Global Existence of Smooth Solutions to a Cross-Diffusion System - PowerPoint PPT Presentation

Global Existence of Smooth Solutions to a Cross-Diffusion System Tuoc V. Phan University of Tennessee - Knoxville, TN TexAMP 2013 at Rice University Oct. 25 - 27, 2013 Joint work with Luan T. Hoang (Texas Tech U.) and Truyen V. Nguyen (U. of Akron) SKT Cross-Diffusion, W 1 , p -estimate, Global solution T. V. Phan (U. of Tennessee - Knoxville) - TexAMP 2013

SKT cross-diffusion system Let Ω ⊂ R n be open, smooth, bounded and n ≥ 2. Consider the Shigesada-Kawasaki-Teramoto system of equations  u t = ∆[( d 1 + a 11 u + a 12 v ) u ] + u ( a 1 − b 1 u − c 1 v ) , Ω × ( 0 , ∞ ) ,     v t = ∆[( d 2 + a 21 u + a 22 v ) v ] + v ( a 2 − b 2 u − c 2 v ) , Ω × ( 0 , ∞ ) ,   with homogenous Newman boundary conditions and u ( · , 0 ) = u 0 ( · ) ≥ 0 , v ( · , 0 ) = v 0 ( · ) ≥ 0 in Ω . This system models the segregation phenomena of two competing species. u and v denote the population densities of two species. d k , a k , b k , c k > 0 and a ij ≥ 0 are constants; a 11 , a 22 are self-diffusion coefficients and a 12 , a 21 are cross-diffusion coefficients. SKT Cross-Diffusion, W 1 , p -estimate, Global solution T. V. Phan (U. of Tennessee - Knoxville) - TexAMP 2013

Divergence form The PDE of the SKT system can be written in the divergence form: U t = ∇ · [ J ( U ) ∇ U ] + F ( U ) , where � � � � u d 1 + 2 a 11 u + a 12 v a 12 v U = , J ( U ) = , d 2 + a 21 u + 2 a 22 v v a 21 u and � � u ( a 1 − b 1 u − c 1 v ) F ( U ) = . v ( a 2 − b 2 u − c 2 v ) SKT Cross-Diffusion, W 1 , p -estimate, Global solution T. V. Phan (U. of Tennessee - Knoxville) - TexAMP 2013

Local well-posedness: H. Amann Theorem Theorem (H. Amann, 1990) Let p 0 > n and U 0 ∈ W 1 , p 0 (Ω) 2 with non-negative entry. Then, there exists maximal existence time t max > 0 such that the SKT system  U t = ∇ · [ J ( U ) ∇ U ] + F ( U ) , Ω × ( 0 , ∞ ) ,    ∂ U  = 0 , ∂ Ω × ( 0 , ∞ ) ,  ∂� ν    U ( · , 0 ) = U 0 , Ω ,   has unique, local non-negative solution U = ( u , v ) T with U ∈ C ([ 0 , t max ); W 1 , p 0 (Ω) 2 ) ∩ C ∞ (Ω × ( 0 , t max )) 2 . Moreover, if t max < ∞ then � � lim � U ( · , t ) � W 1 , p 0 (Ω) × W 1 , p 0 (Ω) = ∞ . � � t → t − max SKT Cross-Diffusion, W 1 , p -estimate, Global solution T. V. Phan (U. of Tennessee - Knoxville) - TexAMP 2013

Global or finite time blow-up solution? The solution for the STK system when J is a FULL 2 × 2 matrix, i.e. � � d 1 + 2 a 11 u + a 12 v a 12 v J ( U ) = , d 2 + a 21 u + 2 a 22 v a 21 u exists globally in time or has finite time blow up? Vastly Unknown . We restrict our study on the case when a 21 = 0, that is � � d 1 + 2 a 11 u + a 12 v a 12 v J ( U ) = . 0 d 2 + 2 a 22 v Let us call the SKT system with this J : Triangular SKT System. SKT Cross-Diffusion, W 1 , p -estimate, Global solution T. V. Phan (U. of Tennessee - Knoxville) - TexAMP 2013

Triangular SKT: Known results The Triangular SKT System, i.e.  = ∇ · [ J ( U ) ∇ U ] + F ( U ) , Ω × ( 0 , ∞ ) , U t    ∂ U  = 0 , ∂ Ω × ( 0 , ∞ ) ,  ∂� ν    U ( · , 0 ) = U 0 , Ω ,   with � � d 1 + 2 a 11 u + a 12 v a 12 v J ( U ) = , 0 d 2 + 2 a 22 v has global solution when n ≤ 9. Y. Lou, W.-M. Ni and J. Wu (1998): n = 2. D. Le, L. Nguyen, T. Nguyen (2003); Y. Choi, R. Lui, Y. Yamada (2004): n ≤ 5. T. P . (2008): n ≤ 9. Many other results: Restrictive conditions on the coefficients. SKT Cross-Diffusion, W 1 , p -estimate, Global solution T. V. Phan (U. of Tennessee - Knoxville) - TexAMP 2013

Today main result Theorem (L. Hoang, T. Nguyen and T. P . – 2013) Let Ω ⊂ R n be open, bounded for any n ≥ 2 , and let U 0 ∈ [ W 1 , p 0 (Ω)] 2 with p 0 > n. Then, the solution U = ( u , v ) T of the Triangular SKT system  U t = ∇ · [ J ( U ) ∇ U ] + F ( U ) , Ω × ( 0 , ∞ ) ,     ∂ U = 0 , ∂ Ω × ( 0 , ∞ ) ,  ∂� ν    U ( · , 0 ) = U 0 , Ω ,   where � � d 1 + 2 a 11 u + a 12 v a 12 v J ( U ) = d 2 + 2 a 22 v 0 exists uniquely, globally in time and � 2 � 2 � � C ([ 0 , ∞ ); W 1 , p 0 (Ω)) C ∞ (Ω × ( 0 , ∞ )) . U ∈ ∩ SKT Cross-Diffusion, W 1 , p -estimate, Global solution T. V. Phan (U. of Tennessee - Knoxville) - TexAMP 2013

Ideas of the proof Let T > 0 be the maximal time existence and assume T < ∞ , we prove by contradiction that � � � � � � � u ( · , t ) � W 1 , p 0 (Ω) + � v ( · , t ) < ∞ . lim � � � � � W 1 , p 0 (Ω) t → T − Sufficient to establish the bound (0 < ǫ ≪ 1) �∇ v � L p (Ω × ( ǫ, T )) + � u � L p (Ω × ( ǫ, T )) ≤ C ( T ) , p > n + 2 . Important known estimates: (i) Maximum Principle (Lou-Ni-Wu, 2003): The PDE of v is v t = ∇ · [( d 2 + 2 a 22 v ) ∇ v ] + v ( a 2 − b 2 u − c 2 v ) . � � max Ω v 0 , a 2 . However, M.P Therefore, 0 ≤ v ≤ max . is not c 2 available for u , b/c u t = ∇ · [( d 1 + 2 a 11 u + a 12 v ) ∇ u + a 12 u ∇ v ] + u ( a 1 − b 1 u − c 1 v ) . (ii) T. P . (2008): �∇ v � L 4 (Ω × ( 0 , T )) ≤ C ( T ) . SKT Cross-Diffusion, W 1 , p -estimate, Global solution T. V. Phan (U. of Tennessee - Knoxville) - TexAMP 2013

Key iteration lemma The PDE of u : u t = ∇ · [( d 1 + 2 a 11 u + a 12 v ) ∇ u + a 12 u ∇ v ] + u ( a 1 − b 1 u − c 1 v ) . Lemma Let p > 2 and assume that �∇ v � L p (Ω T ) ≤ C ( p , T ) . � � p ( n + 1 ) p , Then for each q ∈ with q � ∞ , we have ( n + 2 − p ) + � u � L q (Ω T ) ≤ C ( p , q , T ) . Main question: If u ∈ L q (Ω T ) , can we derive the estimate �∇ v � L q (Ω T ) ≤ C ( q , p , T ) ? SKT Cross-Diffusion, W 1 , p -estimate, Global solution T. V. Phan (U. of Tennessee - Knoxville) - TexAMP 2013

Regularity problem The PDE of v : v t = ∇ · [( d 2 + 2 a 22 v ) ∇ v ] + v ( a 2 − c 2 v ) − b 2 uv , in Ω × ( 0 , T ) . Goal: To establish � � �∇ v � L p (Ω × ( 0 , T )) ≤ C 1 + � u � L p (Ω × ( 0 , T )) . Difficulties: (i) Main term ( d 2 + 2 a 22 v ) depends on solution. Therefore, its oscillation is not small. (ii) The equation is not invariant under either of the scalings v ( x , t ) → v ( θ x , θ 2 t ) v ( x , t ) → v ( x , t ) or , λ, θ > 0 . λ θ (iii) The equation is not invariant under the change of coordinates. SKT Cross-Diffusion, W 1 , p -estimate, Global solution T. V. Phan (U. of Tennessee - Knoxville) - TexAMP 2013

Equations with double scaling parameters Denote Ω T = Ω × ( 0 , T ) , we study the equation: ∇ · [( 1 + λα w ) A ∇ w ] + θ 2 w ( 1 − λ w ) − λθ cw  = in Ω T , w t    ∂ w   = on ∂ Ω × ( 0 , T ) ,  0  ∂� ν    w ( · , 0 ) = w 0 ( · ) Ω .  in  Here, θ, λ > 0 and α ≥ 0 are constants, c ( x , t ) is a nonnegative measurable function, A = ( a ij ) : Ω T → M n × n is symmetric, measurable and ∃ Λ > 0 such that Λ − 1 | ξ | 2 ≤ ξ T A ( x , t ) ξ ≤ Λ | ξ | 2 for a.e. ( x , t ) ∈ Ω T and for all ξ ∈ R n . SKT Cross-Diffusion, W 1 , p -estimate, Global solution T. V. Phan (U. of Tennessee - Knoxville) - TexAMP 2013

Calder´ on - Zygmund type estimates Theorem (L. Hoang, T. Nguyen and T. P ., 2013) Let p > 2 . Then there exists a number δ = δ ( p , Λ , n , α ) > 0 such that if Ω is a Lipschitz domain with the Lipschitz constant ≤ δ and [ A ] BMO (Ω T ) ≤ δ , then for any weak solution w of ∇ · [( 1 + λα w ) A ∇ w ] + θ 2 w ( 1 − λ w ) − λθ cw  = in Ω T , w t    ∂ w    = on ∂ Ω × ( 0 , T ) , 0  ∂� ν    w ( · , 0 ) = w 0 ( · ) Ω .  in  satisfying 0 ≤ w ≤ λ − 1 in Ω T , we have �� θ � p � ˆ ˆ |∇ w | p dxdt ≤ C | c | p dxdt λ ∨ � w � L 2 (Ω T ) + Ω × [¯ t , T ] Ω T for every ¯ t ∈ ( 0 , T ) . Here C > 0 is a constant depending only on Ω , ¯ t, p, Λ , α and n and independent of θ, λ . SKT Cross-Diffusion, W 1 , p -estimate, Global solution T. V. Phan (U. of Tennessee - Knoxville) - TexAMP 2013

Main steps in the proof (interior estimates) P erturbation technique (Caffarelli–Peral): Comparing the solution of w t = ∇ · [( 1 + λα w ) A ∇ w ] + θ 2 w ( 1 − λ w ) − λθ cw in Q 6 (1) with that of the reference equation h t = ∇ · [( 1 + λα h )¯ A B 4 ( t ) ∇ h ] + θ 2 h ( 1 − λ h ) Q 4 , in (2) where ¯ A B 4 ( t ) is the average of A ( · , t ) over B 4 , that is, 1 ˆ ¯ A B 4 ( t ) := A ( x , t ) dx . | B 4 | B 4 Notice that h is a weak solution of (2) iff ¯ h := λ h is a weak solution of h t = ∇ · [( 1 + α ¯ ¯ h )¯ A B 4 ( t ) ∇ ¯ h ] + θ 2 ¯ h ( 1 − ¯ h ) Q 4 . in SKT Cross-Diffusion, W 1 , p -estimate, Global solution T. V. Phan (U. of Tennessee - Knoxville) - TexAMP 2013

Gradient estimate of solutions for the reference equation Lemma Let ¯ h be a weak solution of h t = ∇ · [( 1 + α ¯ ¯ h )¯ A B 4 ( t ) ∇ ¯ h ] + θ 2 ¯ h ( 1 − ¯ h ) in Q 4 satisfying 0 ≤ ¯ h ≤ 1 in Q 4 . Then L ∞ ( Q 3 ) ≤ C ( n , Λ , α ) 1 ˆ h | 2 dxdt . �∇ ¯ h � 2 |∇ ¯ | Q 4 | Q 4 Key Ideas: De Giorgi - Nash - Moser. SKT Cross-Diffusion, W 1 , p -estimate, Global solution T. V. Phan (U. of Tennessee - Knoxville) - TexAMP 2013