Solvability of Brinkman-Forchheimer equations of flow in double-diffusive convection
|
|
- Ἠλίας Παπαγεωργίου
- 5 χρόνια πριν
- Προβολές:
Transcript
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
Example Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότερα2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραThe Pohozaev identity for the fractional Laplacian
The Pohozaev identity for the fractional Laplacian Xavier Ros-Oton Departament Matemàtica Aplicada I, Universitat Politècnica de Catalunya (joint work with Joaquim Serra) Xavier Ros-Oton (UPC) The Pohozaev
Διαβάστε περισσότεραGlobal nonlinear stability of steady solutions of the 3-D incompressible Euler equations with helical symmetry and with no swirl
Around Vortices: from Cont. to Quantum Mech. Global nonlinear stability of steady solutions of the 3-D incompressible Euler equations with helical symmetry and with no swirl Maicon José Benvenutti (UNICAMP)
Διαβάστε περισσότεραMath 446 Homework 3 Solutions. (1). (i): Reverse triangle inequality for metrics: Let (X, d) be a metric space and let x, y, z X.
Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequalit for metrics: Let (X, d) be a metric space and let x,, z X. Prove that d(x, z) d(, z) d(x, ). (ii): Reverse triangle inequalit for norms:
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότεραSection 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Διαβάστε περισσότεραPartial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραSOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr T t N n) Pr max X 1,..., X N ) t N n) Pr max
Διαβάστε περισσότεραSCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Διαβάστε περισσότεραNumerical Analysis FMN011
Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =
Διαβάστε περισσότεραCHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD
CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.
Διαβάστε περισσότεραNowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
Διαβάστε περισσότεραSOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr (T t N n) Pr (max (X 1,..., X N ) t N n) Pr (max
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότερα12. Radon-Nikodym Theorem
Tutorial 12: Radon-Nikodym Theorem 1 12. Radon-Nikodym Theorem In the following, (Ω, F) is an arbitrary measurable space. Definition 96 Let μ and ν be two (possibly complex) measures on (Ω, F). We say
Διαβάστε περισσότεραforms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with
Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We
Διαβάστε περισσότεραSrednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
Διαβάστε περισσότεραAppendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee
Appendi to On the stability of a compressible aisymmetric rotating flow in a pipe By Z. Rusak & J. H. Lee Journal of Fluid Mechanics, vol. 5 4, pp. 5 4 This material has not been copy-edited or typeset
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότερα( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
Διαβάστε περισσότεραΑπόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Διαβάστε περισσότεραPULLBACK D-ATTRACTORS FOR THE NON-AUTONOMOUS NEWTON-BOUSSINESQ EQUATION IN TWO-DIMENSIONAL BOUNDED DOMAIN. Xue-Li Song.
DISCRETE AND CONTINUOUS doi:10.3934/dcds.2012.32.991 DYNAMICAL SYSTEMS Volume 32, Number 3, March 2012 pp. 991 1009 PULLBACK D-ATTRACTORS FOR THE NON-AUTONOMOUS NEWTON-BOUSSINESQ EQUATION IN TWO-DIMENSIONAL
Διαβάστε περισσότερα5. Choice under Uncertainty
5. Choice under Uncertainty Daisuke Oyama Microeconomics I May 23, 2018 Formulations von Neumann-Morgenstern (1944/1947) X: Set of prizes Π: Set of probability distributions on X : Preference relation
Διαβάστε περισσότεραFractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
Διαβάστε περισσότερα6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Διαβάστε περισσότεραA Note on Intuitionistic Fuzzy. Equivalence Relation
International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com
Διαβάστε περισσότεραProblem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ.
Chemistry 362 Dr Jean M Standard Problem Set 9 Solutions The ˆ L 2 operator is defined as Verify that the angular wavefunction Y θ,φ) Also verify that the eigenvalue is given by 2! 2 & L ˆ 2! 2 2 θ 2 +
Διαβάστε περισσότεραThe challenges of non-stable predicates
The challenges of non-stable predicates Consider a non-stable predicate Φ encoding, say, a safety property. We want to determine whether Φ holds for our program. The challenges of non-stable predicates
Διαβάστε περισσότεραLecture 34 Bootstrap confidence intervals
Lecture 34 Bootstrap confidence intervals Confidence Intervals θ: an unknown parameter of interest We want to find limits θ and θ such that Gt = P nˆθ θ t If G 1 1 α is known, then P θ θ = P θ θ = 1 α
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραArithmetical applications of lagrangian interpolation. Tanguy Rivoal. Institut Fourier CNRS and Université de Grenoble 1
Arithmetical applications of lagrangian interpolation Tanguy Rivoal Institut Fourier CNRS and Université de Grenoble Conference Diophantine and Analytic Problems in Number Theory, The 00th anniversary
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότεραTakeaki Yamazaki (Toyo Univ.) 山崎丈明 ( 東洋大学 ) Oct. 24, RIMS
Takeaki Yamazaki (Toyo Univ.) 山崎丈明 ( 東洋大学 ) Oct. 24, 2017 @ RIMS Contents Introduction Generalized Karcher equation Ando-Hiai inequalities Problem Introduction PP: The set of all positive definite operators
Διαβάστε περισσότεραORDINAL ARITHMETIC JULIAN J. SCHLÖDER
ORDINAL ARITHMETIC JULIAN J. SCHLÖDER Abstract. We define ordinal arithmetic and show laws of Left- Monotonicity, Associativity, Distributivity, some minor related properties and the Cantor Normal Form.
Διαβάστε περισσότεραD Alembert s Solution to the Wave Equation
D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique
Διαβάστε περισσότεραParametrized Surfaces
Parametrized Surfaces Recall from our unit on vector-valued functions at the beginning of the semester that an R 3 -valued function c(t) in one parameter is a mapping of the form c : I R 3 where I is some
Διαβάστε περισσότεραLecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Διαβάστε περισσότεραECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραMA 342N Assignment 1 Due 24 February 2016
M 342N ssignment Due 24 February 206 Id: 342N-s206-.m4,v. 206/02/5 2:25:36 john Exp john. Suppose that q, in addition to satisfying the assumptions from lecture, is an even function. Prove that η(λ = 0,
Διαβάστε περισσότεραMATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)
1. MATH43 String Theory Solutions 4 x = 0 τ = fs). 1) = = f s) ) x = x [f s)] + f s) 3) equation of motion is x = 0 if an only if f s) = 0 i.e. fs) = As + B with A, B constants. i.e. allowe reparametrisations
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραQuadratic Expressions
Quadratic Expressions. The standard form of a quadratic equation is ax + bx + c = 0 where a, b, c R and a 0. The roots of ax + bx + c = 0 are b ± b a 4ac. 3. For the equation ax +bx+c = 0, sum of the roots
Διαβάστε περισσότεραPractice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Διαβάστε περισσότεραCongruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Διαβάστε περισσότεραSolutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραSpace-Time Symmetries
Chapter Space-Time Symmetries In classical fiel theory any continuous symmetry of the action generates a conserve current by Noether's proceure. If the Lagrangian is not invariant but only shifts by a
Διαβάστε περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότεραA Two-Sided Laplace Inversion Algorithm with Computable Error Bounds and Its Applications in Financial Engineering
Electronic Companion A Two-Sie Laplace Inversion Algorithm with Computable Error Bouns an Its Applications in Financial Engineering Ning Cai, S. G. Kou, Zongjian Liu HKUST an Columbia University Appenix
Διαβάστε περισσότεραSecond Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Διαβάστε περισσότεραSolution Series 9. i=1 x i and i=1 x i.
Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότερα1 String with massive end-points
1 String with massive end-points Πρόβλημα 5.11:Θεωρείστε μια χορδή μήκους, τάσης T, με δύο σημειακά σωματίδια στα άκρα της, το ένα μάζας m, και το άλλο μάζας m. α) Μελετώντας την κίνηση των άκρων βρείτε
Διαβάστε περισσότεραSTABILITY FOR RAYLEIGH-BENARD CONVECTIVE SOLUTIONS OF THE BOLTZMANN EQUATION
STABILITY FOR RAYLEIGH-BENARD CONVECTIVE SOLUTIONS OF THE BOLTZMANN EQUATION L.Arkeryd, Chalmers, Goteborg, Sweden, R.Esposito, University of L Aquila, Italy, R.Marra, University of Rome, Italy, A.Nouri,
Διαβάστε περισσότερα1. Introduction and Preliminaries.
Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.yu/filomat Filomat 22:1 (2008), 97 106 ON δ SETS IN γ SPACES V. Renuka Devi and D. Sivaraj Abstract We
Διαβάστε περισσότεραEXISTENCE OF POSITIVE SOLUTIONS FOR SINGULAR FRACTIONAL DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equations, Vol. 28(28), No. 146, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE
Διαβάστε περισσότερα= λ 1 1 e. = λ 1 =12. has the properties e 1. e 3,V(Y
Stat 50 Homework Solutions Spring 005. (a λ λ λ 44 (b trace( λ + λ + λ 0 (c V (e x e e λ e e λ e (λ e by definition, the eigenvector e has the properties e λ e and e e. (d λ e e + λ e e + λ e e 8 6 4 4
Διαβάστε περισσότεραw o = R 1 p. (1) R = p =. = 1
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:
Διαβάστε περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότεραExercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Διαβάστε περισσότεραb. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
Διαβάστε περισσότεραMINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS
MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS FUMIE NAKAOKA AND NOBUYUKI ODA Received 20 December 2005; Revised 28 May 2006; Accepted 6 August 2006 Some properties of minimal closed sets and maximal closed
Διαβάστε περισσότεραLecture 13 - Root Space Decomposition II
Lecture 13 - Root Space Decomposition II October 18, 2012 1 Review First let us recall the situation. Let g be a simple algebra, with maximal toral subalgebra h (which we are calling a CSA, or Cartan Subalgebra).
Διαβάστε περισσότεραLecture 2. Soundness and completeness of propositional logic
Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness
Διαβάστε περισσότεραDESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Διαβάστε περισσότεραP AND P. P : actual probability. P : risk neutral probability. Realtionship: mutual absolute continuity P P. For example:
(B t, S (t) t P AND P,..., S (p) t ): securities P : actual probability P : risk neutral probability Realtionship: mutual absolute continuity P P For example: P : ds t = µ t S t dt + σ t S t dw t P : ds
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότεραSOME PROPERTIES OF FUZZY REAL NUMBERS
Sahand Communications in Mathematical Analysis (SCMA) Vol. 3 No. 1 (2016), 21-27 http://scma.maragheh.ac.ir SOME PROPERTIES OF FUZZY REAL NUMBERS BAYAZ DARABY 1 AND JAVAD JAFARI 2 Abstract. In the mathematical
Διαβάστε περισσότεραPARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Διαβάστε περισσότεραResearch Article Existence of Positive Solutions for m-point Boundary Value Problems on Time Scales
Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 29, Article ID 189768, 12 pages doi:1.1155/29/189768 Research Article Existence of Positive Solutions for m-point Boundary
Διαβάστε περισσότεραThe semiclassical Garding inequality
The semiclassical Garding inequality We give a proof of the semiclassical Garding inequality (Theorem 4.1 using as the only black box the Calderon-Vaillancourt Theorem. 1 Anti-Wick quantization For (q,
Διαβάστε περισσότεραDerivation of Optical-Bloch Equations
Appendix C Derivation of Optical-Bloch Equations In this appendix the optical-bloch equations that give the populations and coherences for an idealized three-level Λ system, Fig. 3. on page 47, will be
Διαβάστε περισσότεραPOSITIVE SOLUTIONS FOR A FUNCTIONAL DELAY SECOND-ORDER THREE-POINT BOUNDARY-VALUE PROBLEM
Electronic Journal of Differential Equations, Vol. 26(26, No. 4, pp.. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp POSITIVE SOLUTIONS
Διαβάστε περισσότεραF19MC2 Solutions 9 Complex Analysis
F9MC Solutions 9 Complex Analysis. (i) Let f(z) = eaz +z. Then f is ifferentiable except at z = ±i an so by Cauchy s Resiue Theorem e az z = πi[res(f,i)+res(f, i)]. +z C(,) Since + has zeros of orer at
Διαβάστε περισσότερα