Solvability of Brinkman-Forchheimer equations of flow in double-diffusive convection

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1 Solvability of Brinkman-Forchheimer equations of flow in double-diffusive convection Mitsuharu ÔTANI Waseda University, Tokyo, JAPAN One Forum, Two Cities: Aspect of Nonlinear PDEs 29 August, 211 Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 1 / 46

2 Introduction (BF) Double-difusive convection flow based upon Brinkman-Forchheimer equations u t = ν u u u au p + gt + hc + f 1 in {t > }, T t + u T = T + f 2 in {t > }, C t + u C = C + ρ T + f 3 in {t > }, (BF) (π) u = in {t > }, u = ; T = ; C =, u t= = u (x) ; T t= = T (x) ; C t= = C (x), (u() = u(s ) ; T() = T(S ) ; C() = C(S ), ) u(x, t) : solenoidal velocity of the fluid, T(x, t) : temperature, u t = u t, T t = T t, C t = C t, 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 R N : bounded domain, f 1, f 2, f 3 : external forces Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 2 / 46

3 Introduction Navier-Stokes Equations (NS) (π) u t = ν u u u p + f( t ) in {t > }, u = in {t > }, u = u t= = u (x), (u() = u(s )) u(x, t) : solenoidal velocity of the fluid, p(x, t) : pressure. u t = u t, 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 (, S ; L 2 ()) (NS) π N = 3 : S periodic solution for small f L 2 (, S ; L 2 ()) Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 3 / 46

4 Main Results Theorem 1 Main Results For all N 3 and for any u H 1 σ(), T, C H 1(), f 1 Lloc 2 ([, ); L2 ()), f 2, f 3 Lloc 2 ([, ); L2 ()), (BF) has a unique (global) solution U = (u, T, C) t satisfying u t, Au L 2 (, S ; L 2 σ()), where A : Stokes Operator T t, C t, T, C L 2 (, S ; L 2 ()), u C([, S ]; H 1 σ()), T, C C([, S ]; H 1 ()) S (, ). Theorem 2 For all N 3 and for any f 1 L 2 (, S ; L 2 ()), f 2, f 3 L 2 (, S ; L 2 ()), (BF) π has a S -periodic solution U = (u, T, C) t satisfying u t, Au L 2 (, S ; L 2 σ()), where A : Stokes Operator T t, C t, T, C L 2 (, S ; L 2 ()), u C([, S ]; H 1 σ()), T, C C([, S ]; H 1()). Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 4 / 46

5 Proof of Teorem 1 Preliminaries Proof of Theorem 1 Local Existence Reduce our problem to an abstract Cauchy Problem. Apply an abstract Theorem ( Ô, JDE 1982) to the problem. Existence of Global Solution in time Establish some a priori estimates. Uniquness of the Solution Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 5 / 46

6 Proof of Teorem 1 Preliminaries Function Spaces C σ () = { u = (u 1, u 2,..., u N ); u j C (), j = 1, 2,..., N, u = }, L 2 σ() = the completion of C σ () under the L 2 ()-norm, L 2 () = (L 2 ()) N, H j () = (H j ()) N, ( j = 1, 2) P = the orthogonal projection from L 2 () onto L 2 σ(), H 1 () = the completion of C () under the H1 ()-norm, H 1 σ() = H 1 () L 2 σ(), The Stokes operator A() is defined as follows: A() = P with domain D(A()) = H 2 () H 1 σ(). Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 6 / 46

7 Proof of Teorem 1 Abstract Formulation Reduction to an abstract Cauchy Problem (BF) u t = ν u au p + gt + hc in {t > }, T t + u T = T in {t > }, C t + u C = C + ρ T in {t > }. Operate P to u t = ν u au p + gt + hc, then we have u t = ν P u au + PgT + PhC and take u ν P u au PgT PhC U = T, φ(u) = T, B(U) = u T C C u C ρ T Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 7 / 46

8 Proof of Teorem 1 Abstract Formulation Reduction (CP) du + φ(u(t)) + B(U(t)) = t (, T) dt U() = U = (u, T, C ) t inner product H() = L 2 σ() L 2 () L 2 () (U 1, U 2 ) H = (u 1, u 2 ) L 2 σ + (T 1, T 2 ) L ρ 2 (C 1, C 2 ) L 2 φ(u) = ν 2 u L 2 σ 2 T 2 L ρ 2 C 2 L if U D(φ) 2 + if U H \ D(φ) D(φ) = H 1 σ() H 1 () H1 () Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 8 / 46

9 Proof of Teorem 1 Abstract Result Local Solvability Theorem (M.Ô, 1982(JDE)) Assume (A.1) For each L (, + ), the set {U H; φ(u) + U 2 H L} is compact in H, (A.2) B( ) is φ demiclosed: If U k U in C([, S ]; H) and φ(u k ) φ(u), B(U k ) b weakly in L 2 (, S ; H), then b = B(U) holds. (A.3) monotone increasing function l( ), a( ) L 2 (, S ), k (, 1) s.t. B(U) 2 H k φ(u) 2 H + l(φ(u) + U 2 H ) c(t), t [, S ], U D( φ), Let U D(φ) and f (t) L 2 (, S ; H), then there exists S (, S ) such that du (CP) + φ(u(t)) + B(U(t)) = f (t), U() = U has a local solution U(t) dt on [, S ] satisfying U C([, S ]; H), U t, φ(u) L 2 (, S ; H) Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 9 / 46

10 Proof of Teorem 1 Local Existence Check of (A.1) {U H; φ(u) + U 2 H L} = {u H; ν 2 u 2 L T 2 L ρ 2 C 2 L 2 + u 2 L 2 + T 2 L ρ 2 C 2 L 2 L} From Rellich-Kondrachev s theorem, the level set is compact in H() = L 2 σ() L 2 () L 2 (). Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 1 / 46

11 Proof of Teorem 1 Local Existence Assume Check of (A.2) 1/3 u k u in C([, S ]; L 2 σ()), T k T in C([, S ]; L 2 ()), C k C in C([, S ]; L 2 ()), ν P u k ν P u in L 2 (, S ; L 2 σ()), T k T in L 2 (, S ; L 2 ()), C k C in L 2 (, S ; L 2 ()). Let h 1, h 2, h 3 be weak limit as au k PgT k PhC k h 1 in L 2 (, S ; L 2 σ()), u k T k h 2 in L 2 (, S ; L 2 ()), u k C k ρ T k h 3 in L 2 (, S ; L 2 ()). Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 11 / 46

12 Proof of Teorem 1 Local Existence Check of (A.2) 2/3 then we have to show h 1 = au PgT PhC for a.e. t [, S ], h 2 = u T for a.e. t [, S ], h 3 = u C ρ T for a.e. t [, S ]. Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 12 / 46

13 Proof of Teorem 1 Local Existence Check of (A.2) 3/3 Take ϕ C ( (, S )) u k T k, ϕ = u k T k, ϕ u k T k, ϕ = u k T k, ϕ ut, ϕ = u T, ϕ Let us recall the assumption we imposed: u k T k h 2 in L 2 (, S ; L 2 ()). So we obtain h 2 = u T f or a.e. t [, S ] Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 13 / 46

14 Proof of Teorem 1 Local Existence Check of (A.3) 1/2 B(U) = (au PgT PhC, u T, u C ρ T) t, U = (u, T, C) t B(U) 2 H 3(a2 u 2 + g 2 T 2 L 2 σ L + h 2 C 2 2 L ) + u 2 T ( u 2 C 2 + ρ 2 T 2 ) 2 3ρ 2 C U 2 H + (ε ) φ(u) 2 H + γφ(u)3 since = 1 u 2 T 2 dx C u 2 L 6 T L 6 T L 2 ε T 2 L 2 + C ε u 4 L 2 T 2 L 2, u 2 C 2 dx C u 2 L 6 C L 6 C L 2 ε C 2 L 2 + C ε u 4 L 2 C 2 L 2, hold and by Young s inequality we get C ε u 4 L 2 T 2 L 2 + C ε u 4 L 2 C 2 L 2 C( u 6 L 2 + T 6 L 2 + C 6 L 2 ) γφ(u) 3 Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 14 / 46

15 Proof of Teorem 1 Local Existence Check of (A.3) 2/2 And So we obtain 2 3ρ ( u 2 C 2 + ρ 2 T 2 ) 2 T 2 dx + 2 u 2 C 2 dx ρ 2 Estimates for the Nonlinear Term B(U) 2 H C U 2 H + (ε ) φ(u) 2 H + γφ(u)3 Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 15 / 46

16 Proof of Teorem 1 Global Existence A priori estimate 1 (BF) u t = νp u au + PgT + PhC T t + u T = T in {t > }, C t + u C = C + ρ T in {t > }, First of all, multiplying T t + u T = T by T and integrating over, we have (L) = T t T + T)T = (u 1 d T 2 + u ( 1 2 dt 2 T 2 ) (R) = T T = T 2 1 d 2 dt T 2 L + T 2 2 L = 2 Awhence priori estimate follows 1 S sup T 2 L + 2 t S T 2 L dt T 2 2 L, 2 S >. Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 16 / 46

17 Proof of Teorem 1 Global Existence A priori estimate 2 Multiplying C t + u C = C + ρ T by C and integrating over, we get 1 d C 2 + C 2 = u C C + ρ T C 2 dt = u ( 1 2 C2 ) + ρ T C ρ T L 2 C L C 2 L + ρ2 2 2 T 2 L, 2 S S C(t) 2 L + C 2 2 L dt C 2 2 L + ρ 2 T 2 2 L dt 2 Hence A priori estimate 2 sup C 2 L + 2 t S S C 2 L 2 dt C 2 L 2 + ρ 2 T 2 L 2, S >. Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 17 / 46

18 Proof of Teorem 1 Global Existence A priori estimate 3 Multiplying u t = νp u au + PgT + PhC by u t and integrating over, we have u t 2 L + νau u 2 t + au u t = (PgT + PhC)u t, u t 2 L + ν d 2 2 dt u 2 L + a d 2 2 dt u 2 L γ u 2 t L 2 ( T L 2 + C L 2) 1 2 u t 2 L + γ ( T 2 2 L + C 2 2 L ). 2 Hence u t 2 L 2 + ν d dt u 2 L 2 + a d dt u 2 L 2 γ ( T L 2, C L 2) which implies A priori estimate 3 sup u 2 L + 2 t S S S u t 2 L dt + Au 2 2 L dt γ ( u 2 L 2, T L 2, C L 2), S >. Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 18 / 46

19 Proof of Teorem 1 Global Existence A priori estimate 4 Multiplying T t + u T = T by T and integrating over, we get T 2 L + T T 2 t = T u T, Hence A priori estimate 4 sup T 2 L + 2 t S T 2 L + 1 d 2 2 dt T 2 L T 2 L + u 2 T 2. 2 ( u 2 T 2 dx ε T 2 L + C 2 ε u 4 L T 2 2 L ) 2 S 1 2 T 2 L 2 + γ u 4 L 2 T 2 L 2. T 2 L 2 dt γ ( T L 2, u L 2, T L 2, C L 2), S >. Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 19 / 46

20 Proof of Teorem 1 Global Existence A priori estimate 5 Multiplying C t + u C = C + ρ T by C and integrating over, we get C 2 L + C C 2 t = C u C + ρ C T, C 2 L + 1 d 2 2 dt C 2 L C 2 L + u 2 C 2 + ρ 2 T 2 2 L 2 ( u 2 C 2 dx ε C 2 L + C 2 ε u 4 L C 2 2 L ) 2 A priori estimate 5 sup C 2 L + 2 t S S 3 4 C 2 L 2 + γ u 4 L 2 C 2 L 2 + ρ 2 T 2 L 2. C 2 L 2 dt γ ( C L 2, T L 2, u L 2), S >. Global Existence So, the every local solution can be continued globally in time. Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 2 / 46

21 Proof of Teorem 1 Uniqueness Uniquness(1/4) Let V 1 and V 2 are the two solutions of (BF) with same initial data: and put Then (w, τ, θ) t satisfy V 1 = (u 1, T 1, C 1 ) t, V 2 = (u 2, T 2, C 2 ) t V = V 1 V 2 = (w, τ, θ) t, V() = V =. (W) w,t = ν P w aw + P gτ + P hθ, τ t = τ u 1 T 1 + u 2 T 2, θ t = θ + ρ τ u 1 C 1 + u 2 C 2. Multiplying (W) by w, τ, θ, respectively, and integrating over, we have w t wdx = ν w w aw 2 + τg w + θh w dx, (1) τ t τdx = τ τ (u 1 T 1 u 2 T 2 )τdx, (2) θ t θdx = θ θ (u 1 C 1 u 2 C 2 )θ + ρθ τdx. (3) Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 21 / 46

22 Proof of Teorem 1 Uniqueness Uniquness(2/4) From (1), we obtain 1 d 2 dt w 2 + ν w 2 + a w 2 g τ L 2 σ L 2 σ L 2 σ L 2 w L 2 σ + h θ L 2 w L 2 σ 1 2 ( g + h ) w L 2 σ 2 g τ 2 L h θ 2 L. (4) 2 From (2) and v 2 v L 3 L 2 v L 6, we have 1 d 2 dt τ 2 L + τ 2 2 L = {u 1 τ τ w T 2 τ}dx 2 = u τ2 dx + T 2 wτdx τ L 6 T 2 L 2 w L τ 2 L 2 + γ T 2 2 L 2 w L 2 σ w L τ 2 L 2 + ν 4 w 2 L 2 + γ ν T 2 4 L 2 w 2 L 2 σ. (5) Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 22 / 46

23 Proof of Teorem 1 Uniqueness Uniquness(3/4) By the argument similar to that for (5), from (3) we obtain 1 d 2 dt θ 2 L + θ 2 2 L = u 1 θ θdx + w C 2 θdx ρ τ θdx 2 = u θ2 dx + C 2 wθdx ρ τ θdx C 2 L 2 w L 6 θ 1 2 L 2 θ 1 2 L 2 + ρ τ L 2 θ L 2 ρ2 ν 4 w 2 L 2 + γ ρ 2 ν C2 2 L 2 θ L 2 θ L θ 2 L 2 + ρ 2 τ 2 L 2 ρ2 ν 4 w 2 L 2 + ρ 2 τ 2 L θ 2 L 2 + γ2 ρ 4 ν 2 C2 4 L 2 θ 2 L 2. (6) Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 23 / 46

24 Proof of Teorem 1 Uniqueness Put y(t) = w(t) 2 L 2 σ Uniquness(4/4) and sum up (4), (5) and (6) 1 2ρ 2, then we get Since + τ(t) 2 L ρ 2 θ(t) 2 L 2 1 d 2 dt y(t) γy(t) + γ ν T 2 4 L w 2 2 L + γ2 2 2ρ 6 ν 2 C2 4 L θ 2 2 L 2 by Gronwall s inequality, we have whence follows the uniqueness. γ( T 2 4 L 2 + C 2 4 L 2 + 1)y(t). ξ(t) = γ( T 2 4 L 2 + C 2 4 L 2 + 1) L 1 (, T) V(t) 2 H V H exp( t ξ(s)ds) =, Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 24 / 46

25 Conclusion Conclusion Theorem 1 For all N 3 and for any u H 1 σ(), T, C H 1(), f 1 Lloc 2 ([, ); L2 ()), f 2, f 3 Lloc 2 ([, ); L2 ()), (BF) has a unique (global) solution U = (u, T, C) t satisfying u t, Au L 2 (, S ; L 2 σ()), where A : Stokes Operator T t, C t, T, C L 2 (, S ; L 2 ()), u C([, S ]; H 1 σ()), T, C C([, S ]; H 1 ()) S (, ). Generally speaking, it is difficult to show the existence of global solution of Navier-Stokes equations in 3 dimensional space. The absence of nonlinear convective term of the 1st equation in our problem enables us to prove the global existence in 3D space, even if similar nonlinear convective terms apear in 2nd and 3rd equations. Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 25 / 46

26 Periodic Problem Proof of Theorem 2 Reduce our problem to an Abstract Periodic Problem Introduce Approximation Problems Apply an abstract Theorem ( Ô, JDE 1984) to approximation problems Establish some a priori estimates Convergence of solutions of approximation problems Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 26 / 46

27 Periodic Problem Reduction to an Abstract Problem Reduction (PP) du(t) + φ(u(t)) + B(U(t)) = F(t) t (, S ) dt U() = U(S ) H() = L 2 σ() L 2 () L 2 () (U 1, U 2 ) H = (u 1, u 2 ) L 2 σ + (T 1, T 2 ) L ρ (C 1, C 2 2 ) L 2 ν φ(u) = 2 u L 2 σ 2 T 2 L ρ L 2 + if U H \ D(φ) if U D(φ) D(φ) = H 1 σ() H 1 () H1 () φ(u) = ( ν P u, T, C ) t B(U) = (au PgT PhC, u T, u C ρ T ) t F(t) = ( f 1 (t), f 2 (t), f 3 (t)) t Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 27 / 46

28 Proof of Theorem 2 Abstract Result Abstract Result for Periodic Problem Theorem (M.Ô, 1984(JDE)) Assume (A.1) For each L (, + ), the set {U H; φ(u) + U 2 H L} is compact in H, (A.2) B( ) is φ demiclosed: If U k U in C([, S ]; H) and φ(u k ) φ(u), B(U k ) b weakly in L 2 (, S ; H), then b = B(U) holds. (A.3) monotone increasing function l( ), k (, 1) s.t. B(U) 2 H k φ(u) 2 H + l( U H)(φ(U) + 1) 2, U D( φ), (A.4) α, K > s.t. ( φ(u) B(U), U) H + αφ(u) K, U D( φ) Let f (t) L 2 (, S ; H), then there exists a solution of (PP) satisfying U C π ([, S ]; H), U t, φ(u) L 2 (, S ; H) Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 28 / 46

29 Proof of Theorem 2 Approximation Approximation Problems We can not apply our abstract result directly, since B(U) 2 H C U 2 H + (ε ) φ(u) 2 H + γφ(u)3 We introduce the following Approximation Problems u t = ν P u au + P g[t] ε + P h[c] ε + P f 1, T t + u T = T ε T p 2 T + f 2, (BF) ε C t + u C = C + ρ T ε C p 2 C + f 3, u() = u(s ), T() = T(S ), C() = C(S ), [T] ε = T if T 1/ε, = sign T 1/ε if T > 1/ε. ν φ ε (U) = 2 u L 2 σ 2 T 2 L ρ 2 C 2 L + ε 2 p T p L + ε p 3ρ 2 p C p L if U D(φ p ε ) + if U H \ D(φ ε ) B ε (U) = (au Pg[T] ε Ph[C] ε, u T, u C ρ T ) t Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 29 / 46

30 Proof of Theorem 2 Solvability of Approximation Problems Check of (A3) 1/2 B ε (U) 2 H u T 2 L 2 + u C 2 L 2 u T 2 L = (u i D 2 i T)(u j D j T)dx = u i T D i (u j D j T)dx = u i T u j D i D j T dx + u i T D i (u j )D j T dx C u L 6 T L 12 u L 4 T L 2 + C u L 4 T L 12 u L 6 T L 2 u L 6 T L 12 u L 4 T L 2 ε T 2 L 2 + C ε u 2 L 2 T 2 L 12 u 2 L 4 ε T 2 L 2 + u 4 L 2 + C ε T 4 L 12 u 4 L 4 ε T 2 L u 4 L 2 + C ε u 4 L 2 T 16 L 12 ( u 4 L 4 C u L 2 u 3 L 2 ) Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 3 / 46

31 Proof of Theorem 2 Solvability of Approximation Problems Check of (A3) 2/2 u L 4 T L 12 u L 6 T L 2 ε u 2 L 2 + C ε u 2 L 4 T 2 L 12 T 2 L 2 Let p 12, then we have ε u 2 L 2 + T 4 L 2 + C ε T 4 L 12 u 4 L 4 ε u 2 L 2 + T 4 L 2 + u 4 L 2 + C ε u 4 L 2 T 16 L 12 B ε (U) 2 H C(ε ) φ ε(u) 2 H + C ε U 4 H φ ε(u) 2 ε (, ε ). Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 31 / 46

32 Proof of Theorem 2 Solvability of Approximation Problems Check of (A4) (A4) α, K > s.t. ( φ(u) + B(U), U ) H αφ(u) K, U D( φ) φ ε (U) = ( ν P u, T + ε T p 2 T, C + ε C p 2 C ) t ( φ ε (U), U ) H = ν u 2 L 2 σ 2φ ε (U) + T 2 L ρ 2 C 2 L + ε T p 2 L + ε p 3ρ 2 C p L p B ε (U) = (au Pg[T] ε Ph[C] ε, u T, u C ρ T ) t ( B ε (U), U ) H a u 2 g u L 2 σ L 2 σ [T] ε L 2 h u L 2 σ [C] ε L 2 1 ρ T L 2 C L 2 a 2 u 2 L 2 σ C ε T 2 L ρ 2 C 2 L 2 Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 32 / 46

33 Proof of Theorem 2 A priori estimates A proori estimates 1 T t + u T = T ε T p 2 T + f 2, T dx S dt S S T 2 L dt + ε T p 2 L dt C p ( f 2 L 2 (,S ;L 2 ())) t [, S ] s.t. K T(t ) L 2 T(t ) L 2 C ( f 2 L 2 (,S ;L 2 ()))/S t t dt max t S T(t) L 2 C ( f 2 L 2 (,S ;L 2 ()), S ) Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 33 / 46

34 Proof of Theorem 2 A priori estimates A proori estimates 2 C t + u C = C + ρ T ε C p 2 C + f 3, C dx 1 ρ 2 S dt 1 S C 2 ρ 2 L dt + ε S C p 2 ρ 2 L dt C p ( f 2 L 2 (,S ;L 2 ()), f 3 L 2 (,S ;L 2 ())) t [, S ] s.t. K C(t ) L 2 C(t ) L 2 C /S t t dt max t S C(t) L 2 C ( f 2 L 2 (,S ;L 2 ()), f 3 L 2 (,S ;L 2 ()), S, ρ) Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 34 / 46

35 Proof of Theorem 2 A priori estimates A proori estimates 3 u t = ν P u au + P g[t] ε + P h[c] ε + P f 1, u dx S dt S (a uν 2 L 2 σ + u 2 )dt L 2 σ S ( g [T] ε L 2 + h [C] ε L 2 + f 1 L 2 ) u L 2 σ dt C ( f 1 L 2, f 2 L 2 (,S ;L 2 ()), f 3 L 2 (,S ;L 2 ())) t [, S ] s.t. K u(t ) L 2 σ u(t ) 2 C L 2 /S σ t t dt max t S u(t) L 2 σ C ( f 1 L 2, f 2 L 2 (,S ;L 2 ()), f 3 L 2 (,S ;L 2 ()), S, ρ) Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 35 / 46

36 Proof of Theorem 2 A priori estimates A proori estimates 4 u t = ν P u au + P g[t] ε + P h[c] ε + P f 1, u t dx S dt Eq. S S u t 2 L 2 σ dt C ( f 1 L 2, f 2 L 2, f 3 L 2) P u 2 L 2 σ dt C t [, S ] s.t. u(t ) 2 C L 2 /S σ t t dt max t S u(t) L 2 σ C ( f 1 L 2, f 2 L 2, f 3 L 2, S, ρ) Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 36 / 46

37 Proof of Theorem 2 A priori estimates A proori estimates 5-i T t + u T = T ε T p 2 T + f 2, T dx 1 d 2 dt T(t) 2 L + T(t) 2 2 L + ε(p 1) T p 2 T(t) 2 2 L dx 2 ( u T L 2 + f 1 L 2 ) T(t) L 2 u T 2 L 2 T 2 L 4 u 2 L 4 K T 1/2 L 2 T 3/2 ε T 2 L + C 2 ε T 2 L u 2 u 6 2 L 2 σ L 2 σ t [, S ] s.t. T(t ) L 2 C /S t t dt max t S T(t) L 2 + S u 1/2 L 2 L 2 σ u 3/2 L 2 σ T(t) 2 L 2 dt C. Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 37 / 46

38 Proof of Theorem 2 A priori estimates A proori estimates 5-ii T t + u T = T ε T p 2 T + f 2, T t dx 1 2 d dt T(t) 2 L 2 + T t (t) 2 L 2 + ε p d dt T(t) p L p ( u T L 2 + f 1 L 2 ) T t (t) L 2 u T 2 L ε T 2 2 L + C 2 ε T 2 L u 2 u 6 2 L 2 σ L 2 σ S dt Eq. S T t (t) 2 L 2 dt C S ε 2 T p 2 T(t) 2 L dt C 2 Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 38 / 46

39 Proof of Theorem 2 A priori estimates A proori estimates 6-i C t + u C = C + ρ T ε C p 2 C + f 3, T dx 1 d 2 dt C(t) 2 L + C(t) 2 2 L + ε(p 1) C p 2 C(t) 2 2 L dx 2 ( u C L 2 + f 1 L 2 + ρ T L 2 ) C(t) L 2 u C 2 L 2 C 2 L 4 u 2 L 4 K C 1/2 L 2 C 3/2 ε C 2 L + C 2 ε C 2 L u 2 u 6 2 L 2 σ L 2 σ t [, S ] s.t. C(t ) L 2 C /S t t dt max t S C(t) L 2 + S u 1/2 L 2 L 2 σ u 3/2 L 2 σ C(t) 2 L 2 dt C. Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 39 / 46

40 Proof of Theorem 2 A priori estimates A proori estimates 6-ii C t + u C = C + ρ T ε C p 2 C + f 2, C t dx 1 2 d dt C(t) 2 L 2 + C t (t) 2 L 2 + ε p d dt C(t) p L p ( u C L 2 + f 1 L 2 + ρ T L 2 ) C t (t) L 2 u C 2 L ε C 2 2 L + C 2 ε C 2 L u 2 u 6 2 L 2 σ L 2 σ S dt Eq. S C t (t) 2 L 2 dt C S ε 2 C p 2 C(t) 2 L dt C 2 Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 4 / 46

41 Proof of Theorem 2 Convergence Convergence 1 A priori estimates 1 max U ε (t) H + max φ ε(u ε (t)) + t S t S S ( U ε t (t) 2 H + φ ε(u ε (t)) 2 H )dt C, U ε = (u ε, T ε, C ε ) t max φ ε(u ε (t)) C {U ε (t)} ε (,1) forms a precompact set in H t [, S ] t S S U ε t (t) 2 H dt C {U ε (t)} ε (,1) is equi-continuous in C π ([, S ]; H) Ascoli stheorem U n (t) = U ε n (t) (ε n as n ) s.t. U n U = (u, T, C ) t strongly in C π ([, S ]; H) as n U n t U t = U = (u t, T t, C t ) t weakly in L 2 (, S ; H) as n. Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 41 / 46

42 Proof of Theorem 2 Convergence Convergence 2 A priori estimates 2 ε T ε p L p (,S ;L p ()) + ε T ε p 2 T ε L 2 (,S ;L 2 ()) C, ε T ε p 2 T ε L p ε n T ε n p 2 T ε n g weakly in L 2 (, S ; L 2 ()) p p 1 (,S ;L p 1 ()) = ε 1 p g = ( ε T ε p L p (,S ;L p ())) p 1 p Similarly ε n C ε n p 2 C ε n weakly in L 2 (, S ; L 2 ()) φ εn (U ε n ) φ(u) weakly in L 2 (, S ; H) [T] εn T, [C] εn C strongly in C π ([, S ]; L 2 ()) as ε B εn (U ε n ) B(U) weakly in L 2 (, S ; L 2 ()) U gives a solution of (PP). Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 42 / 46

43 Concluding Remarks Related Results Concluding Remarks(1/4) Our main theorem holds true also for unbounded domains. For bounded domain case, U(t) = (u(t), T(t), C(t)) t zero. decays exponentially to Existence of global attractors and exponential attractors Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 43 / 46

44 Concluding Remarks More general initial data Concluding Remarks(2/4) B α,p = {u D( φ); t α u (I + t φ) 1 u H L p (, 1)}, where f L p (,1) = ( 1 f (t) p 1 t dt)1/p 1 < p <, L (, 1) = L (, 1). Theorem (M.Ô, 1982(JDE)) Assume (A.1), (A.2) and (A.3) α monotone increasing function l( ), l ( ) s.t. B(U) 2 H l( U H){ε φ(u) 2 H + l (1/ε) φ(u) 2(1 α) 1 2α + 1}, ε >, U D( φ). Let U B α,2 and f (t) L 2 (, S ; H), then there exists S (, S ) such that (CP) has a local solution U(t) C([, S ]; H) on [, S ] satisfying t 1/2 α U t, t 1/2 α φ(u) L 2 (, S ; H); t 1/2 α φ(u) 1/2 L q (, S ) q [2, ] Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 44 / 46

45 Concluding Remarks More general initial data Concluding Remarks(3/4) Estimates for the Nonlinear Term B(U) 2 H C U 2 H + (ε ) φ(u) 2 H + γφ(u)3 (U 1, U 2 ) H = (u 1, u 2 ) L 2 σ + (T 1, T 2 ) L ρ (C 1, C 2 2 ) L is replaced by (ε + 2 3ρ2 kρ 2 3 ) is replaced by (ε + 2 3k ) k Then (A.3) α is satisfied with 3 = 2(1 α) 1 2α α = 1/4 U = (u, T, c ) B α,2 u D(A 1/4 ), T, C D(( ) 1/4 ) (u, T, c ) D(A 1/4 ) D(( ) 1/4 ) D(( ) 1/4 ),!sol.u C(, S ; H) s, t. t 1/2 α φ(u) 1/2 L q (, S ) q [2, ] Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 45 / 46

46 Concluding Remarks More general initial data Concluding Remarks(4/4) q = 4, α = 1/4 t (1/2 1/4)4 1 φ(u) 4/2 L 1 (, S ) φ(u) 2 L 1 (, S ) Uniquness q =, α = 1/4 t 1/4 φ(u) L (, S ) U(S ) D(φ) u(s ) H 1 σ(); T(S ), C(S ) H 1 ()!Global solution Mitsuharu ÔTANI (Waseda University, Tokyo, JAPAN) Double-Diffusive Convection One Forum, Two Cities 46 / 46