Global solvability of some double-diffusive convection systems . - PowerPoint PPT Presentation

. Global solvability of some double-diffusive convection systems . Mitsuharu ˆ O TANI Waseda University, Tokyo, JAPAN DIMO2013 September 10, 2013 Mitsuharu ˆ O TANI ( Waseda University, Tokyo, JAPAN) Double-Diffusive Convection DIMO2013-Diffuse Interface Models 1 / 44

Introduction (BF) . Double-difusive convection flow based upon Brinkman-Forchheimer equations .  u t = ν ∆ u − u · ∇ u ⧹ − a u − ∇ p + g T + h C + f 1 in Ω × { t > 0 } , ⧹      T t + u · ∇ T = ∆ T + f 2 in Ω × { t > 0 } ,       C t + u · ∇ C = ∆ C + ρ ∆ T + f 3 in Ω × { t > 0 } ,      (BF) D ( N )  ∇ · u = 0 in  Ω × { t > 0 } , ( π )    u = 0 ; T = 0 ( ∂ T ∂ n = 0 ); C = 0 ( ∂ T  ∂ n = 0 ) on ∂ Ω × { t > 0 }       u | t = 0 = u 0 ( x ) ; T | t = 0 = T 0 ( x ) ; C | t = 0 = C 0 ( x ) ,        ( u (0) = u ( S ) ; T (0) = T ( S ) ; C (0) = C ( S ) , )  . u t = ∂ u ∂ t , T t = ∂ T ∂ t , C t = ∂ C u ( x , t ) : solenoidal velocity of the fluid, ∂ t , T ( x , t ) : temperature, C ( x , t ) : concentration of solute (salt for oceanography), p ( x , t ) : pressure, g , h , ρ, a : constant vector term derived from gravity, Soret coefficient, and Darcy coefficient Ω ⊂ R N : bounded domain, f 1 , f 2 , f 3 : external forces Mitsuharu ˆ O TANI ( Waseda University, Tokyo, JAPAN) Double-Diffusive Convection DIMO2013-Diffuse Interface Models 2 / 44

Introduction Navier-Stokes Equations .  u t = ν ∆ u − u · ∇ u − ∇ p + f ( t ) in Ω × { t > 0 } ,       ∇ · u = 0 in Ω × { t > 0 } ,   (NS) ( π )   u | ∂ Ω = 0       u | t = 0 = u 0 ( x ) , ( u (0) = u ( S ))   . u t = ∂ u u ( x , t ) : solenoidal velocity of the fluid, ∂ t , p ( x , t ) : pressure. . Known Results . (NS) N = 2 : ∃ unique global solution (NS) N = 3 : ∃ unique local solution, ∃ unique global small solution (NS) π N = 2 : ∃ S − periodic solution for any f ∈ L 2 (0 , S ; L 2 ( Ω )) (NS) π N = 3 : ∃ S − periodic solution for small f ∈ L 2 (0 , S ; L 2 ( Ω )) . Mitsuharu ˆ O TANI ( Waseda University, Tokyo, JAPAN) Double-Diffusive Convection DIMO2013-Diffuse Interface Models 3 / 44

Known Results Dirichlet Boundary Condition Known Results: Dirichlet BC . Theorem 1 ( Terasawa- ˆ O (2010)) . For all N ≤ 3 and for any u 0 ∈ H 1 σ ( Ω ) , T 0 , C 0 ∈ H 1 0 ( Ω ) , loc ([0 , ∞ ); L 2 ( Ω )) , (BF) D has a unique (global) f 1 ∈ L 2 loc ([0 , ∞ ); L 2 ( Ω )) , f 2 , f 3 ∈ L 2 solution U = ( u , T , C ) t satisfying  u t , A u ∈ L 2 (0 , S ; L 2 σ ( Ω )) , where A : Stokes Operator       T t , C t , ∆ T , ∆ C ∈ L 2 (0 , S ; L 2 ( Ω )) ,       u ∈ C ([0 , S ]; H 1 T , C ∈ C ([0 , S ]; H 1 σ ( Ω )) , 0 ( Ω )) ∀ S ∈ (0 , ∞ ) .   . . Theorem 2 ( Uchida- ˆ O (2013)) . For all N ≤ 3 and for any f 1 ∈ L 2 (0 , S ; L 2 ( Ω )) , f 2 , f 3 ∈ L 2 (0 , S ; L 2 ( Ω )) , (BF) D π has a S -periodic solution U = ( u , T , C ) t satisfying  u t , A u ∈ L 2 (0 , S ; L 2 σ ( Ω )) , where A : Stokes Operator       T t , C t , ∆ T , ∆ C ∈ L 2 (0 , S ; L 2 ( Ω )) ,       u ∈ C ([0 , S ]; H 1 T , C ∈ C ([0 , S ]; H 1 σ ( Ω )) , 0 ( Ω )) .   . Mitsuharu ˆ O TANI ( Waseda University, Tokyo, JAPAN) Double-Diffusive Convection DIMO2013-Diffuse Interface Models 4 / 44

Neumann Boundary Condition Neumann BC . Double-difusive convection flow with Neumann BC .  u t = ν ∆ u − a u − ∇ p + g T + h C + f 1 in Ω × { t > 0 } ,       T t + u · ∇ T = ∆ T + f 2 in Ω × { t > 0 } ,         C t + u · ∇ C = ∆ C + ρ ∆ T + f 3 in Ω × { t > 0 } ,      (BF) N   ∇ · u = 0 in Ω × { t > 0 } , ( π )      u = 0 ; ∂ T ∂ n = 0; ∂ T  ∂ n = 0 on ∂ Ω × { t > 0 }        u | t = 0 = u 0 ( x ) ; T | t = 0 = T 0 ( x ) ; C | t = 0 = C 0 ( x ) ,        ( u (0) = u ( S ) ; T (0) = T ( S ) ; C (0) = C ( S ) , )  . Mitsuharu ˆ O TANI ( Waseda University, Tokyo, JAPAN) Double-Diffusive Convection DIMO2013-Diffuse Interface Models 5 / 44

Neumann Boundary Condition Preliminaries Function Spaces Ω : bounded domain in R N , Q = Ω × (0 , S ) , σ ( Ω ) = { u = ( u 1 , u 2 , · · · , u N ) t ; u j ∈ C ∞ 0 ( Ω ) ∀ j = 1 , 2 , · · · , N , ∇ · u = 0 } , C ∞ H 1 ( Ω ) = ( H 1 ( Ω )) N = ( W 1 , 2 ( Ω )) N , L 2 ( Ω ) = ( L 2 ( Ω )) N , L 2 σ ( Ω ) = The closure of C ∞ σ ( Ω ) under the L 2 ( Ω ) -norm, H 1 σ ( Ω ) = The closure of C ∞ σ ( Ω ) under the H 1 ( Ω ) -norm, H 0 = L 2 ( Ω ) × L 2 ( Ω ) × L 2 ( Ω ) , H = L 2 σ ( Ω ) × L 2 ( Ω ) × L 2 ( Ω ) , C π ([0 , S ]; H ) = { U ∈ C ([0 , S ]; H ); U (0) = U ( S ) } , P Ω = The orthogonal projection from L 2 ( Ω ) onto L 2 σ ( Ω ) , A = −P Ω ∆ : The Stokes operator with domain D ( A ) = H 2 ( Ω ) ∩ H 1 σ ( Ω ) , D ( A N ) = { u ∈ H 2 ( Ω ); ∂ u with domain on ∂ Ω } , A N = − ∆ ∂ n = 0 A α , A α D and A α N denote the fractional powers of A , A D and A N of order α. Mitsuharu ˆ O TANI ( Waseda University, Tokyo, JAPAN) Double-Diffusive Convection DIMO2013-Diffuse Interface Models 6 / 44

Neumann Boundary Condition Main Results Main Results: Initial Boundary Value Problem . Theorem 3 (Uchida- ˆ O (2013)) . Let N ≤ 3 and ( f 1 , f 2 , f 3 ) t ∈ L 2 (0 , S ; H 0 ) . Then for each initial data U 0 = ( u 0 , T 0 , C 0 ) t ∈ D ( A α ) × D ( A α N ) with α ∈ [1 / 4 , 1 / 2] , (BF) N admits a N ) × D ( A α unique solution U = ( u , T , C ) t ∈ C ([0 , S ]; H ) satisfying U (0) = U 0 and t 1 / 2 − α ∂ t u , t 1 / 2 − α A u ∈ L 2 (0 , S ; L 2  σ ( Ω )) ,       t 1 / 2 − α ||∇ u || L 2 ( Ω ) ∈ L p  ∗ (0 , S ) for all p ∈ [2 , ∞ ] ,     (#) α   t 1 / 2 − α ∂ t T , t 1 / 2 − α ∂ t C , t 1 / 2 − α ∆ T , t 1 / 2 − α ∆ C ∈ L 2 (0 , S ; L 2 ( Ω )) ,         t 1 / 2 − α ||∇ T || L 2 ( Ω ) , t 1 / 2 − α ||∇ C || L 2 ( Ω ) ∈ L p   ∗ (0 , S ) for all p ∈ [2 , ∞ ] ,  ∗ (0 , S ) = (∫ S | f ( t ) | p t − 1 dt ) 1 / p for 1 ⩽ p < ∞ and where L p ∗ = L p ( dt / t ) , i.e., || f || L p 0 L ∞ ∗ = L ∞ . . Mitsuharu ˆ O TANI ( Waseda University, Tokyo, JAPAN) Double-Diffusive Convection DIMO2013-Diffuse Interface Models 7 / 44

Neumann Boundary Condition Main Results Main Results: Periodic Problem . Theorem 4 (Uchida- ˆ O (2013)) . Let N ≤ 3 and ( f 1 , f 2 , f 3 ) t ∈ L 2 (0 , S ; H 0 ) such that ∫ { } f 2 , f 3 ∈ f ; f ( x , t ) dxdt = 0 . (1) Q π admits a solution U = ( u , T , C ) t ∈ C π ([0 , S ]; H ) satisfying Then (BF) N ∂ t u , A u ∈ L 2 (0 , S ; L 2  σ ( Ω )) ,        u ∈ C ([0 , S ]; H 1 σ ( Ω )) ,     (#) 1 / 2   ∂ t T , ∂ t C , ∆ T , ∆ C ∈ L 2 (0 , S ; L 2 ( Ω )) ,          T , C ∈ C ([0 , S ]; H 1 ( Ω )) .   . . Remark 1 . Condition (1) is the necessary condition for the existence of the periodic solution of (BF) N . . Mitsuharu ˆ O TANI ( Waseda University, Tokyo, JAPAN) Double-Diffusive Convection DIMO2013-Diffuse Interface Models 8 / 44

Neumann Boundary Condition Abstract Formulation Reduction to an Abstract Problem Let φ be a proper lower semi-continuous convex function from H into ( −∞ , + ∞ ] . Define the effective domain of φ by D ( φ ) = { U ∈ H ; φ ( U ) < + ∞} and the subdifferential of φ by ∂φ ( U ) = { f ∈ H ; φ ( V ) − φ ( U ) ⩾ ( f , V − U ) H for all V ∈ H } with domain D ( ∂φ ) = { U ∈ H ; ∂φ ( U ) � ∅ } . Then A = ∂φ becomes a maximal monotone operator. It is well known that J λ U = ( I + λ A ) − 1 U → U , as λ → + 0 for all U ∈ D ( A ) . Then for α ∈ (0 , 1) , p ∈ [1 , ∞ ] , by measuring how fast J λ U converges to U , we can define a nonlinear interpolation class B α, p ( A ) associated with A by B α, p ( A ) = { U ∈ D ( A ); t − α | U − J t U | H ∈ L p ∗ (0 , 1) } . We often use the notation � � � t − α | U − J t U | H | U | B α, p ( A ) = ∗ (0 , 1) . � � � L p If A is non-negative self-adjoint operator, then D ( A α ) = B α, 2 ( A ) . In the later arguments, it will be shown that the leading terms ( A , A N , A N ) t can be given as the subdifferential of a suitable lower semi-continuous convex function on H . Mitsuharu ˆ O TANI ( Waseda University, Tokyo, JAPAN) Double-Diffusive Convection DIMO2013-Diffuse Interface Models 9 / 44