Global Existence and Asymptotic Behavior for a Compressible Energy - PowerPoint PPT Presentation

Global Existence and Asymptotic Behavior for a Compressible Energy Transport Model Yong LI Department of Mathematical Sciences, Tsinghua University BEIJING, P. R. CHINA Joint with Ling Hsiao

The energy transport model is a degenerate quasi-linear cross diffusion parabolic system with principal part in divergence form, the common form is governed by  ∂   ∂tn ( µ, T ) − div J n = 0 ,      ∂ (1) ∂tU ( µ, T ) − div J w = −∇ V · J n + W ( µ, T ) ,        λ 2 △ V = n − b ( x ) . Where µ , T are chemical potential of the electrons and the electron temperature respectively, V is the electrostatic po- tential, n ( µ, T ) is the electron density, U ( µ, T ) is the density of the internal energy, λ is the scaled Debye length, b ( x ) is the doping profile which represents the background of the device, 2

W ( µ, T ) is the energy relaxation term satisfying W ( µ, T )( T − T 0 ) ≤ 0 , the positive constant T 0 is the lattice temperature, J n is the carrier flux density and J w is the energy flux density, which are given by  � � � � � µ � + ∇ V − 1    J n = L 11 ∇ + L 12 ∇ ,   T T T � � � � � µ �  + ∇ V − 1    J w = L 21 ∇ + L 22 ∇ ,  T T T L is the diffusion matrix, � L 11 � L 12 L = . L 21 L 22 3

We consider the following relations for n ( µ, T ) and U ( µ, T ), which derived from the Boltzmann statistics, � µ � 1 γ − 1 exp n ( µ, T ) = T , T � µ � 1 γ γ − 1 exp U ( µ, T ) = γ − 1 T . T The energy relaxation term W ( µ, T ) is given by � µ � T 0 − T 1 1 γ − 1 exp W ( µ, T ) = γ − 1 T , T ε the diffusion matrix L is a positive semidefinite matrix,   γT 1 � µ � γ γ − 1   γ − 1 exp   L = ( L ij ) = εT  .  γ 2 T 2 γT T γ − 1 ( γ − 1) 2 4

We could rewritten the above energy transport model as the following form:  n t − ε ∇ · [ ∇ ( nT ) − n � E ] = 0 ,     � nT � γT � �     γ − 1[ ∇ ( nT ) − n �  − ε ∇ · E ]  γ − 1 t (2)  E − n ( T − T 0 )   = ε [ n � E − ∇ ( nT )] · �  ε ( γ − 1) ,       λ 2 ∆ V = n − b ( x ) , � E = ∇ V. Where ε is the energy relaxation time, and the adiabatic ex- ponent γ > 1. The diffusion matrix L satisfying   γT 1 γ − 1     L = ( L ij ) = εnT  .  γ 2 T 2 γT ( γ − 1) 2 γ − 1 γ 2 ( γ − 1) 2 T 2 . Eigenvalues: µ 1 = 0 , µ 2 = 1 + 5

◮ Derivation The model (2) can be derived from the non-isentropic Euler- Poisson system in semiconductors or plasma,  n t + ∇ · ( n� u ) = 0 ,     � �  E − n� u  u ) + ∇ P = n � u ) t + ∇ · ( n� ( n�  u τ p ,  E − W − W 0 u · �  W t + ∇ · ( � uW + � uP ) = n�  ,   τ w    λ 2 ∆ V = n − b ( x ) , �  E = ∇ V, nT u | 2 + W 0 = nT 0 with P = nT, W = n | � γ − 1 ; γ − 1 , τ p , τ w are the momentum and energy relaxation times. This model was introduced in ♦ K. Bl¨ otekjær, IEEE Trans. Electron. Devices , ED-17 (1970), 38-47. ♦ Markowich, Ringhofer and Schmeiser, Semiconductors Equations , 1990. 6

Define τ 2 = τ p << 1 , τ w and consider the following variables transformations, t → t τ w = ε u → τ� τ p = ετ, τ , � u, τ ( n, T, � E ) → ( n, T, � E ) . We rewrite the system as  n τ t + ∇ · ( n τ � u τ ) = 0 ,       E τ − n τ � u τ u τ ⊗ � u τ ) + ∇ ( n τ T τ ) = n τ �  ( τ 2 n τ � u τ ) t + ∇ · ( τ 2 n τ �  ,   ε     � τ 2 � τ 2 u τ | 2 + n τ T τ � u τ � u τ + γn τ T τ u τ | 2 � 2 n τ | � 2 n τ | � t + ∇ · γ − 1 �  γ − 1    � τ 2 �   u τ | 2 + n τ ( T τ − T 0 )  E τ · � u τ − = n τ � 2 n τ | �  ,   ε ( γ − 1)     E τ = n τ − b ( x ) . λ 2 ∇ · � 7

Taking the formal limit as τ → 0, we obtain the compressible energy transport model,  n t − ε ∇ · [ ∇ ( nT ) − n �  E ] = 0 ,    � nT � γT  � �    γ − 1[ ∇ ( nT ) − n �  − ε ∇ · E ]  γ − 1 t (3) E − n ( T − T 0 )   = ε [ n � E − ∇ ( nT )] · �  ε ( γ − 1) ,        λ 2 ∇ · � E = n − b ( x ) . This formal limit has been rigorously justified by ♦ I. Gasser and R. Natalini, Quart.Appl.Math. , LVII, No.2(1999), 269-282. ♦ Y. Li, Acta Math. Sci. , (in press, 2007). 8

◮ Some Previous Results: (The diffusion matrix L is (uniformly) positive definite) • Numerical results – Chen-Shu-Dutton, Degond-J¨ ungel-Pietra, Jerome-Shu, Souissi-Gnudi, Holst-J¨ ungel-Pietra,... • Steady state – Degond-G ´ e nieys-J¨ ungel, Griepentrog, Xie, Chen-Hsiao • Global existence –The diffusion matrix L is uniformly positive definite Degond-G ´ e nieys-J¨ ungel, J¨ ungel, Al ` ı etc. – The diffusion matrix L just positive definite Chen-Hsiao, Chen-Hsiao-Li 9

Consider the one-dimensional case of model (2) ( ε = λ = T 0 = 1)   n t − [( nT ) x − nE ] x = 0 , J ( x, t ) = nE − ( nT ) x ,       ( nT ) t − { γT [( nT ) x − nE ] } x (4)  = ( γ − 1)[ nE − ( nT ) x ] E − n ( T − 1) ,       E x = n − b ( x ) . The insulated boundary conditions and initial values J ( x, t ) | x =0 , 1 = 0 , T x ( x, t ) | x =0 , 1 = 0 , E (0 , t ) = 0 , (5) n ( x, 0) = n I ( x ) , T ( x, 0) = T I ( x ) , x ∈ Ω , where the initial density n I ( x ) is chosen to satisfy � Ω ( n I − b )( x ) dx = 0 . Then the boundary conditions are n x ( x, t ) | x =0 , 1 = 0 , T x ( x, t ) | x =0 , 1 = 0 , E ( x, t ) | x =0 , 1 = 0 . (6) 10

Introduce the flux as nu = nE − ( nT ) x . So (4) can be rewritten as  n t + ( nu ) x = 0 ,    ( nT ) t + ( γnTu ) x = ( γ − 1) nuE − n ( T − 1) , (7)    E x = n − b ( x ) . The model (7) can be understood as the ”full” compressible Navier-Stokes equations (with a temperature equation),  n t + ( nu ) x = 0 ,       ( nu ) t + ( γnu 2 ) x + ( nT ) x     = γ ( nTu x ) x − [2 + b ( x )] nu + 2 nE − n x    T t + uT x + ( γ − 1) Tu x = ( γ − 1) u 2 − ( T − 1) ,        E x = n − b ( x ) . 11

◮ The isothermal stationary solution We consider a typical stationary solution ( N , 1 , V ).  ∇N − N∇V = 0 , x ∈ Ω ,   ∆ V = N − b ( x ) , x ∈ Ω ,   ∇V · γ | ∂ Ω = 0 , γ : the unit outer normal vector Assume that 0 < C ≤ b ( x ) ≤ C and b ∈ L ∞ (Ω) , Theorem 1 then the problem has a solution ( N , V ) , and satisfing 0 < C ≤ N ( x ) ≤ C, x ∈ Ω , c ≤ V ( x ) ≤ c, x ∈ Ω , | ∆ V ( x ) | , |∇V ( x ) | , |∇N ( x ) | , | ∆ N ( x ) | ≤ C, x ∈ Ω , where C is a positive constant and c, c are constants. 12

◮ The GlobalExistence and exponential decay Suppose 0 < C ≤ b ( x ) ≤ C , b ( x ) ∈ C 2 (Ω) and Theorem 2 ( n I ( x ) , T I ( x )) ∈ H 3 (Ω) . There exists a positive constant δ 1 , such that if � n I ( x ) − N ( x ) � H 3 + � T I ( x ) − 1 � H 3 ≤ δ 1 , then the problem (4)-(5) has a unique global solution ( n, T, E ) in Ω × [0 , ∞ ) satisfying � n ( · , t ) − N ( · ) � H 3 + � T ( · , t ) − 1 � H 3 + � E ( · , t ) − V x ( · ) � H 3 ≤ c 0 ( � n ( · , 0) − N ( · ) � H 3 + � T ( · , 0) − 1 � H 3 ) exp ( − ηt ) for some positive constants c 0 and η . In our theorem, the doping profile b ( x ) can be Remark 3 roughly large. 13

◮ A-priori estimates The 1-D compressible Energy Transport Model (7),  n t + ( nu ) x = 0 ,    ( nT ) t + ( γnTu ) x = ( γ − 1) nuE − n ( T − 1) ,    E x = n − b ( x ) , with nu = nE − ( nT ) x . Setting n = N + N ρ, T = 1 + θ, E = E + ϕ, E = V x , where the stationary solution is ( N , 1 , E ). Introduce the entropy function s as follows θ = (1 + ρ ) γ − 1 (1 + s ) − 1 . 14

Then the model (7) can be rewritten as   ρ t + [(1 + ρ ) u ] x = −E (1 + ρ ) u,      s t + us x = − (1 + ρ ) γ − 1 (1 + s ) − 1     (1 + ρ ) γ − 1 (8)  γ − 1  (1 + ρ ) γ − 1 | u | 2 + ( γ − 1) E (1 + s ) u,   +       ϕ x = N ρ, 1 1 + ρ [(1 + ρ ) γ (1 + s )] x − E [(1 + ρ ) γ − 1 (1 + s ) − 1] . u = ϕ − The corresponding initial data and the boundary conditions, ρ ( x, 0) = N ( x ) − 1 n I ( x ) − 1 , (9) s ( x, 0) = N ( x ) γ − 1 T I ( x ) n I ( x ) − ( γ − 1) − 1 , ρ x | x =0 , 1 = 0 , s x | x =0 , 1 = 0 , ϕ | x =0 , 1 = 0 . (10) 15