# Riemann Problems for Two Dimensional Euler Systems ... 2011/11/16 آ The so-called Riemann...

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SIAM Conference on Analysis of PDEs (PD11),

November 14-17, 2011,

San Diego Marriott Mission Valley, San Diego

MS74: Multidimensional Conservation Laws and

Related Applications - Part III

Riemann Problems for Two

Dimensional Euler Systems

Yuxi Zheng

Yeshiva University

Euler Equation

ρt +∇ · (ρu) = 0, (ρu)t +∇ · (ρu⊗ u+ pI) = 0, (ρE)t +∇ · (ρEu+ pu) = 0.

(1)

Cauchy problems: Open

We look for self-similar solutions

(u, v, p, ρ)(ξ, η)

for

ξ = x

t , η =

y

t .

The Euler system becomes the self-similar form

−ξρξ − ηρη + (ρu)ξ + (ρv)η = 0 −ξ(ρu)ξ − η(ρu)η + (ρu2 + p)ξ + (ρuv)η = 0 −ξ(ρv)ξ − η(ρv)η + (ρuv)ξ + (ρv2 + p)η = 0 −ξ(ρE)ξ − η(ρE)η + (ρuE + pu)ξ + (ρvE + pv)η = 0.

(2)

Any initial data has to be

(u, v, p, ρ)(0, x, y) = Independent of radius

The so-called Riemann problem.

It is a boundary-value problem: Initial data be-

comes boundary data at inifinity.

Riemann problems: 2-D

Figure 1: Four−constant Riemann problem

constant state 3

constant state 1

constant state 4

x

y

constant state 2

State 2

State 1

The characteristics are tangent to the sonic curve. of four forward rarefaction waves in the 1990 paper.

η

P

Q su

bs on

ic

a

� � � �

ξ

Figure 2. Proposed structure for the interaction

Z

�� �� �� ��

����

Numerical simulations:

1. Tong Zhang, Guiqiang Chen, Shuli Yang;

2. Lax Liu;

3. Schulz-Rinne, Collins, and Glaz;

4. Tadmore , Kurganov;

5. Glimm, J., Xiaomei Ji; Jiequan Li, Xiaolin Li;

Peng Zhang; Tong Zhang; Yuxi Zheng;

6. W.C. Sheng et al

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C on

fig ur

at io

n 1

Density T=0.2

Figure 3: Density contour lines, courtesy of Lax and Liu.

4R (Two forward, two backward)

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+1.0

First, two wave interaction:

Hodograph method: paper with Jiequan Li, ARMA

(vol193(2009)).

Direct method (with Xiao Chen)(IUMJ, 59(2010))

with Jiequan Li and Zhicheng Yang, (JDE (2010)).

Then: Simple wave (With Tong Zhang and Jiequan

Li: CMP, v.267(2006))

Then: Semi-hyperbolic patches

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+1.0

Semi-hyperbolic patch in channel flow

1. With Kyungwoo Song, on pressure gradient

system, DCDS (2009)

2. With Mingjie Li (Ph.D student from CNU)

ARMA 2011

Recently, in Tianyou Zhang’s thesis, we estab-

lish regularity via a new existence approach for

the pressure gradient system

ut + px = 0, vt + py = 0, pt + pux + pvy = 0,

Or

( pt

p )t − pxx − pyy = 0.

In the self- similar plane

(p− ξ2)pξξ − 2ξηpξη + (p− η2)pηη +1p(ξpξ + ηpη)

2 − 2(ξpξ + ηpη) = 0. In polar coordinates,

(p− r2)prr + p

r2 pθθ +

p

r pr +

1

p (rpr)

2 − 2rpr = 0.

Sonic line.: r2 − p = 0;

Hyperbolic where r2 − p > 0; Elliptic in the region r2 − p < 0.

D’Alembert type decomposition:

∂+∂−p = q(∂+p− ∂−p)∂−p, ∂−∂+p = q(∂−p− ∂+p)∂+p,

where

∂± = ∂θ ± λ−1∂r

λ =

√

p

r2(r2 − p)

q = r2

4p(r2 − p) .

Introducing R = ∂+p, S = ∂−p,W = (p, R, S),

pθ = R+S 2 ,

Rθ − λ−1Rr = q(S −R)R, Sθ + λ

−1Sr = q(R− S)S.

For the semi-hyperbolic patch solution constructed

in Song and Zheng, pr may blow up. We do not

expect better regularity. In numerical simulations,

demarcation line between hyperbolic region and

elliptic region is composed of the sonic line and

a shock curve. Although the sonic curve changes

dramatically close to the shock, for a large portion

away from connection point with shock it is quite

smooth and pressure p changes smoothly across

it. We mainly focus on this part to propose our

problem for today’s talk.

Problem 1: Given a sonic curve r = f(θ), θa <

θ < θb, with

f ′ ≥ γ > 0,

denoted by Γ. We look for solutions to the PGE in

a class of functions with given first-order deriva-

tive pθ(f −1(r), r) = a0(r)2 on Γ. We assume

a20 ≥ α > 0.

As a result pr = 2f − pθf ′ ≡ a1(r) is also known on the sonic curve. We assume pθθ is bounded at the

sonic line.

We show that the precise boundary conditions

are

W |Γ = (r2, a20, a20),

(pθ, √

r2 − pRθ, √

r2 − pSθ)|Γ = (a20,− a20a1 2 ,

a20a1 2 ).

Using the hyperbolicity quantity t = √

r2 − p and r as new independent variables, and introducing

new dependent variables

U(t, r) = R(t, r)− a20(r)− a1(r)t, V (t, r) = S(t, r)− a20(r) + a1(r)t,

The system is transformed into

Ut + 2tλ−1

S+2rλ−1Ur = b1(U, V ),

Vt − 2tλ −1

R−2rλ−1Vr = b2(U, V ),

where b1(U, V ) and b2(U, V ) contain a singular

term U−Vt and depend on a0 and a1.

b1(U, V ) = 1 2

(

U−V t

)

+

(

2t2qR S+2rλ−1 −

1 2

)

(

U−V t +2a1

)

−f(a0, a1, V ), b2(U, V ) =

1 2

(

V−U t

)

+

(

2t2qS R−2rλ−1 −

1 2

)

(

V−U t − 2a1

)

+g(a0, a1, U),

and

f(a0, a1, V ) = 2tλ−1

S+2rλ−1 (

∂r(a20 + ta1) )

,

g(a0, a1, U) = 2tλ−1

R−2rλ−1 (

∂r(a20 − ta1) )

.

The boundary conditions are

U(0, r) = V (0, r) = Ut(0, r) = Vt(0, r) = 0,

(ra ≤ r ≤ rb.)

By establishing an iteration scheme we establish

the existence of a solution by showing that itera-

tion sequence converges under a weighted metric

through characteristic analysis. The main result

is stated as follows:

Theorem. Assume a0 and a1 are in C 3([ra, rb])

and D(δ0) is a strong determinate domain, then

there exists a δ > 0 such that on D(δ), the degen-

erate hyperbolic problem has a classical solution

in S.

Proof.

Now λ−1(t, r) = tr√ r2−t2

and q(t, r) = r 2

4(r2−t2)t2.

Let

D(δ0) := {(t, r)|0 ≤ t ≤ δ0, r1(t) ≤ r ≤ r2(t)},

where r1(t), r2(t) are continuously differentiable

on 0 ≤ t ≤ δ0, r1(0) = ra, r2(0) = rb and r1 < r2 for 0 ≤ t ≤ δ0.

Definition a strong determinate domain D(δ0).

Solutions of the Cauchy problem will be ob-

tained within the class S consisting of all continu- ously differentiable functions F =

(

f1 f2

)

: D(δ) →

R 2 satisfying the following properties:

(S1) fi(0, r) = ∂tfi(0, r) = 0 (i = 1,2),

(S2) ∥

∥

∥

f1 t2

∥

∥

∥

0 + ∥

∥

∥

f2 t2

∥

∥

∥

0 ≤ M ,

(S3) ∥

∥

∥

∂rf1 t2

∥

∥

∥

0 + ∥

∥

∥

∂rf2 t2

∥

∥

∥

0 ≤ M ,

(S4) Fr is Lipschtz continuous with respect to r

with

∥

∥

∥

∥

∂2rrf1 t2

∥

∥

∥

∥

0 +

∥

∥

∥

∥

∂2rrf2 t2

∥

∥

∥

∥

0 ≤ M ,

where || · ||0 is the superem norm over domain D(δ) and 0 < δ ≤ δ0. We will denote W the larger class containing only continuous functions

on D(δ) which satisfy the first two conditions

(S1) and (S2). Both S and W are subsets of C0(D(δ);R2).

We define a new weighted metric on S and W :

d(F,G) :=

∥

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