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The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic Region

Bo-Chih Huang

Department of Mathematics, National Central University, Taiwan

January 22, 2016 2016 Annual Workshop on Differential Equations Joint work with John M. Hong, Shih-Wei Chou(NCU) and Chien-Chang Yen(Fu Jen Catholic University).

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Outline

• 1. The hydrodynamic escape phenomenon in planetary atmospheres
• 2. The hydrodynamic escape model without conduction
• 3. Global transonic solution for initial-boundary value problem for HEP
• 4. The hydrodynamic escape region

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Thermal escape process (Neutral processes)

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Layer of the atmoesphere

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

The hydrodynamic escape model

The three-dimensional inviscid hydrodynamic equations for a single-constituent atmosphere is govern by the following system of equations: ✩ ✬ ✬ ✬ ✫ ✬ ✬ ✬ ✪ ❇tρ ∇ ☎ ♣ρuq ✏ 0, ❇t♣ρuq ∇ ☎ ♣♣ρuq ❜ uq ∇P ✏ ✁GMρ ⑤r⑤3 r, ❇tE ∇ ☎ ♣♣E Pquq ✏ ✁GMρ ⑤r⑤3 ♣u ☎ rq q ∇ ☎ ♣κ∇Tq. (2.1) Here the total energy E and the state equation are given respectively by E ✏ 1 2ρ⑤u⑤2 ρe ✏ 1 2ρ⑤u⑤2 P γ ✁ 1, P ✏ ρRT. (2.2) ρ: gas density u: gas flow velocity P: the pressure G: the universal gravitational constant M: the mass of planet E: the total energy T: the temperature γ: the adiabatic constant with 1 ➔ γ ➔ 5④3 r: the position vector form center of the planet to the particle of gas q ✏ q♣rq: the heat profile κ ✏ κ♣Tq: thermal conductivity

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

The hydrodynamic escape model

We are concerned with in spherical symmetric space-time model of system (2.1) without conduction, that is, set κ ✏ 0 ✩ ✬ ✬ ✫ ✬ ✬ ✪ ❇t

• ρx2✟

❇x

• ρux2✟

✏ 0, ❇t

• ρux2✟

❇x

• ρu2x2 Px2✟

✏ ✁GMρ 2Px, ❇t

• Ex2✟

❇x

• ♣E Pqux2✟

✏ ✁GMρu qx2. (2.3) Let m :✏ ρu, U :✏ ♣ρ, m, EqT, using relation (2.2), system (2.3) can be written in compact form Ut f ♣Uqx ✏ h♣xqg♣x, Uq, (2.4) where h♣xq ✏ ✁ 2

x and

f ♣Uq ✏ ✁ m, 3 ✁ γ 2 m2 ρ ♣γ ✁ 1qE, m ρ ✁ γE ✁ γ ✁ 1 2 m2 ρ ✠✠T , g♣x, Uq ✏ ✁ m, m2 ρ GM 2x ρ, m ρ ✁ γE ✁ γ ✁ 1 2 m2 ρ ✠ GM 2x m ✁ xq 2 ✠T .

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Motivation and Previous Works

Motivation: [1] Feng Tian, Thermal Escape From Super Earth Atmospheres in the Habitable Zones of M Stars, The Astrophysical Journal 703 (2009) , pp. 905–909. [2] H. D. Sterk, S. Rostrup, and F. Tian, A fast and accurate algorithm for computing radial transonic flows, J. Comput. Appl. Math., 223 (2009) , pp. 916–928. [3] X. Zhu, D. F. Strobel, and J. T. Erwin, The density and thermal structure of Plutos atmosphere and associated escape processes and rates, Icurus, 228 (2014) , pp. 301–314. Previsous Works: [4]J. M. Hong, C.-C. Yen, B.-C. Huang, Characterization of the transonic stationary solutions of the hydrodynamic escape problem, SIAM

• J. Appl. Math., 74 (2014), pp. 1709–1741.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

The hydrodynamic escape model

The complete model of hydrodynamic escape problem (HEP) model is given by the following free boundary value problem (HEP) ✩ ✬ ✬ ✬ ✬ ✬ ✫ ✬ ✬ ✬ ✬ ✬ ✪ Ut f ♣Uqx ✏ h♣xqg♣x, Uq, ♣x, tq P Σ ✑ rxB, xTs ✂ r0, ✽q, U♣x, 0q ✏ U0♣xq P Ω, x P rxB, xTs, ρ♣xB, tq ✏ ρB♣tq, m♣xB, tq ✏ mB♣tq, t → 0, ρ ✞ ✞

Σ, m

ρ ✞ ✞

Σ → 0, Kn♣x, Uq

✞ ✞

Σ ↕ 1,

(2.5) where the exobase of the atmosphere x ✏ xT (as well as Σ), is need to be decided, Kn♣x, Uq stands for the Knudsen number.

Difficult to solve

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

The hydrodynamic escape model

The complete model of hydrodynamic escape problem (HEP) model is given by the following free boundary value problem (HEP) ✩ ✬ ✬ ✬ ✬ ✬ ✫ ✬ ✬ ✬ ✬ ✬ ✪ Ut f ♣Uqx ✏ h♣xqg♣x, Uq, ♣x, tq P Σ ✑ rxB, xTs ✂ r0, ✽q, U♣x, 0q ✏ U0♣xq P Ω, x P rxB, xTs, ρ♣xB, tq ✏ ρB♣tq, m♣xB, tq ✏ mB♣tq, t → 0, ρ ✞ ✞

Σ, m

ρ ✞ ✞

Σ → 0, Kn♣x, Uq

✞ ✞

Σ ↕ 1,

(2.5) where the exobase of the atmosphere x ✏ xT (as well as Σ), is need to be decided, Kn♣x, Uq stands for the Knudsen number.

Difficult to solve

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

The Knudsen number

The Knudsen number is useful for determining whether statistical mechanics or the continuum mechanics formulation of fluid dynamics should be used. It is defined as Kn :✏ ℓ④H, the ratio of the mean free path of the molecules (the average distance traveled by a moving molecules between successive collisions) ℓ ✏ 1 ❄ 2τn , to the density scale height (the increase in altitude for which the atmospheric density decreases by a factor of e) H ✏ kBT GMm④x2 ,

• f the atmosphere. Here τ ✓ 1014π cm2 is the collision cross section, n is the

number density, m is the mass of a molecule, T is the temperature of the gas and kB is the Boltzmann constant.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

The Knudsen number (Cont.)

Assuming the gas is ideal, that is, P ✏ ρRT ✏ nkBT. The Knudsen number can be computed as follows: Kn♣x, Uq ✏ ℓ H ✏ GMpm ❄ 2τnx2kBT ✏ GMpm ❄ 2τx2P ✏ γGMpm ❄ 2τx2ρc2 . The hydrodynamic equations are applied well where Kn ➔ 1 so that many collisions occur over relevant length scales keeping the gas in thermal

• equilibrium. If Kn ➙ 1, the continuum assumption of fluid mechanics maybe no

longer a good approximation since there are few collisions in this level to inhibit a molecule from escaping.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

The hydrodynamic escape model

The initial-boundary value problems of (2.4), subject to the initial and boundary data near the sonic states, are as follows. (IBVP) ✩ ✬ ✫ ✬ ✪ Ut f ♣Uqx ✏ h♣xqg♣x, Uq, ♣x, tq P Π ✑ rxB, ✽q ✂ r0, ✽q, U♣x, 0q ✏ U0♣xq P Ω, ρ♣xB, tq ✏ ρB♣tq, m♣xB, tq ✏ mB♣tq, t → 0, (2.6) where U0♣xq ✏ ♣ρ0♣xq, m0♣xq, E0♣xqqT and Ω is an open domain centered at some sonic state Us ✑ ♣ρs, ms, Esq P T :✏ ✦ ♣ρ, m, Eq ✞ ✞ ✞ m ✏ ρ ❞ γ♣γ ✁ 1q ✁E ρ ✁ u2 2 ✠✮ . We impose the following conditions: (A1) ρ0♣xq, m0♣xq, E0♣xq and ρB♣tq, mB♣tq are bounded positive functions with small total variations, and there exists ̺ → 0 small enough such that ρ0♣xq ➙ ̺ and ρB♣tq ➙ ̺ for ♣x, tq P Π; (A2) min

t➙0 tmB♣tq✉ → ♣1 ǫqT.V.tU0♣xq✉ ♣1 ǫ ǫ2q2C for 0 ➔ ǫ ➔ 1 2 and

some positive constant C; (A3) q♣xq P W 1,1rxB, ✽q.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Glimm method

Glimm method are the main tool help us to study the initial-boundary problem, which are following steps: (i) The construction of approximate solutions for Riemann Problem; (ii) Wave interaction estimate between two Riemann problems; (iii) Showing that the convergence of approximate solution; (iv) Showing that the limit of approximate solution is indeed a weak solution.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Definitions

Definition 1 Consider the initial-boundary value problem in (2.6). We say that a measurable function U♣x, tq is a weak solution of (2.6) if ➻

x→xB ,t→0

tUφt f ♣Uqφx h♣xqg♣x, Uqφ✉ dxdt ➺ ✽

xB

U0♣xqφ♣x, 0qdx

• ➺ ✽

f ♣U♣xB, tqqφ♣xB, tqdt ✏ 0, for any test function φ P C 1

0 ♣Πq.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Definitions

Definition 2 Let Ω be a convex subset of R3. A pair ♣η♣Uq, ω♣Uqq is an entropy pair of (2.3) provided that η is convex on Ω and dω ✏ dηdf

• n

Ω. Furthermore, a measurable function U is an entropy solution of (2.6) if U is a weak solution of (2.6) and satisfies ➻

x→xB ,t→0

tηφt ωφx dη ☎ hgφ✉ dxdt ➺ ✽

xB

η♣U0♣xqqφ♣x, 0qdx

• ➺ ✽

ω♣U♣xB, tqqφ♣xB, tqdt ➙ 0, for every entropy pair ♣η♣Uq, ω♣Uqq and any positive test function φ P C 1

0 ♣Πq.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Riemann and Boundary-Riemann problems

The Riemann problem of (2.4), is given by RG♣x0, t0; gq : ✩ ✫ ✪ Ut f ♣Uqx ✏ h♣xqg♣x, Uq, ♣x, tq P D♣x0, t0q, U♣x, t0q ✏ ✧ UL, if x0 ✁ ∆x ➔ x ➔ x0, UR, if x0 ➔ x ➔ x0 ∆x. (3.1) and the boundary-Riemann problem of is given by BRG♣xB, t0; gq : ✩ ✫ ✪ Ut f ♣Uqx ✏ h♣xqg♣x, Uq, ♣x, tq P DB♣xB, t0q, U♣x, t0q ✏ UR, xB ↕ x ↕ xB ∆x, ρ♣xB, tq ✏ ρB♣tq, m♣xB, tq ✏ mB♣tq, t0 ➔ t ➔ t0 ∆t. (3.2) where m, U, f and g are given in (2.4), D♣x0, t0q :✏ t♣x, tq ⑤ ⑤x ✁ x0⑤ ➔ ∆x, t0 ➔ t ➔ t0 ∆t✉, (3.3) DB♣xB, t0q :✏ t♣x, tq ⑤ xB ➔ x ➔ xB ∆x, t0 ➔ t ➔ t0 ∆t✉. (3.4) and UL ✏ ♣ρL, mL, ELq, UR ✏ ♣ρR, mR, ERq are constant states.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Classical Riemann and Boundary-Riemann problems

System (2.4) is strictly hyperbolic system whose Jacobian flux df has three distinct real eigenvalues: λ1♣Uq :✏ u ✁ c♣Uq, λ2♣Uq :✏ u, λ3♣Uq :✏ u c♣Uq, where u ✏ m ρ , c♣Uq ✏ ❞ γ♣γ ✁ 1q ✁E ρ ✁ u2 2 ✠ . and the corresponding eigenvectors R1♣Uq ✏ ♣✁1, c ✁ u, uc ✁ HqT, R2♣Uq ✏ ♣1, u, 1

2u2qT,

R3♣Uq ✏ ♣1, c u, uc HqT, where H ✏ H♣Uq ✏ γE ρ ✁ γ ✁ 1 2 u2.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Classical Riemann and Boundary-Riemann problems

By setting source term g ✑ 0 in (3.1) and (3.2), we have the corresponding classical Riemann problem RC♣x0, t0q and boundary-Riemann problems BRC♣xB, t0q respectively. Theorem 3.1 (Lax) Consider classical Riemann problem RC♣x0, t0q and classical boundary-Riemann problem BRC♣xB, t0q. Let UL P Ω. Then there is a neighborhood r Ω ⑨ Ω of UL such that if UR P r Ω, then RC♣x0, t0q has a unique solution consisting of at most four constant states separated by shocks or rarefaction waves and contact

• discontinuity. Moreover, under condition

(E) Weak solution U ✏ ♣ρ, m, Eq is the entropy solution of BRC♣xB, t0q if U has the least total variation in ρ within all weak solutions of BRC♣xB, t0q. there exists ρB → 0, mB → 0 such that ♣ UB ✏ ♣ρB, mB, 0q P r Ω and admits a unique self-similar solution U satisfying U♣xB, tq ✏ ♣ UB.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Generalized solvers of Riemann and Boundary-Riemann problems

Once we obtain solutions r U of RC♣x0, t0q and BRC♣xB, t0q. Next, we construct the approximate solutions by U♣x, tq ✏ r U♣x, tq s U♣x, tq. Denote v♣xq ✏ GM

2x .

Consider the linearized system around r U with initial data s U♣x, 0q ✏ 0: ★ s Ut ♣A♣x, tq s Uqx ✏ B♣x, tq s U C♣x, tq, s U♣x, 0q ✏ 0, ♣x, tq P D♣x0, t0q. (3.5) where A♣x, tq ✏ df ♣ r Uq ✏ ✔ ✕ 1

γ✁3 2 ˜

u2 ♣3 ✁ γq˜ u γ ✁ 1

γ✁1 2 ˜

u3 ✁ ˜ u r H r H ✁ ♣γ ✁ 1q˜ u2 γ ˜ u ✜ ✢ , B♣x, tq ✏ h♣xqgU♣x, r Uq ✏ h♣xq ✔ ✕ 1 v 2 ✁ ˜ u2 2˜ u

γ✁1 2 ˜

u3 ✁ ˜ u r H r H ✁ ♣γ ✁ 1q˜ u2 ✁ v 2 γ ˜ u ✜ ✢ , C♣x, tq ✏ h♣xqg♣x, r Uq ✏ h♣xq ✔ ✕ ˜ m ˜ ρ♣˜ u2 v 2q ˜ m♣ r H v 2q ✁ xq

2

✜ ✢ .

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Generalized solvers of Riemann and Boundary-Riemann problems

We obtain the perturbation ¯ U given in the form ¯ U♣x, tq ✏ ♣S♣x, t, ˜ Uq ✁ I3q ˜ U ñ U♣x, tq ✏ S♣x, t, ˜ Uq ˜ U, (3.6) where S♣x, t, ˜ Uq ✏ ✔ ✕ eh˜

ut cosh♣hvtq

veh˜

ut sinh♣hvtq

eh˜

ut cosh♣hvtq

S31 S32 S33 ✜ ✢ , (3.7) with S31 ✏ ✁xq 2γ ˜ m ♣eγh˜

ut ✁ 1q ✁

v 3 v 2 ✁ ♣γ ✁ 1q2 ˜ u2 ♣veγh˜

ut ✁ veh˜ ut cosh♣hvtq ✁ ♣γ ✁ 1q˜

ueh˜

ut sinh♣hvtqq,

S32 ✏ ♣γ ✁ 1q˜ u♣v 2 ♣γ ✁ 1q˜ u2q 2♣v 2 ✁ ♣γ ✁ 1q2 ˜ u2q ♣eγh˜

ut ✁ eh˜ ut cosh♣hvtqq

v♣2v 2 ✁ ♣3γ ✁ 2q♣γ ✁ 1q˜ u2q 2♣v 2 ✁ ♣γ ✁ 1q2 ˜ u2q eh˜

ut sinh♣hvtq,

S33 ✏ ✁♣γ ✁ 1q♣v 2 ♣γ ✁ 1q˜ u2q v 2 ✁ ♣γ ✁ 1q2 ˜ u2 eγh˜

ut

• γv

v 2 ✁ ♣γ ✁ 1q2 ˜ u2 ♣veh˜

ut cosh♣hvtq ♣γ ✁ 1q˜

ueh˜

ut sinh♣hvtqq.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Residual of the approximate solutions for RG♣x0, t0; gq

Theorem 3.2 Let Γ ⑨ Π and U be the approximate solutions of the Riemann problem RG♣x0, t0; gq. Let φ P C 1

0 ♣Γq be a test function. Suppose U ✏ r

U s

• U. Then

the residuals of U can be estimated by R♣U, D♣x0, t0q, φq ✏ ➺ x0∆x

x0✁∆x

♣Uφq♣x, tq ✞ ✞ ✞

t✏t0∆t t✏t0

dx ➺ t0∆t

t0

♣f ♣Uqφq♣x, tq ✞ ✞ ✞

x✏x0∆x x✏x0✁∆xdt

O♣1q ✂ ♣∆tq2♣∆xq ♣∆tq3 ♣∆tq2 osc.

D♣x0,t0qt r

U✉ ✡ ⑥φ⑥✽, (3.8) where osc

Λ tw✉ means the oscillation of a function w in the set Λ, and D♣x0, t0q

are given in (3.3).

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Residual of the approximate solutions for BRG♣xB, t0; gq

Theorem 3.3 Let Γ ⑨ Π and UB be the approximate solutions of the boundary-Riemann problem BRG♣xB, t0; gq. Let φ P C 1

0 ♣Γq be a test function. Suppose

UB ✏ r UB s

• UB. Then the residuals of UB can be estimated by

R♣UB, D♣xB, t0q, φq ✏ ➺ xB ∆x

xB

♣UBφq♣x, tq ✞ ✞ ✞

t✏t0∆t t✏t0

dx ➺ t0∆t

t0

♣f ♣UBqφq♣x, tq ✞ ✞ ✞

x✏xB ∆x x✏xB

dt O♣1q ✂ ♣∆tq2♣∆xq ♣∆tq3 ♣∆tq2 osc

D♣xB ,t0qt r

UB✉ ✡ ⑥φ⑥✽, where osc

Λ tw✉ means the oscillation of a function w in the set Λ, and

D♣x0, t0q, D♣xB, t0q are given in (3.3) and (3.4) respectively.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Courant-Friedrichs-Lewy condition

Let us choose the spatial resolution ∆x → 0 and the temporal step ∆t → 0 small enough which satisfying the C-F-L condition λ✝ :✏ ∆x ∆t → sup

♣ρ,m,EqPΩ

★ m ρ ❞ γ♣γ ✁ 1q ✁E ρ ✁ m2 2ρ ✠✰ . (3.9) to make sure that the solutions in each Riemann cell will not interact with each

• ther.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Random choice process and mesh points

Let us discretize the domain Π ✑ rxB, ✽q ✂ r0, ✽q into xk ✏ xB k∆x, tn ✏ n∆t, k, n ✏ 0, 1, 2, ☎ ☎ ☎ , where ∆x, ∆t are small positive constants satisfying the C-F-L condition (??). The nth time strip Tn is denoted by Tn :✏ rxB, ✽q ✂ rtn, tn1q, n ✏ 0, 1, 2, ☎ ☎ ☎ . Suppose, that the approximate solution Uθ,∆x♣x, tq has been constructed in Tn, then we choose a random number θn P ♣✁1, 1q and define the initial data Un

k ✑ ♣ρn k, mn k, E n k q in Tn by

Un

k :✏ Uθ,∆x♣x2k θn∆x, t✁ n q, k ✏ 1, 2, ☎ ☎ ☎ .

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Random choice process and mesh points

x t U0 U0

1

xB “ x0 U0

2

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Random choice process and mesh points

x t U0 U0

1

xB “ x0 U0

2

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Random choice process and mesh points

x t t1 xB “ x0 x2 x4

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Random choice process and mesh points

x t t1 U1 xB “ x0 U1

1

U1

2

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Random choice process and mesh points

x t t1 t2 xB “ x0 U1 U1

1

U1

2

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Random choice process and mesh points

x t t1 t2 xB “ x0 U2

1

U2

2

t3 U2

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Random choice process and mesh points

x t t1 t2 xB “ x0 U2 U2

1

t3 p U2 U2

2

U2

3

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Random choice process and mesh points

x t t1 t2 xB “ x0 U3

1

U3

2

t3 t3 U3

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Mech curves

Let xk ✏ xB k∆x, tn ✏ n∆t, k, n ✏ 0, 1, 2, ☎ ☎ ☎ . In order to obtain the desired estimates, it is convenient to consider not horizontal lines, but rather, curves consisting of line segments joining ♣x2k, tnq to both ♣x2k2, tn✟1q. Let diamond denote ”Γk,n” denote the diamond region centered at ♣x2k, tnq with vertices N ✏ ♣x2k θn1∆x, tn1q, E ✏ ♣x2k θn∆x, tnq, W ✏ ♣x2k2 θn∆x, tnq, S ✏ ♣x2k θn✁1∆x, tn✁1q,

• r

N ✏ ♣x2k θn1∆x, tn1q, E ✏ ♣x2k✁2 θn∆x, tnq, W ✏ ♣x2k θn∆x, tnq, S ✏ ♣x2k θn✁1∆x, tn✁1q, where ✁1 ➔ θn ➔ 1 for all n are the random numbers.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Mech curves

• N

E W S

(Loading)

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Random choice process and mesh points

x t t1 t2 xB “ x0 pρ1

B, m1 B, E1 Bq

pρ0

B, m0 B, E0 Bq

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Wave interaction estimate

ε α β r UL UM r UR r UL ` s UL r UR ` s UR UL UR N W E S ε α β r UL UM r UR r UL ` s UL r UR ` s UR UL UR N W E S x2k tn

Classical wave strengths in the diamond region Γk,n.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Wave interaction estimate

Theorem 3.2 Let UL, UR and UM be constant states in some neighborhood contained in Ω with UL ✏ r UL s UL and UR ✏ r UR s

• UR. Suppose that the classical wave

strengths of the incoming generalized waves across the boundaries WS and SE

• f Γk,n are

α♣UM, r UL; xL, tn✁1q ✏ ♣α1, α2, α3q and β♣ r UR, UM; xR, tn✁1q ✏ ♣β1, β2, β3q, respectively, and the classical wave strength of the outgoing generalized waves across the boundary WNE is ε♣UR, UL; xM, tnq ✏ ♣ε1, ε2, ε3q, see figure above. Then there exist constants C ✶, C ✷

k,n such that

⑤ε⑤ ↕ ♣1✁ζ∆tq♣⑤α⑤⑤β⑤qC ✶D♣α, βqC ✷

k,n∆t∆xO♣1q♣∆tq3, as ⑤α⑤⑤β⑤ Ñ 0,

(3.10) where ζ ✏ ♣3✁γq⑤uM ⑤

xM

.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Wave interaction estimate near boundary

Case pIIq Case pIIIq p Un`1

B

p Un`1

B

ε2 ε2 β1 β1 α2 α2

p Un

B

p Un

B

tn`1

α3 α3

tn

Un`1

R

Un`1

R

Un

M

Un

M

Un`1

B

Un`1

B

Un

B

Un

B

Un

R

Un

R

ε0 ε0 α0 α0 ε3 ε3 Wave strengths in the region near the boundary x ✏ xB.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Wave interaction estimate near boundary

Theorem 3.3 Let ♣ Un

B, Un B as defined previously, and Un1 B

:✏ r Un1

B

s Un1

B

, where Un

M, Un R, Un1 R

are defined as in Theorem 3.1. respectively, see Figure 5. Let us also denote ♣ ♣ Un

B, Un Mq :✏ r♣ ♣

Un

B, Un B, Un Z, Un Mq④♣α0, α2, α3qs,

♣ ♣ Un1

B

, Un1

R

q :✏ r♣ ♣ Un1

B

, Un1

B

, Un1

Z

, Un1

R

q④♣ε0, ε2, ε3qs represent the solutions of BRG♣ ♣ Un

B, Un M; xB, tnq and BRG♣ ♣

Un1

B

, Un1

R

; xB, tn1q,

• respectively. Suppose that ♣Un

M, Un Rq :✏ r♣Un M, Un Rq④♣β1qs is the 1-wave of

RG♣x2, tnq right next to ♣ ♣ Un

B, Un Mq on the nth time strip, see Figure. 5. Then

there exists a constant C such that ⑤ε⑤ ↕ ⑤α β11⑤ C ✁ ➳

App

⑤αi⑤⑤β1⑤ ⑤β1⑤ ⑤ρn1

B

✁ ρn

B⑤ ⑤mn1 B

✁ mn

B⑤

✠ , (3.11) where 1 ✏ ♣1, 1, 1q.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Glimm functional

Given a mesh curve J, we define the following functionals L, and Q as L♣Jq :✏ ➳ t⑤αi⑤ : αi crosses J✉ K1 ✁ ⑤β1⑤ ➳

kPB♣Iq

lk

B

✠ , Q♣Jq :✏ ➳ t⑤αi⑤⑤αi✶⑤ : αi, αi✶ cross J and approach✉, ln

B :✏ ⑤ρn1 B

✁ ρn

B⑤ ⑤mn1 B

✁ mn

B⑤ ⑤E n1 B

✁ E n

B⑤.

We define F♣Jq :✏ L♣Jq KQ♣Jq where K, K1 → 1 are constants. B♣Jq :✏ tn : PxB ,n ✏ ♣xB, tn ∆t④2q P J✉, ln

B

is evaluated at the mesh point PxB ,n, and the presence of ⑤β1⑤ depends on β1 crosses J and locates in some boundary triangle region.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Decay of the Glimm functional

Let Jn, n ✏ 1, 2, ☎ ☎ ☎ denote the mesh curves located on the time strip Tn, which contains all mesh points ♣xk θn✁1∆x, tn✁1q at time t ✏ tn✁1. For suitable choice of constants K, K1, we can show that F♣Jnq ↕ F♣J1q ♣1 ǫ ǫ2q2C O♣1q♣∆xq2 ↕ ♣1 ǫqT.V.tU0✉ ♣1 ǫ ǫ2q2C O♣1q♣∆xq2, (3.12) mθ,∆x♣x, tnq ➙ mB♣tnq ✁ L♣Jnq ➙ min

t➙0 tmB♣tq✉ ✁ ♣1 ǫqL♣J1q ✁ ♣1 ǫ ǫ2q2C → 0.

In particular, uθ,∆x♣x, tnq → 0, ❅ x P rxB, ✽q. Therefore the inequality (3.12) leads to T.V.Jt r Uθ,∆x✉ ↕ O♣1qL♣Jq ↕ O♣1qF♣Jq ↕ ♣1 ǫqT.V.tU0♣xq✉ ♣1 ǫ ǫ2q2C O♣1q♣∆xq2 (3.13)

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Generalized Glimm scheme and its stability

Theorem 3.4 For fixed K and 0 ➔ ǫ ➔ 1

• 2. Let Uθ,∆x be an approximate solution of (2.6) by

the generalized Glimm scheme. Then under conditions (A1)-(A3), for any given constant state q U there exist positive constants d, depending on the radius r of Ω, such that if sup

xPrxB ,✽q

⑤U0♣xq ✁ q U⑤ ↕ r 2, T.V.tU0♣xq✉ ↕ d, sup

tPR ⑤mB♣tq ✁ q

m⑤ ↕ r 2 ♣1 ǫ ǫ2q2C hold for (2.6) wtih some constatn C. Then Uθ,∆x♣x, tq is well-defined for t ➙ 0 and sufficiently small ∆x → 0.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Generalized Glimm scheme and its stability

Theorem 3.4 (Cont.) Furthermore, Uθ,∆x♣x, tq has uniform total variation bound and satisfies the following proterties: (i) ⑥Uθ,∆x ✁ q U⑥L✽ ↕ r ♣1 ǫ ǫ2q2C. (ii) T.V.tUθ,∆x♣☎, tq✉ ↕ r 2 ♣1 ǫ ǫ2q2C. (iii) ➺ ✽

xB

⑤Uθ,∆x♣x, t2q ✁ Uθ,∆x♣x, t1q⑤dx ↕ O♣1q♣⑤t2 ✁ t1⑤ ∆tq. (iv) The velocity uθ,∆x♣x, tq → 0, ❅ ♣x, tq P Π. (v) The density ρθ,∆x♣x, tq ➙ ̺, ❅ ♣x, tq P Π, where ̺ is the constant in (A1).

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

The existence of global entropy solution for IBVP

Theorem 3.5 Assume that (A1)-(A3) hold. Let tUθ,∆x✉ be a family of approximate solutions for (2.6) by the GGS. Then there exist a subsequence tUθ,∆xi ✉ of tUθ,∆x✉ and a measurable function U such that (i) Uθ,∆xi ♣x, tq Ñ U♣x, tq in L1

loc as ∆xi Ñ 0;

(ii) for any continuous function f , we have f ♣x, t, Uθ,∆xi q Ñ f ♣x, t, Uq in L1

loc

as ∆xi Ñ 0. Theorem 3.6 Assume that tUθ,∆x✉ is the familie of approximate solutions for (2.6), which are constructed by the GGS with any random sequence θ and have uniformly (in θ and ∆x) bounded variation. Then there exists a subsequence t∆xi✉ converging to zero such that, for almost random sequence θ, U♣x, tq ✏ lim

∆xi Ñ0 Uθ,∆xi ♣x, tq,

which is an entropy solution to (2.6) respectively.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Estimation of the approximate solutions

For any ♣x, tq P Π, let k ✏ t x

∆x ✉, n ✏ t t ∆t ✉ 1 and let Dk,n denote the

Riemann cell containing the point ♣x, tq. By (3.7) and the random choice process, the approximate solution in nth time step satisfies ⑤Uθ,∆x♣x, n∆tq ✁ Uθ,∆x♣x, ♣n ✁ 1q∆tq⑤ ↕ ✞ ✞ S♣y, ♣n ✁ 1q∆t, r Uθ,∆x♣y, ♣n ✁ 1q∆tqq ✁ I ✟ r Uθ,∆x♣y, ♣n ✁ 1q∆tq ✞ ✞ ⑤ r Uθ,∆x♣y, ♣n ✁ 1q∆tq ✁ Uθ,∆x♣x, ♣n ✁ 1q∆tq⑤, for some y P Dk,n such that Uθ,∆x♣x, n∆tq ✏ S♣y, ♣n ✁ 1q∆t, r Uθ,∆x♣y, ♣n ✁ 1q∆tqq r Uθ,∆x♣y, ♣n ✁ 1q∆tq. Lemma 4.1 Let U♣x, tq ✏ ♣ρ♣x, tq, m♣x, tq, E♣x, tqqT be the solution of (2.6) constructed in Theorem 3.5., with initial data U0♣xq ✏ ♣ρ0♣xq, m0♣xq, E0♣xqqT and boundary data ♣ρB♣tq, mB♣tqqT. Then for any ♣x, tq P Π, we have ⑤ρ♣x, tq ✁ ρ0♣xq⑤ ↕ T.V.tρ0♣xq✉ T.V.tρB♣tq✉, ⑤m♣x, tq ✁ m0♣xq⑤ ↕ T.V.tm0♣xq✉ T.V.tmB♣tq✉, ⑤E♣x, tq ✁ E0♣xq⑤ ↕ T.V.tE0♣xq✉ T.V.tE♣xB, tq✉ 2λ✁1

✝ ⑥q⑥L1rxB ,✽q.

(4.1)

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Estimation of the approximate solutions

Theorem 4.2 Suppose the transonic initial data U0 ✏ ♣ρ0, m0, E0qT satisfying ♣A1q ✒ ♣A3q where ρ0, E0 are decreasing, m0 is increasing and Kn♣xB, 0q ➔ 1, that is, x2

Bρ0♣xBqc0♣xBq2 → γGMpm

❄ 2 τ . (4.2) Let U ✏ ♣ρ, m, EqT be the solution of (2.6) constructed by Theorem 3.5., then there exists x✝ P ♣xB, ✽q and Σ1 ✑ rxB, x✝s ✂ r0, ✽q such that Kn♣x, tq ↕ 1, ❅ ♣x, tq P Σ1.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Estimation of the approximate solutions

Recall that the Mach number of U is defined as Ma♣Uq :✏ ⑤u⑤ c Theorem 4.3 Suppose the transonic initial data U0 ✏ ♣ρ0, m0, E0qT satisfying ♣A1q ✒ ♣A3q where ρ0, E0 are decreasing, m0 is increasing and u0♣xBq ➔ c0♣xBq. Let U ✏ ♣ρ, m, EqT be the solution of (2.6) constructed by Theorem 3.5., then there exists x✝✝ P ♣xB, ✽q and Σ2 ✑ rxB, x✝✝s ✂ r0, ✽q such that the characteristic speeds of the solution U♣x, tq in Π③Σ2 are positive. Finally, according to Theorem 4.2. and Theorem 4.3. we define the hydrodynamic region of HEP (2.3) by Σ ✑ Σ1 ❳ Σ2. The wave speeds of the solution U♣x, tq, constructed in Theorem 3.5., are positive in the region Π③Σ. Meanwhile, the Knudsen number Kn♣x, Uq ↕ 1 in the region Σ. Therefore, we

• btain an entropy solution U♣x, tq of (2.3) has both mathematical and physical

significance in the hydrodynamic region Σ.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

• 1. Time asymptotic stability for the smooth transonic solution.
• 2. Hot Jupiter Model: HEP with tidal force effect (extrasolar planet).
• 3. Two species problem (Isotope ratio).
• 4. A better model for escape phenomenon (Hybrid kinetic-fluid model).

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

• 1. Time asymptotic stability for the smooth transonic solution.
• 2. Hot Jupiter Model: HEP with tidal force effect (extrasolar planet).
• 3. Two species problem (Isotope ratio).
• 4. A better model for escape phenomenon (Hybrid kinetic-fluid model).

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

• 1. Time asymptotic stability for the smooth transonic solution.
• 2. Hot Jupiter Model: HEP with tidal force effect (extrasolar planet).
• 3. Two species problem (Isotope ratio).
• 4. A better model for escape phenomenon (Hybrid kinetic-fluid model).

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

• 1. Time asymptotic stability for the smooth transonic solution.
• 2. Hot Jupiter Model: HEP with tidal force effect (extrasolar planet).
• 3. Two species problem (Isotope ratio).
• 4. A better model for escape phenomenon (Hybrid kinetic-fluid model).

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region

Thank you for your attention.

Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic