On the Dependence of Euler Equations on Physical Parameters - PowerPoint PPT Presentation

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou On the Dependence of Euler Equations on Physical Parameters Cleopatra Christoforou Department of Mathematics , University of Houston Joint Work with: Gui-Qiang Chen, Northwestern University Yongqian Zhang, Fudan University 12th International Conference on Hyperbolic Problems: Theory, Numerics, Applications June 9-13, 2008

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou OUTLINE: 1 Introduction/ Motivation 2 Our approach 3 Applications to Euler Equations

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou Intro Hyperbolic Systems of Conservation Laws in one-space dimension: � ∂ t U + ∂ x F ( U ) = 0 x ∈ R (1) U (0 , x ) = U 0 , where U = U ( t , x ) ∈ R n is the conserved quantity and F : R n → R n smooth flux . Admissible/Entropy weak solution: U ( t , x ) in BV . in D ′ ∂ t η ( U ) + ∂ x q ( U ) ≤ 0

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou Motivation Examples: Isothermal Euler equations ∂ t ρ + ∂ x ( ρ u ) = 0 (2) ∂ t ( ρ u ) + ∂ x ( ρ u 2 + κ 2 ρ ) = 0 where ρ is the density and u is the velocity of the fluid. Isentropic Euler equations ∂ t ρ + ∂ x ( ρ u ) = 0 (3) ∂ t ( ρ u ) + ∂ x ( ρ u 2 + κ 2 ρ γ ) = 0 γ > 1 where γ > 1 is the adiabatic exponent. Relativistic Euler equations: � ( p + ρ c 2 ) � � � u 2 u ( p + ρ c 2 ) ∂ t c 2 − u 2 + ρ + ∂ x = 0 c 2 − u 2 c 2 � � � � u 2 u ( p + ρ c 2 ) ( p + ρ c 2 ) ∂ t + ∂ x c 2 − u 2 + p = 0 , c 2 − u 2 (4) where c < ∞ is the speed of light.

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou Motivation Question: As γ → 1 and c → ∞ , can we pass from the isentropic to the isothermal and from the relativistic to the classical ?

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou Motivation Question: As γ → 1 and c → ∞ , can we pass from the isentropic to the isothermal and from the relativistic to the classical ? In general, the Question is: How do the admissible weak solutions depend on the physical parameters?

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou Approach Systems of Conservation Laws in one-space dimension: � ∂ t W µ ( U ) + ∂ x F µ ( U ) = 0 x ∈ R (5) U (0 , x ) = U 0 , where W µ , F µ : R n → R n are smooth functions that depend on a parameter vector µ = ( µ 1 , . . . , µ k ), µ i ∈ [0 , µ 0 ], for i = 1 , . . . , k . and W 0 ( U ) = U . Formulate an effective approach to establish L 1 estimates of the type: � U µ ( t ) − U ( t ) � L 1 ≤ C TV { U 0 } · t · � µ � (6) • U µ is the entropy weak solution to (5) for µ � = 0 constructed by the front tracking method. • U ( t ) := S t U 0 , S is the Lipschitz Standard Riemann Semigroup associated with (5) for µ = 0. • � µ � is the magnitute of the parameter vector µ .

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou Approach Error estimate Let S be a Lipschitz continuous semigroup: S : D × [0 , ∞ ) �→ D , � t �S h w ( τ ) − w ( τ + h ) � L 1 �S t w (0) − w ( t ) � L 1 ≤ L lim inf d τ, (7) h h → 0+ 0 where L is the Lipschitz constant of the semigroup and w ( τ ) ∈ D . The above inequality appears extensively in the theory of front tracking method: e.g. (i) the entropy weak solution by front tracking coincides with the trajectory of the semigroup S if the semigroup exists, (ii) uniqueness within the class of viscosity solutions, etc.... References: Bressan et al.

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou Approach Front-Tracking Method For δ > 0, let U δ,µ be the δ -approximate solution to � ∂ t W µ ( U ) + ∂ x F µ ( U ) = 0 for µ � = 0 U (0 , x ) = U 0 , (i) U δ 0 piecewise constant, � U δ 0 − U 0 � L 1 < δ . (ii) U δ,µ are globally defined piecewice constant functions with finite number of discontinuities. (iii) The discontinuities are of three types: • shock fronts, • rarefaction fronts with strength less than δ , • non-physical fronts with total strength � | α | < δ . (iv) U δ,µ → U µ in L 1 loc as δ → 0+. References: Bressan, Dafermos, DiPerna, Holden–Risebro.

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou Approach Description Approach Apply the error estimate on w = U δ,µ : � t �S h U δ,µ ( τ ) − U δ,µ ( τ + h ) � L 1 �S t U δ 0 − U δ,µ ( t ) � L 1 ≤ L lim inf d τ, h h → 0+ 0 The aim is to estimate �S h U δ,µ ( τ ) − U δ,µ ( τ + h ) � L 1 (8) which is equivalent to solving the Riemann problem of (5) when µ = 0 for τ ≤ t ≤ τ + h with data � U δ,µ ( τ, x ) x < ¯ x ( U L , U R ) = (9) U δ,µ ( τ, x ) x > ¯ x over each front of U δ,µ at time τ , i.e. find S h ( U L , U R ). Then compare it with the same front of U δ,µ ( τ + h ). We solve the Riemann problem at all non-interaction times of U δ,µ .

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou Approach Description If we can show: � ¯ x + a |S h U δ,µ ( τ ) − U δ,µ ( τ + h ) | dx = O (1) h ( � µ � | U L − U R | + δ ) , ¯ x − a (10) then summing over all fronts of U δ,µ ( τ ), �S t U δ,µ (0) − U δ,µ ( t ) � L 1 ≤ � t � ¯ � x + a 1 |S h U δ,µ ( τ ) − U δ,µ ( τ + h ) | dx d τ ≤ L h 0 ¯ x − a fronts x =¯ x ( τ ) (11) (12)

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou Approach Description If we can show: � ¯ x + a |S h U δ,µ ( τ ) − U δ,µ ( τ + h ) | dx = O (1) h ( � µ � | U L − U R | + δ ) , ¯ x − a (10) then summing over all fronts of U δ,µ ( τ ), �S t U δ,µ (0) − U δ,µ ( t ) � L 1 ≤ � t � ¯ � x + a 1 |S h U δ,µ ( τ ) − U δ,µ ( τ + h ) | dx d τ ≤ L h 0 x − a ¯ fronts x =¯ x ( τ ) � � t � TVU δ,µ ( τ ) d τ + δ t = O (1) � µ � 0 (11)

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou Approach Description If we can show: � ¯ x + a |S h U δ,µ ( τ ) − U δ,µ ( τ + h ) | dx = O (1) h ( � µ � | U L − U R | + δ ) , ¯ x − a (10) then summing over all fronts of U δ,µ ( τ ), �S t U δ,µ (0) − U δ,µ ( t ) � L 1 ≤ � t � ¯ � x + a 1 |S h U δ,µ ( τ ) − U δ,µ ( τ + h ) | dx d τ ≤ L h 0 ¯ x − a fronts x =¯ x ( τ ) � � t � TVU δ,µ ( τ ) d τ + δ t = O (1) � µ � 0 = O (1)( � µ � TV { U 0 } + δ ) · t (11)

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou Approach Description If we can show: � ¯ x + a |S h U δ,µ ( τ ) − U δ,µ ( τ + h ) | dx = O (1) h ( � µ � | U L − U R | + δ ) , ¯ x − a (10) then summing over all fronts of U δ,µ ( τ ), �S t U δ,µ (0) − U δ,µ ( t ) � L 1 ≤ � t � ¯ � x + a 1 |S h U δ,µ ( τ ) − U δ,µ ( τ + h ) | dx d τ ≤ L h 0 ¯ x − a fronts x =¯ x ( τ ) � � t � TVU δ,µ ( τ ) d τ + δ t = O (1) � µ � 0 = O (1)( � µ � TV { U 0 } + δ ) · t (11) As δ → 0+, we obtain � U ( t ) − U µ ( t ) � L 1 = O (1) TV { U 0 } · t � µ � (12) where U := S t U 0 is the entropy weak solution to (5) for µ = 0.

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou Approach Description Remarks Note that U := S t U 0 is unique within the class of viscosity solutions. (Bressan et al). Thus, as µ → 0 U µ → S t U 0 L 1 . in • Temple: existence using that the nonlinear functional in Glimm’s scheme depends on the properties of the equations at µ = 0. • Bianchini and Colombo: consider S F , S G and show S F is Lipschitz w.r.t. the C 0 − norm of DF .

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou Applications: Isothermal Euler equations Isothermal Euler equations: ∂ t ρ + ∂ x ( ρ u ) = 0 (13) ∂ t ( ρ u ) + ∂ x ( ρ u 2 + κ 2 ρ ) = 0 where ρ is the density and u is the velocity of the fluid. • Nishida [1968] : Existence of entropy solution for large initial data via the Glimm’s scheme. • Colombo-Risebro [1998] : Construction of the Standard Riemann Semigroup for large initial data. Existence, stability and uniqueness within viscosity solutions. ⋆ Let S be the Lipschitz Standard Riemann Semigroup generated by Isothermal Euler Equations (13).

On the Dependence of Euler Equations on Physical Parameters, Cleopatra Christoforou Applications: Isothermal Euler equations Isentropic → Isothermal Euler equations 1. Isentropic Euler Equations: ∂ t ρ + ∂ x ( ρ u ) = 0 (14) ∂ t ( ρ u ) + ∂ x ( ρ u 2 + p ( ρ )) = 0 of a perfect polytropic fluid p ( ρ ) = κ 2 ρ γ , where γ > 1 is the adiabatic exponent. Existence results: when ( γ − 1) TV { U 0 } < N (i) Nishida-Smoller by Glimm’s scheme, [1973] (ii) Asakura by the front tracking method [2005].