b ) , ( t , x ) b ( t , x , u ( t , x ) , R b u ( t , x )) ∈ L ∞ ([0 , C ∞ b ) , and by (4.4), ( t , x ) f ( t , x , u ( t , x ) , R f u ( t , x )) ∈ Y ∞ . Set u ( t , x ) ≡
0. By Theorem 3.2, we can recursively define u n ∈ X ∞ by the following linearequation: ∂ t u n + a ( u n − , R a u n − )( − ∆ ) u n + b ( u n − , R b u n − ) · ∇ u n = f ( u n − , R f u n − ) (4.6) ubject to the initial value u n (0) = ϕ ∈ W ∞ .We first assume that γ =
0. For simplicity of notation, we set a n ( t , x ) : = a ( t , x , u n − ( t , x ) , R a u n − ( t , x )) , b n ( t , x ) : = b ( t , x , u n − ( t , x ) , R b u n − ( t , x )) , f n ( t , x ) : = f ( t , x , u n − ( t , x ) , R f u n − ( t , x )) . By the maximum principle (see Theorem 2.3) and in view of γ =
1, it is easy to see that k u n ( t ) k ∞ k ˜ u n ( t ) k ∞ + Z t k f ( s , · , u n − ( s , · ) , R f u n − ( s , · )) k ∞ d s k ˜ u n (0) k ∞ + Z t ( C f , k u n − ( s ) k ∞ + k h k ∞ )d s k ϕ k ∞ + k h k ∞ + C f , Z t k u n − ( s ) k ∞ d s , which yields by Gronwall’s inequality thatsup t ∈ [0 , k u n ( t ) k ∞ e C f , ( k ϕ k ∞ + k h k ∞ ) = : K . (4.7)By Theorem 2.4 and (2.6), there exist β ∈ (0 ,
1) and constant K > K such thatfor all n ∈ N , sup t ∈ [0 , | u n ( t ) | C β + sup t ∈ [0 , |R a u n ( t ) | C β + sup t ∈ [0 , |R b u n ( t ) | C β K . Thus, by (4.3) we have | a n ( t , x ) − a n ( t , y ) | k∇ x a k L ∞ K | x − y | + K (cid:16) k∇ u a k L ∞ K + k∇ u a k L ∞ K (cid:17) | x − y | β , (4.8) | b n ( t , x ) − b n ( t , y ) | k∇ x b k L ∞ K | x − y | + K (cid:16) k∇ u b k L ∞ K + k∇ u b k L ∞ K (cid:17) | x − y | β , (4.9)where k · k L ∞ K denotes the sup-norm in L ∞ ([0 , × R d × B mK × R k ),For k = , , , · · · , set w ( k ) n ( t , x ) : = ∇ k u n ( t , x ) . By the chain rule, we have ∂ t w ( k ) n + a n ( − ∆ ) w ( k ) n + b n · ∇ w ( k ) n = g ( k ) n , where g (0) n = f n and for k > g ( k ) n : = ∇ k f n − k X j = k !( k − j )! j ! (cid:16) ∇ j a n · ∇ k − j ( − ∆ ) u n + ∇ j b n · ∇ k − j ∇ u n (cid:17) . By (4.8), (4.9) and Lemma 3.1, we have for all p > t ∈ [0 , k w ( k ) n k X , pt C k , p (cid:18) k∇ k ϕ k − p , p + k g ( k ) n k Y , pt (cid:19) , (4.10)where C k , p is independent of n .For k =
0, by (4.10), (4.4) and (4.7), we have k u n ( t ) k pp + Z t k u n ( s ) k p , p d s C k ϕ k p − p , p + C Z t k f n ( s ) k pp d s C k ϕ k p − p , p + Z t (cid:16) C f , ( K γ + k u n − ( s ) k p + k h ( s ) k p (cid:17) p d s C k ϕ k p − p , p + C Z t k u n − ( s ) k pp d s + C Z t k h ( s ) k pp d s . y Gronwall’s inequality, one gets sup n ∈ N sup t ∈ [0 , k u n ( t ) k pp C p , and therefore, for all p >
1, sup n ∈ N k u n k X , p C p . Now for any k = , , · · · , since by the chain rules, g ( k ) n only contains the powers of all derivativesup to k -order of u n , R a u n , R b u n and R f u n , by induction method and using H¨older’s inequality, itis easy to see that for all k ∈ N and p > n ∈ N k u n k X k , p C k , p . (4.11)Below we write w n , m ( t , x ) : = u n ( t , x ) − u m ( t , x ) . Then ∂ t w n , m + a n ( − ∆ ) w n , m + b n · ∇ w n , m = g n , m , where g n , m : = f n − f m + ( a m − a n )( − ∆ ) u m + ( b m − b n ) · ∇ u m . By Lemma 3.1 again, we have for all p > t ∈ [0 , k w n , m k X , pt C k g n , m k Y , pt . Here and below, C > n , m . Using (4.11) and (4.3), we have k g n , m k Y , pt C (cid:16) k f n − f m k Y , pt + k a n − a m k Y , pt + k b n − b m k Y , pt (cid:17) C (cid:16) k∇ u f k L ∞ K + k∇ r f k L ∞ K + k∇ u a k L ∞ K + k∇ r a k L ∞ K + k∇ u b k L ∞ K + k∇ r b k L ∞ K (cid:17) k w n − , m − k Y , pt . Hence, sup s ∈ [0 , t ] k w n , m ( s ) k pp C Z t k w n − , m − ( s ) k pp d s . Taking sup-limits and by Fatou’s lemma, we obtainlim n , m →∞ sup s ∈ [0 , t ] k w n , m ( s ) k pp C Z t lim n , m →∞ sup s ∈ [0 , r ] k w n , m ( s ) k pp d r . So, lim n , m →∞ sup s ∈ [0 , k w n , m ( s ) k pp = , which together with (4.11) and the interpolation inequality yields that for all k ∈ N and p > n , m →∞ sup s ∈ [0 , k w n , m ( s ) k pk , p = . Thus, there exists a u ∈ X ∞ such that for all k ∈ N and p > n , m →∞ sup s ∈ [0 , k u n ( s ) − u ( s ) k pk , p = . Taking limits for (4.6), one sees that u solves equation (4.1).Now we want to drop γ = R >
0, let χ R ∈ C ∞ ( R d ) be a nonnegativecuto ff function with χ R ( u ) = | u | R and χ R ( u ) = | u | > R +
1. Set f R ( t , x , u , r ) : = f ( t , x , u , r ) χ R ( u ) et u R ∈ X ∞ solve ∂ t u R + a ( u R , R a u R )( − ∆ ) u R + b ( u R , R b u R ) · ∇ u R = f R ( u R , R f u R ) . Noticing that by (2.9), 2 h ( − ∆ ) u R , u R i R m = ( − ∆ ) | u R | + E ( u R , u R ) , we have 2 ∂ t | u R | + a ( u R , R a u R )( − ∆ ) | u R | + b ( u R , R b u R ) · ∇| u R | = h u R , f R ( u R , R f u R ) i R m − a ( u R , R a u R ) E ( u R , u R ) ( . ) C f ( | u R | + . Thus, by the maximal principle, we have k u R ( t ) k ∞ k ϕ k ∞ + C f Z t ( k u R ( s ) k ∞ + s , which implies that for all R > t ∈ [0 , k u R ( t ) k ∞ e C f ( k ϕ k ∞ + C f ) . The proof is finished by taking R : = [ e C f ( k ϕ k ∞ + C f )] / . (cid:3)
5. F ully nonlinear and nonlocal equation : P roof of T heorem Lemma 5.1.
Let a ∈ L ∞ ([0 , C b ( R d )) be bounded below by a > and b ∈ L ∞ ([0 , C b ( R d )) .Let u : [0 , × R d → R d belong to X , p for some p > and satisfy ∂ t u = a ( − ∆ ) − (cid:3) u + b · ( ∇ u − ( ∇ u ) t ) , (5.1) where (cid:3) : = div ∇ − ∇ div . Then we have k u k X , p + k U k X , p C (cid:16) k u (0) k , p + k U (0) k , p (cid:17) , where U : = ∇ u − ( ∇ u ) t .Proof. By equation (5.1), one sees that ∂ t u = − a ( − ∆ ) u − a ( − ∆ ) − ∇ div u + b · U , and ∂ t U = − a ( − ∆ ) U + b · ∇ U + ( ∇ b ) · U − [( ∇ b ) · U ] t + A , where A : = ( ∇ a ) t · ( − ∆ ) − (cid:3) u − (( − ∆ ) − (cid:3) u ) t · ∇ a . By Lemma 3.1, there exists a constant C > t ∈ [0 , k u k X , pt C k u (0) k − p , p + C k a ( − ∆ ) − ∇ div u k Y , pt + C k b · U k Y , pt C k u (0) k − p , p + C k a k ∞ k div u k Y , pt + C k b k ∞ k U k Y , pt , and k U k X , pt C k U (0) k − p , p + C k ( ∇ b ) · U + U · ( ∇ b ) t + A k Y , pt C k U (0) k − p , p + C ( k∇ a k ∞ + k∇ b k ∞ ) k∇ u k Y , pt . In particular, for all t ∈ [0 , k u ( t ) k pp + Z t k u ( s ) k p , p d s C k u (0) k p , p + C Z t k div u ( s ) k pp d s + C Z t k U ( s ) k pp d s C k u (0) k p , p + Ct sup s ∈ [0 , t ] k div u ( s ) k pp + sup s ∈ [0 , t ] k U ( s ) k pp ! , (5.2)and k U ( t ) k pp + Z t k U ( s ) k p , p d s C k U (0) k p , p + C Z t k u ( s ) k p , p d s . (5.3)On the other hand, noticing that div (cid:3) u = , we have ∂ t div u = ∇ a · ( − ∆ ) − (cid:3) u + div( b · U ) . Hence, k div u ( t ) k p k div u (0) k p + k∇ a k ∞ Z t k ( − ∆ ) − (cid:3) u ( s ) k p d s + k b k ∞ Z t k∇ U ( s ) k p d s + k∇ b k ∞ Z t k U ( s ) k p d s ( . ) C (cid:16) k div u (0) k p + k U (0) k , p (cid:17) + C Z t k u ( s ) k p , p d s ! / p . (5.4)Now substituting (5.3) and (5.4) into (5.2), we obtain that for all t ∈ [0 , k u ( t ) k pp + Z t k u ( s ) k p , p d s C (cid:16) k u (0) k p , p + k U (0) k p , p (cid:17) + C t Z t k u ( s ) k p , p d s , where C , C are independent of k u (0) k , p and k U (0) k , p . Choosing t : = / (2 C ), we arrive atsup t ∈ [0 , t ] k u ( t ) k pp + Z t k u ( s ) k p , p d s C (cid:16) k u (0) k p , p + k U (0) k p , p (cid:17) . So, for some C > k u k p X , pt + k U k p X , pt C (cid:16) k u (0) k p , p + k U (0) k p , p (cid:17) . In particular, Z t t / (cid:16) k u ( s ) k p , p + k U ( s ) k p , p (cid:17) d s C (cid:16) k u (0) k p , p + k U (0) k p , p (cid:17) . Thus, there is at least one point s ∈ [2 t / , t ] such that k u ( s ) k p , p + k U ( s ) k p , p C t (cid:16) k u (0) k p , p + k U (0) k p , p (cid:17) . Now starting from s , as above, one can prove that for the same t , k u ( · + s ) k p X , pt + k U ( · + s ) k p X , pt C (cid:16) k u ( s ) k p , p + k U ( s ) k p , p (cid:17) C t (cid:16) k u (0) k p , p + k U (0) k p , p (cid:17) . Repeating the above proof, we obtain the desired estimate. (cid:3)
We are now in a position to give
Proof of Theorem 1.1 : We divide the proof into three steps. (Step 1) . In this step we consider the following fully non-linear and nonlocal parabolicequation: ∂ t u = F ( t , x , ∇ u , − ( − ∆ ) u ) , u (0) = ϕ. As introduced in the introduction, let R w = ( − ∆ ) − div w . (5.5) or any ϕ ∈ U ∞ = ∩ k , p U k , p , where U k , p is defined by (2.4), by Theorem 4.1, there exists a unique w ∈ X ∞ solving the following parabolic system: ∂ t w = − ( ∂ q F )( w , R w )( − ∆ ) w + ( ∇ w F )( w , R w ) ∇ w + ∇ x F ( w , R w )subject to w (0) = ∇ ϕ . Define u ( t , x ) : = ϕ ( x ) + Z t F ( s , x , w ( s , x ) , R w ( s , x ))d s and h ( t , x ) : = ∇ u ( t , x ) − w ( t , x ) . Then we have ∂ t h = ( ∂ q F )( w , R w )( ∇R w + ( − ∆ ) w ) + ( ∇ w F )( w , R w )(( ∇ w ) t − ∇ w ) = ( ∂ q F )( w , R w )( − ∆ ) − (cid:3) h + ( ∇ w F )( w , R w )( ∇ h − ( ∇ h ) t )subject to h (0) =
0, where (cid:3) : = div ∇ − ∇ div. By Lemma 5.1, we have h = ⇒ w = ∇ u . Thus, by (5.5), ∂ t u ( t , x ) = F ( t , x , ∇ u ( t , x ) , R∇ u ( t , x )) = F ( t , x , ∇ u ( t , x ) , − ( − ∆ ) u ( t , x )) . By the maximum principle (see Theorem 2.3), we have k u ( t ) k ∞ k ϕ k ∞ + Z t k F ( s , · , , k ∞ d s . (5.6)In particular, u ∈ C ([0 , U ∞ ). (Step 2) . Now we consider the general case. Set u ( t , x ) =
0. Let u n ∈ C ([0 , U ∞ ) bedefined recursively by the following equation: ∂ t u n = F ( t , x , u n − , ∇ u n , − ( − ∆ ) u n ) , u n (0) = ϕ. (5.7)By (5.6) and (1.6), we have k u n ( t ) k ∞ k ϕ k ∞ + Z t k F ( s , · , u n − ( s , · ) , , k ∞ d s k ϕ k ∞ + κ Z t ( k u n − ( s ) k ∞ + s . By Gronwall’s inequality, we get k u n ( t ) k ∞ e κ ( k ϕ k ∞ + κ ) = : K . (5.8)On the other hand, by taking gradients with respect to x for equation (5.7), we have ∂ t ∇ u n = − ∂ q F ( t , x , u n − , ∇ u n , − ( − ∆ ) u n )( − ∆ ) ∇ u n + ∇ w F ( t , x , u n − , ∇ u n , − ( − ∆ ) u n ) ∇ u n + ∂ u F ( t , x , u n − , ∇ u n , − ( − ∆ ) u n ) ∇ u n − + ∇ x F ( t , x , u n − , ∇ u n , − ( − ∆ ) u n ) . By the maximum principle again and (1.9), (1.10) with γ K , =
0, we have k∇ u n ( t ) k ∞ k∇ ϕ k ∞ + Z t k ∂ u F ( s , x , u n − , ∇ u n , − ( − ∆ ) u n ) ∇ u n − k ∞ d s + Z t k∇ x F ( s , x , u n − , ∇ u n , − ( − ∆ ) u n ) k ∞ d s k∇ ϕ k ∞ + C Z t (cid:16) k∇ u n − ( s ) k ∞ + k∇ u n ( s ) k ∞ + (cid:17) d s , where C is independent of n . By Gronwall’s inequality, we getsup n sup t ∈ [0 , k∇ u n ( t ) k ∞ < + ∞ . (5.9)Moreover, by (1.8), (1.9), (1.10), (5.8), Theorem 2.4 and Lemma 2.1, as in the proof of Theorem4.1, we have for all p > k∇ u n k X , pt C (cid:18) k∇ ϕ k − p , p + k∇ u n − k Y , pt + k∇ u n k Y , pt + k h k p (cid:19) , which implies by Gronwall’s inequality thatsup n k∇ u n k X , p < + ∞ , (5.10)and furthermore, for all k ∈ N and p > n k∇ u n k X k , p < + ∞ . (5.11)This together with (5.8) gives sup n sup t ∈ [0 , k u n ( t ) k U k , p < + ∞ . (5.12) (Step 3) . Next we want to show that u n converges to some u in C ([0 , U k , p ). For n , m ∈ N ,set v n , m ( t , x ) : = u n ( t , x ) − u m ( t , x ) . Then ∂ t v n , m = − a n , m ( − ∆ ) v n , m + b n , m · ∇ v n , m + f n , m v n − , m − , where a n , m : = Z ( ∂ q F )( u n − , ∇ u n , − ( − ∆ ) ( sv n , m + u m ))d s , b n , m : = Z ( ∇ w F )( u n − , ∇ ( sv n , m + u m ) , − ( − ∆ ) u m )d s , f n , m : = Z ( ∂ u F )( sv n − , m − + u m − , ∇ u m , − ( − ∆ ) u m )d s . By the maximum principle, we have k v n , m ( t ) k ∞ C Z t k v n − , m − ( s ) k ∞ d s , and by Gronwall’s inequality, lim n , m →∞ sup t ∈ [0 , k v n , m ( t ) k ∞ = . (5.13)On the other hand, by Lemma 2.1 and (5.12), we may derive that for all t ∈ [0 , k v n , m k X , pt C k v n − , m − k Y , pt , and so, lim n , m →∞ k v n , m k X , p = . This together with (5.11), the interpolation inequality and (5.13) yields that for all k ∈ N and p >
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