Nonlinear system of mixed type and its application to steady Euler-Poisson system

Transcription

1 The 1st Meeting of Young Researchers in PDEs Nonlinear system of mixed type and its application to steady Euler-Poisson system Myoungjean Bae (POSTECH) -based on collaborations with- B. Duan, J. Xiao, C. Xie, H. Park, Y. Park, and H. Ryu August 18, /23

2 Introduction ρ : (electron) density, u : velocity, p : pressure, Φ : Coulomb potential Steady Euler-Poisson system div(ρu) = 0 div(ρu u+pi) = ρ Φ div(ρub) = ρu Φ Φ = ρ b Bernoulli function ( b > 0 : background charge density) B = 1 2 u 2 + γp (γ 1)ρ, γ > 1 Entropy S = ln p ρ γ Transport equations If (ρ,u,p, Φ) is a C 1 solution, then div(ρus) = div(ρu(b Φ)) = 0 div(ρu)=0 ρu S = ρu (B Φ) = 0 ** We call B Φ pseudo-bernoulli function. 2/23

3 Rankine-Hugoniot conditions Ω n Γ Ω + divq = 0 in Ω. (1) Q L 2 (Ω) C 1 (Ω ± ) C 0 (Ω ± Γ) satisfies Q Dφ dx = 0 for any φ C0 (Ω) Ω Q satisfies (1) in Ω ± and the relation [Q n] Γ := Q n x Ω Γ Q n x Ω + Γ = 0 (2) because Q Dφ = Ω + Ω Γ [Q n] φ φ divq Ω + Ω 3/23

4 Discontinuous transition [ρu n] Γ = [ρ(u n)u+pn] Γ = [ρ(u n)b] Γ = 0 Let τ be a tangential vector field on Γ. { τ [ρ(u n s )u+pn] Γ = 0 ρ(u n)[u τ] Γ = 0 n [ρ(u n)u+pn] Γ = 0 [ρ(u n) 2 +p] Γ = 0 Case 1. If u n 0 on Γ, then [ρu n] Γ = [u τ] Γ = [ρ(u n) 2 +p] Γ = [B] Γ = 0 (shock) Case 2. If u n = 0 on Γ, then [p] = 0 (contact discontinuity) 4/23

5 Example of shocks 5/23

6 Motivation: De Laval Nozzle problem for Euler system Given (ρ,u,p ) with M < 1, A A ex 0 p A t x p max p 0 x Figure: Supersonic Flow And Shock Waves by Courant-Friedrich 6/23

7 1-d solutions for E-P system References:Ascher-Markowich-Pietra-Schmeiser(1991),Luo-Xin(2012) etc. (For b 0 < ρ sonic ) E ρ = ρs b ρ Figure 1 Phase portrait for 0 < b < ρs One dimensional smooth transonic-transonic shock solution is here!! 7/23

8 Main Problem To construct multidimensional smooth transonic-transonic shock configuration for prescribed exit pressure in flat nozzle where proper electric field applies through nozzle. Γ w : u n w = 0 Γ sonic : M = 1 Γ shock M 0 < 1 M < 1 M > 1 M < 1 p = p ex x = s 0 x = s 1 Figure: Smooth transonic-transonic shock configuration Due to Φ = ρ b, a transonic shock problem is a two-phase problem with a free boundary to be determined. 8/23

9 Why is this problem difficult? Analytic property of E-P system Suppose u = 0 i.e., u = ϕ. If B Φ = 0 (pseudo-bernoulli s law), ( div (Φ 1 ) 2 ϕ 2 ) 1 γ 1 ϕ = 0, (3) Φ = (Φ 1 2 ϕ 2 ) 1 γ 1 b0 (4) Flow type M < 1 M = 1 M > 1 (3) Elliptic Degenerate Hyperbolic (3) (4) Elliptic system D.M.-E coupled system H-E coupled system W.-P. Unknown Unknown at all Unknown Note M < 1 ϕ 2 < 2(γ 1) γ+1 Φ 9/23

10 Recent results M-D subsonic flow of E-P system with u = 0 for fixed exit pressure (B.-Duan-Xie, ARMA 2016) 2-D subsonic flow of full E-P system for fixed exit pressure (B.-Duan-Xie, SIMA 2014) 2-D subsonic flow of E-P system with self-gravitation (B.-Duan-Xie, M3AS 2015) Φ = ρ 3-D subsonic flow of E-P system with nonzero swirl(=angular momentum) (B.-Weng, submitted) 2-D supersonic flow of full E-P system (B.-Duan-Xie, preprint) and so on... 10/23

11 Subsonic flow (M < 1 ϕ 2 < 2(γ 1) γ+1 Φ) Problem 1. Subsonic flow for potential flow model Given p ex with p ex p + (L) ( α,γl) 1,α,Γ L small, find a solution (ϕ,φ) to E-P system for potential flow with B.C. given as follows: x R n 1 ϕ = 0 Φ = Φ + Λ R n 1 n w p = p ex Φ = Φ + 0 nw ϕ = 0 = nw Φ L x 1 11/23

12 Theorem (B.-Duan-Xie, ARMA 2016) α (0,1), σ 0 > 0 so that if σ σ 0, then the NLBVP has a unique solution (ϕ,φ) C 1,α (N) C 2 (N) s.t. ϕ ϕ + ( 1 α,corners) 2,α,N + Φ Φ + ( 1 α,corners) 2,α,N C p ex p + (L) ( α, Λ) 1,α,Λ. Essential observation Same coupled terms ( div (Φ 1 ) 2 ϕ 2 ) 1 γ 1 ϕ = 0, Φ = (Φ 1 2 ϕ 2 ) 1 γ 1 b0 Nonlinear boundary condition p = p ex on x 1 = L Φ 1 2 ϕ 2 = p γ 1 γ ex on x 1 = L 12/23

13 Problem 2. 3-D subsonic flow with nonzero vorticity Find axi-symmetric subsonic flow of E-P system with B.C. given as follows: (r,θ) Γ en : u e r = 0 Φ = Φ 0 Λ = B 1 (0) R 2 n w Γ ex : p = p ex 0 Γ w : u n w = nw Φ = 0 L x Helmholtz decomposition for axi-symmetric flow with nonzero swirl u = = ϕ(x,r) }{{} curl free(compressibility) ( 1 r r(rψ)+ x ϕ ) + (h(x,r)e r +ψ(x,r)e θ ) }{{} div free(vorticity) e x +( x ψ + r ϕ)e r + x h(x,r)e θ }{{} nonzero swirl 13/23

14 Corresponding equations derived from E-P system ( div H(S,K+Φ 1 ) 2 u(r,ψ,dψ,dϕ,λ) 2 )u(r,ψ,dψ,dϕ,λ) = 0, }{{} (=:M) Φ = H(S,K+Φ 1 2 u(r,ψ,dψ,dϕ,λ) 2 ) b, (ψe θ ) = T(ρ,S) rs r K+ Λ r 2 r Λ 1 r e θ =: fe θ, r(rψ)+ x ϕ M (S,K,Λ) = 0. Theorem (B.-Weng, Submitted/ArXiv) Given a small perturbation of (b,p ex,s en,k en,λ en ), there exists a 3-D axisymmetric subsonic solution. The uniqueness (in small perturbation regime) is obtained under an additional assumption of boundary data. 14/23

15 Elliptic equation for ψ with a singular coefficient, W = ψ(x,r)e θ Either we find a solution ψ to ( xx + 1 r r(r r ) 1 r 2 ) ψ = f(x,r) in N = (0,L) B1 (0), s.t. ψ(x,r)e θ is C 2,α (N), or, we solve x ψ(0,r) = 0 on Γ en, ψ = 0 on Γ w, x ψ = 0 on Γ ex, ψ = 0 on N {r = 0}, W = f(x,r)e θ in N = (0,L) B 1 (0), x W = 0 on Γ en Γ ex, W = 0 on Γ w, then show that W is in the form of W = ψ(x,r)e θ. Compatibility condition for axisymmetric flow f(x, 0) = 0 15/23

16 Proposition For a fixed α (0,1), suppose that a vector field f(x,r)e θ : N R 3 is C α in N, and that f satisfies the compatibility condition f(x,0) = 0. Then the linear boundary value problem has a unique solution W : N R 3. Furthermore, W is represented as where ψ satisfies the estimate W(x) = ψ(x,r)e θ in N, (5) ψ 2,α,[0,L] [0,1] C f α,n. Note that W is well defined by (5) due to the condition ψ(x,0) = 0 for all x [0,L]. 16/23

17 Idea of Proof The unique existence of C 2,α solution W is easily obtained. 1. Let W = W 1 (x)e 1 +W 2 (x)e 2 +W 3 (x)e 3 be a C 2 solution. Then, V 1 0 in N. for W = W 2 e 2 +W 3 e 3 =: U r e r +U θ e θ in N { U r (x,r,θ) = W 2 (x,rcosθ,rsinθ)cosθ +W 3 (x,rcosθ,rsinθ)sinθ U θ (x,r,θ) = W 2 (x,rcosθ,rsinθ)sinθ +W 3 (x,rcosθ,rsinθ)cosθ. W C 2,α (U r,u θ ) C 2,α ([0,L] [0,1] T) 17/23

18 2. U r and U θ satisfy { L cyl 1 (U r,u θ ) := ( x 2 + r r r + 1 r 2 2 θ ) 1 r 2 Ur 2 r 2 θ U θ = 0, L cyl 2 (U r,u θ ) := ( x r + 1 r r + 1 r 2 2 θ ) 1 r 2 Uθ + 2 r 2 θ U r = f(x,r), 3. For each n N, set U n (x,r,θ) := 1 Then, so, U n θ 2 n 1 2 n k=0 U r ( x,r,θ + 2πk 2 n ) 1 (x,r,θ) := 2 n 1 2 n k=0 U ( ) θ x,r,θ + 2πk 2 n (U n r,u n θ )(x,r,θ) = (U n r,u n θ )(x,r,θ + 2πj 2 n ) 0 j 2n 1, n 1, (L cyl 1,Lcyl 2 )(Un r,un θ ) = (0, f). And, {(U n r,u n θ )} is uniformly bounded in C2,α ([0,L] [0,1] T). 18/23

19 4. By Arzelá-Ascoli Theorem, a subsequence {(U n k r,un k θ )} that converges to U = (Ur,U θ ), where U satisfies (L cyl 1,Lcyl 2 )U = (0, f), and U (x,r,θ) = U (x,r,θ + 2πj 2 nk ) 0 j 2 n k 1, k 1 U (x,r,θ) = U (x,r,θ +2κπ) 0 κ 1. 19/23

20 2-D supersonic flow (M > 1 ϕ 2 > 2(γ 1) γ+1 Φ) Problem 3. Given u en with u en u 0 C4 (Γ 0) small, find a supersonic potential replacements flow (ϕ,φ) of E-P system with B. C. given as follows: x 2 ϕ = 0,ϕ x1 = u en Φ x1 = E en n w Φ = Φ0 0 nw ϕ = 0 = nw Φ L x 1 Figure: D k x 2 u en = 0 on Λ for k = 1,2,3 20/23

21 Fix γ > 1, and set m 0 = J 0 S 1 γ 1 0 for J 0 = ρ 0 u 0. In N = (0,L) ( 1,1), set σ 1 := u en u 0 C 4 ([ 1,1]), σ 2 := b b 0 C 2 (N) + E en E 0 C 4 ([ 1,1]) Theorem (B.-Duan-Xiao-Xie, preprint) Given γ > 1,L > 0,m 0 > 0,u 0 > u sonic (= (γm γ 1 0 ) 1 γ+1 ),E0 > 0,b 0 > 0, S 0 > 1 depending only on γ,l,m 0,u 0,E 0 s.t. the following holds: If S 0 S 0 (J 0 small), then, σ 1, σ 2 > 0 such that if σ 1 σ 1, and σ 2 σ 2, then NLBVP II has a unique solution (ϕ,φ) [H 4 (N) C 2 (N)] 2 satisfying ϕ ϕ 0 H 4 (N) +S 1 2(γ 1) 0 Φ Φ 0 H 4 (N) C(σ 1 +S 1 2(γ 1) 0 σ 2 ). Question Is the condition S 0 S 0 ( J 0 << 1) simply technical? Or, is there a physical reason? 21/23

22 Ongoing works Tranosnic flows of E-P system M-d smooth transonic flow of E-P system (with B. Duan & C. Xie) M-d transonic shock solution of E-P system in flat nozzle (with B. Duan & C. Xie) M-d transonic shock solution of E-P system in convergent nozzle (with Y. Park) Application of Helmholtz decomposition to transonic shock solutions (with Y. Park) M-d supersonic flow of E-P system (with H. Ryu) Application of Helmholtz decomposition Transonic shocks (with Y. Park) Contact discontinuity (with H. Park) 22/23

23 Ongoing works Tranosnic flows of E-P system M-d smooth transonic flow of E-P system (with B. Duan & C. Xie) M-d transonic shock solution of E-P system in flat nozzle (with B. Duan & C. Xie) M-d transonic shock solution of E-P system in convergent nozzle (with Y. Park) Application of Helmholtz decomposition to transonic shock solutions (with Y. Park) M-d supersonic flow of E-P system (with H. Ryu) Application of Helmholtz decomposition Transonic shocks (with Y. Park) Contact discontinuity (with H. Park) THANK YOU!!! 23/23