Topological interpretation of Lax conditions on the wave manifold

for 2x2 quadratic systems of conservation laws

C.Eschenazi UFMGCFB Palmeira PUC-Rio

• References:• [P].Palmeira CFB; Line fields given by eigenspaces of derivatives

of maps from the plane to itself; Proceedings of the VIth International Colloquium on Differential Geometry of Santiago de Compostela (1988) p.177-205.

[MP].Marchesin D, Palmeira CFB; Topology of elementary waves for mixed systems of conservation laws, Journal of Dynamics and Differential Equations, vol. 6, n.3 ,(1994), p.427-446.

[EP]. Eschenazi CS, Palmeira CFB; The structure of composite rarefaction-shock foliations for quadratic systems of conservation laws. Matemática Contemporânea , vol 22,n.1 p113-140, (2002).

[AEMP].Azevedo AV, Eschenazi CS, Marchesin D, Palmeira CFB; Topological construction of non classical Riemann Solutions, (to appear).

• Ut +F(U)x =0 (1)

• U=U(x,t) R2 F=(f, g)

• f(u,v)= v2/2 +(b+1)u2/2 + a1u+a2v. g(u,v)= uv+a3u+a4v.

• Normal form from [P] , [MP], [EP], [AEMP]

• Riemann Problem:

• (1) + U(x,0)= U1 if x<0

• U2 if 0<x

• Solutions must be invariant by (x,t) (ax,at) a>0.

• Goals:

• 1.As U1 moves in R2, count intersections between the shock curve and the sonic surface , in the wave manifold.

• 2.As U1 moves in the hyperbolic region, characterize the arcs of shock curve satisfying the Lax criteria.

• Solutions can be obtained as successions of shocks, rarefactions and composites.

• Shock: U(x,t)= U1 if x<st• U2 if x>st

• with F(U1)-F(U2)=s(U1-U2) (2)

• + Lax conditions: inequalities relating s and the eigenvalues of DF.

• Wave Manifold in R4

• R4 ={(U,U’): U and U’ in R2}

• F(U1)-F(U2)=s(U1-U2) is equivalent to• : f(u,v)-f(u’,v’)=s(u-u’), (3)• g(u,v)-g(u’,v’)=s(v-v’).

• Eliminating s:– [f(u,v)-f(u’,v’)](v-v’) = [g(u,v)-g(u’,v’)](u-u’).

• 2-plane U=U’, and 3 dim. manifold M.

• Solutions of the RP become curves in M.

• M is difeomorphic to BxR, where B is the Moebius band.

• D diagonal U=U’ in R4

• C= M∩D characteristic surface, cylinder.

• C projects in R2 (U plane or U’plane) as the outside of an ellipse E (hyperbolic region). Inside of E= elliptic region.

• Given U =(U1,U2) in M,

• f(u,v)-f(u’,v’)=s(u-u’) and g(u,v)-g(u’,v’)=s(v-v’) • define s as a real valued function in M.

• Given U0 = (u0,v0),

• shock curve by U0 = sh (U0 )= {(U,U’) in M : U=U0}

• shock’ curve by U0 =sh’(U0) = {(U,U’) in M : U’=U0}

• Given U =(U1,U2) in M, sh(U) =sh (U1) and sh’(U) =sh’(U2)

• Sonic surface:• Son= set of critical points of s restricted to shock curves.

• Sonic’ surface:• Son’= set of critical points of s restricted to shock’ curves.

• Given U = (U,U) in C, let λs(U) ≤ λf(U) be the eigenvalues of DF(U).

• If U in E, λs(U) = λf(U), let π: C R2 , π(U,U) = U

• π -1(E)= closed curve, not bounding a disk in C. It separates C into 2 components: Cs and Cf .

• In Cs , s= λs . In Cf , s= λf .

• Given U in M , we denote:• Us= sh(U)∩ C s . Uf= sh(U)∩ C f

• Us’= sh’(U)∩ C s Uf’= sh’(U)∩ C f

• Lax conditions become:

• s(U) < s(Us) and

• s(Us’) < s(U) < s(Uf’)

• Study intersections of sh(U) with C, Son, Son’ as U moves in R2.

• C: U inside E (elliptic region): no intersection• .• U on E: one tangency point.• U out of E(hyperbolic region): 2 points (transversal) • • Son’: studied in [EP].• 0, 2, 4 points, depending on b in (-∞,-1), (-1,0), (0,1) or (1, +∞) • and the position of U in the plane:

I b<-1 : • elliptic region no intersection.• ellipse 2 distinct tangency points• hyperbolic region 4 points.

• I I -1< b < 0•

• I I I 0< b <1 and IV 1< b

The straight lines are the projections of Son∩Son’ – C. It does not exist in case I.

• Son:

• The ellipse is replaced by an open curve, the straight lines are replaced by the projections of the bifurcation lines (lines where the shock curve family in M is singular ).

• I b < -1 • Intersections:• No point in the

triangle, • 2 points in the outer

regions, • 4 points between the

curves and the asymptotes

• II -1 < b < 0

• Also 4 Intersection points between the curve and the asymptote

• III 0 < b < 1 •

• Also 4 Intersection points between the curve and the asymptote

• IV 1< b • Intersections:• Above the curve:2• Below the asymp. 2• In the loop: 0• Between the curve

and the asymp. 4

• Question: What can we tell about Lax conditions as U moves in the hyperbolic region?

• In [AEMP] it is shown that Lax conditions are satisfied in arcs with one extremity in either Cs or Son’, and the other either in Son or at infinity. There is always an arc starting in Cs, called local arc.

• In case IV we have :

• a : L u • b : L u • c: L u • d: L u , n b• e: L u , n n• f: L b

• tri: L u , n b • b tri: L b , n b • rgb: L b• rgrb: L b