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Journal of Hyperbolic Differential Equations Vol. 5, No. 2 (2008) 295­315 c World Scientific Publishing Company

LAX SHOCKS IN MIXED-TYPE SYSTEMS OF CONSERVATION LAWS

ALEXEI A. MAILYBAEV Institute of Mechanics Moscow State Lomonosov University Michurinsky pr. 1, 119192 Moscow, Russia mailybaev@imec.msu.ru DAN MARCHESIN Instituto Nacional de Matematica Pura e Aplicada ­ IMPA ´ Estrada Dona Castorina, 110 22460-320 Rio de Janeiro RJ, Brazil marchesi@impa.br Received 25 Jan. 2007 Accepted 18 Feb. 2008 Communicated by Gui-Qiang Chen Abstract. Small amplitude shocks involving a state with complex characteristic speeds arise in mixed-type systems of two or more conservation laws. We study such shocks in detail in the generic case, when they appear near the codimension-1 elliptic boundary. Then we classify all exceptional codimension-2 states on smooth parts of the elliptic boundary. Asymptotic formulae describing shock curves near regular and exceptional states are derived. The type of singularity at the exceptional point depends on the second and third derivatives of the flux function. The main application is understanding the structure of small amplitude Riemann solutions where one of the initial states lies in the elliptic region. Keywords : Mixed-type system of conservation laws; shock wave; elliptic region; Riemann problem; singularity; exceptional point. Mathematics Sub ject Classification 2000: 35M10, 76L05

1. Intro duction Sho ck waves are responsible for the mathematical interest of the theory of nonlinear conservation laws. When only strictly hyperbolic states (i.e. all characteristic speeds are real and distinct) are involved and under additional technical hypotheses, sho ck waves are usually extremely stable and well behaved, see e.g. [25]. A number of mo dels studied recently contain elliptic regions, see [3­5, 12, 15, 22, 23] and the review in [19]. In these models, some stable shock waves typically contain points near
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the boundary of the elliptic region. Hugoniot curves near regular points of elliptic boundaries were studied in [12] for a specific system with nonhomogeneous quadratic flux, and in [14] for equations reduced to normal form. Related Riemann solutions were discussed in [5, 7, 12, 21]. In this work, we study shock curves for systems of m conservation laws in the neighborho o d of the elliptic boundary, which is defined as the surface where two characteristic speeds coincide. The lo cal structure of sho ck waves is described near regular and exceptional states of the elliptic boundary. Here the exceptional states are the points on the elliptic boundary where the eigenvector is tangent to the boundary. Exceptional points typically exist on the boundaries of elliptic regions, for example, in systems with nonhomogeneous quadratic fluxes. The structure of sho ck wave is very special near exceptional points. Note that the structure of rarefaction waves is also singular near exceptional points, as shown in [16]. In this paper, the classification of exceptional points according to the lo cal behavior of sho ck curves is given. Explicit formulae providing qualitative and quantitative description of shock curve singularities are derived. These formulae use eigenvectors and asso ciated vectors of coincident characteristic speeds as well as the derivatives up to third order of the flux function at the point of the elliptic boundary. The importance of the third derivative of the flux function is remarkable at exceptional states. As a result, the quadratic approximation of the flux function is insufficient for lo cal analysis of shock curves near such states. Our metho d is based on the Liapunov­Schmidt reduction of Rankine­Hugoniot equation written in a specific ("blow-up") coordinate system. This co ordinate system is similar to that used for constructing the wave manifold in [13]. The Lax conditions for sho cks are checked by using bifurcation theory of multiple eigenvalues [24]. The paper is organized as follows. Section 2 contains general information on shock waves. Section 3 studies shock waves with states near regular points of the elliptic boundary. In Sec. 4, a similar analysis is carried out near exceptional points of the elliptic boundary. Section 5 gives a numerical example of a Riemann solution with one initial state inside the elliptic region. The paper ends with a short discussion. 2. Sho ck Waves in Conservation Laws Let us consider a system of m conservation laws in one space dimension x: F (U ) U + = 0, (2.1) t x where U (x, t) Rm is a vector of conserved quantities, and F Rm is a flux function smoothly dependent on U . Let A(U ) = F / U be the m â m Jacobian matrix of the flux function F (U ). Then the system is (strictly) hyperbolic at U if all the eigenvalues of the matrix A(U ) are real (and distinct). The elliptic region consists of the points U where the matrix A(U ) possesses complex eigenvalues.


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In the region of strict hyperbolicity, we list the eigenvalues of A(U ) (characteristic speeds) in increasing order as 1 (U ) < 2 (U ) < ··· < m (U ). A sho ck wave is a discontinuity in a (weak) solution of system (2.1) at x = xs (t). It consists of a left state U- = limxxs (t)-0 U (x, t) and a right state U+ = limxxs (t)+0 U (x, t); the shock speed is = dxs /dt. The left and right states and the speed of the sho ck must satisfy the Rankine­Hugoniot condition F (U+ ) - F (U- ) = (U+ - U- ), (2.2)

following from the requirement that the shock is a weak solution of (2.1) [25]. We will consider as admissible the shocks satisfying the extended Lax conditions 1-sho ck : < Re 1 (U- ), Re 1 (U+ ) < < Re 2 (U+ ); . . . k -sho ck : Re k . . . m-sho ck : Re m
-1 -1

(U- ) < < Re k (U- ), Re k (U+ ) < < Re k

+1

(U+ );

(U- ) < < Re m (U- ), Re m (U+ ) < , (2.3)

where the integer k denotes shock family number. For states U± in the hyperbolic region (when the characteristic speeds are real), inequalities (2.3) are the classical Lax conditions. They are sufficient for stability of small sho cks under some additional technical conditions, see e.g. [25]. When at least one of the states U± lies in the elliptic region (when the characteristic speeds are complex), inequalities (2.3) extend the Lax condition for the case of conservation laws with vanishing diffusion of the form F (U ) 2U U + = , +0. (2.4) t x x2 Under conditions (2.3), if a traveling wave in (2.4) exists for a sho ck wave with certain left and right states U± , then the traveling wave generically exists for sho cks under small perturbations of left and right states satisfying (2.2). Examples show that inequalities (2.3) are not sufficient for the stability of the traveling wave [6, 10]. In this paper, we use extended Lax admissibility conditions in a formal way without checking stability. For fixed U- , Eq. (2.2) determines a set of curves in state space U+ ; we will call them the Hugoniot curve. It is known that, for U- lying in the region of strict hyperbolicity, there are m Hugoniot curves passing through U- ; each curve is tangent to the eigenvector of the matrix A(U- ) at U- . 3. Sho ck Curves Near the Elliptic Boundary In this paper, we study small-amplitude shocks with states near the boundary of the elliptic region. Consider a point U0 on the elliptic boundary. In the generic case, two


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real eigenvalues (characteristic speeds) k and k+1 coincide at U0 forming a 2 â 2 Jordan blo ck [1]. We denote A0 = A(U0 ) and 0 = k (U0 ) = k+1 (U0 ). Then there is a real eigenvector r0 and a generalized eigenvector (asso ciated vector) r1 satisfying the Jordan chain equations A0 r0 = 0 r0 , A0 r1 = 0 r1 + r0 . (3.1)

The generalized eigenvector r1 can be chosen to be orthogonal to r0 , so we assume that r0 · r0 = 1, r0 · r1 = 0, (3.2)

where the dot denotes the standard inner product in Rm (generally r1 = 1). In addition to the (right) vectors r0 , r1 , we define the left eigenvector l0 and generalized eigenvector l1 as (both l0 and l1 are row-vectors) l0 A0 = 0 l0 , l0 r1 = 1, l1 A0 = 0 l1 + l0 , l1 r1 = 0. (3.3) (3.4)

The normalization conditions (3.4) define uniquely l0 and l1 for given r0 , r1 . Furthermore, these vectors satisfy the relations (see e.g. [24]) l0 r0 = 0, l1 r0 = l0 r1 = 1. (3.5)

For small sho cks near the elliptic boundary, we can write U- = U0 + u, U+ = U0 + u + e, = 0 + . (3.6)

Here u Rm , R, and R are small, and the direction vector e Rm has unit norm e = 1. Here we use co ordinates similar to the co ordinates on the wave manifold intro duced in [13]. Note that the coordinates with e and changed by -e and - are identical. For U- = U+ one has = 0 and arbitrary e (this is the essence of the "blow-up" pro cedure used in [13]). As we will see below, these co ordinates facilitate the analysis. By using (3.6), the Rankine­Hugoniot conditions (2.2) can be rewritten as (, e, , u) F (U0 + u + e) - F (U0 + u) - (0 + )e = 0. (3.7)

It is easy to see that the function (, e, , u) is smo oth with respect to all variables. At u = 0 and = = 0, Eq. (3.7) takes the form A0 e - 0 e = 0, (3.8)

which implies that e = r0 (the sign of r0 is irrelevant due to the equivalence (e, ) (-e, - ) mentioned above). Hence, we must look for a solution of the equation (, e, , u) = 0 in the neighborho o d of the point (0,r0 , 0, 0).


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3.1. Asymptotic relation for the Hugoniot curve Let us intro duce the vectors n, q R n · a l0 d2 F (a, r0 ),
m

as (3.9)

q·a

1 1 l1 d2 F (a, r0 )+ l0 d2 F (a, r1 ). 2 2

Here d2 F (a, b) denotes the second-order derivative of the flux function at U0 : d2 F (a, b) = so that F (U0 +U ) = F (U0 )+ A0 U + 12 d F (U, U )+ o( U 2
2 m i,j =1

2F Ui U

j U =U0

ai b j ,

a , b Rm ,

(3.10)

).

(3.11)

Theorem 3.1. Assume that at the el liptic boundary point U0 the nondegeneracy condition n · r0 = 0 (3.12)

is satisfied, then Eq. (3.7) has a unique solution (, u), e(, u) in the neighborhood of (, e, , u) = (0,r0 , 0, 0). It has the form (, u) = 2 2 - n · u + o(2 , u ), n · r0 (3.13) (3.14)

e(, u) = r0 + r1 + eu u + o(, u ), where eu u = G n·u 2 d F (r0 ,r0 ) - d2 F (u, r0 ) , n · r0
T G = (A0 - 0 I + r1 r0 )-1 .

(3.15)

Pro of. Let us consider the Taylor expansion of the function at the point (, e, , u) = (0,r0 , 0, 0). As it was shown above, the function vanishes at (0,r0 , 0, 0) (expression (3.7) is reduced to (3.8)). Using expression (3.11) in (3.7) yields for the first and second order terms (, e, , u) = 12 d F (r0 ,r0 ) +(A0 - 0 I ) h - r0 + d2 F (u, r0 ) 2 1 1 + d3 F (r0 ,r0 ,r0 ) 2 + d3 F (u, r0 ,r0 ) + d2 F (h, r0 ) + d2 F (u, h) 6 2 1 + d3 F (u, u, r0 ) - h + ··· 2 = 0, (3.16)

where h the flux small , the m â

= e - r0 , I is the identity matrix, and d3 F (a, b, c) is the third derivative of function defined in a way analogous to the second derivative in (3.10). For h, and u, Eq. (3.7) with e = 1 has a unique solution (, u), e(, u) if (m + 1) Jacobian matrix 1 d2 F (r0 ,r0 ), A0 - 0 I has full rank. Since the 2


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left null-space of A0 - 0 I is one-dimensional and it is defined by the left eigenvector l0 , this condition is equivalent to l0 d2 F (r0 ,r0 ) = n · r0 = 0. (3.17)

This inequality is satisfied by the assumption (3.12) of the theorem. Multiplying (3.16) by l0 from the left, and using (3.3), (3.5), (3.9), we obtain l0 (, e, , u) = n · r0 + n · u + o(, h ,, u )= 0. 2 2n · u + o(, u ). n · r0 (3.18)

Hence, we find (, u) up to first order terms as (, u) = - (3.19)

In first order approximation, we have r0 · h = 0 (recall that e = r0 + h is the unit norm vector). In order to find h, it is convenient to add (r0 · h)r1 = 0 to the expression (3.16). Then, by using (3.19) and keeping only the first order terms, we write the equation = 0 as n·u 2 T (A0 - 0 I + r1 r0 ) h - r0 + d2 F (u, r0 ) - d F (r0 ,r0 ) = 0. (3.20) n · r0
T T The matrix A0 - 0 I + r1 r0 is nonsingular (r1 r0 is the diadic pro duct matrix). The T inverse matrix G = (A0 - 0 I + r1 r0 )-1 defined in (3.15) satisfies the relations

Gr0 = r1 ,

Gr1 = r0 ,

l0 G = l1 ,

T r0 G = l0 ,

(3.21)

which can be verified by using (3.1)­(3.5). Then one solves Eq. (3.20) in the form (3.14), (3.15). Now let us evaluate the second derivative of (3.16) with respect to as a composite function (i.e. with = (, u) and e = e(, u)). At (, u) = (0, 0), the following equalities hold: = 0, e = r0 and = 0, e = h = r1 . Thus, we obtain


=

12 d F (r0 ,r0 ) +(A0 - 0 I ) h 2



- 2r1 = 0

(3.22)

(the subscripts denote derivatives). Multiplying by l0 from the left and using (3.3), (3.4), we find = 4/(n · r0 ). This provides the co efficient of the 2 term in (3.13).

3.2. Singularities of Hugoniot curves Using expressions (3.13) and (3.14) of Theorem 3.1 in (3.6), we obtain the following asymptotic relations for sho ck states and speed U- = U0 + u, U+ = U0 + u +2 2 - n · u (r0 + r1 ), n · r0 = 0 + . (3.23)

Here we kept only the essential lowest order terms in the expression for U+ . The term eu u is dropped in the parenthesis in (3.23). As we will see below, the geometry


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of the sho ck curve is described by in the interval u , so eu u 2 is a higher order correction term. The vector n defined in (3.9) is normal to the elliptic boundary at U0 , pointing into the hyperbolic region, see e.g. [24]. Thus, the nondegeneracy condition (3.12) implies that the eigenvector r0 is transversal to the elliptic boundary at U0 . Assuming that u is small and fixed (the left state U- is fixed near the elliptic boundary), we have three different forms for the Hugoniot curve in U+ state as shown in Fig. 1. The three cases are distinguished by the sign of the quantity n · u. If u = 0, then the Hugoniot curve has a cusp at the point U- = U0 . This corresponds to a left state U- lying exactly on the elliptic boundary. The cusp consists of two curves that are tangent to the eigenvector r0 and lie in the hyperbolic region. Asymptotically, the Hugoniot curve lies in the plane spanned by the vectors r0 and r1 . If n · u > 0, then the left state U- = U0 + u belongs to the hyperbolic region, and the Hugoniot curve forms a lo op with self-intersection at U+ = U- . The self intersection point corresponds to the sho ck speeds = 0 ± n · u. Asymptotically, the elliptic boundary crosses the loop at the middle. This means that there is a line segment parallel to the eigenvector r0 with one end at U- = U0 + u and opposite nu end at U+ = U0 + u - 2 n··r0 r0 (the point corresponding to = 0), which intersects the elliptic boundary at the middle, see Fig. 1(b). At the point U+ , the vector r1 is tangent to the Hugoniot curve. As U- U0 , the lo op of the Hugoniot curve shrinks forming the cusp singularity in Fig. 1(a). It is easy to see from expressions (3.23) that the angle at the self-intersection point tends to zero asymptotically as 2 r1 n · u. Recall that the tangent vectors to the Hugoniot curve at the selfintersection point U- = U+ are the eigenvectors of the matrix A(U- ). These two eigenvectors merge as U- approaches the elliptic boundary. Note that the lo op in Fig. 1(b) was recognized in [14] by using theory of normal forms. If n · u < 0, then the left state U- = U0 + u belongs to the elliptic region, and the Hugoniot curve do es not pass through U- , so it do es not have self-intersection

(a)

(b)
-

(c) on or near the elliptic boundary. The elliptic region

Fig. 1. Hugoniot curves for fixed left states U lies below the gray surface.


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points. The whole curve lies in the hyperbolic region. Asymptotically, the point U+ of the curve nearest to the elliptic boundary corresponds to = 0. The seg ment between U- and U+ is parallel to r0 and asymptotically intersects the elliptic boundary at the middle; it is also orthogonal to the Hugoniot curve at U+ , see Fig. 1(c). We see that, for small sho cks involving a state inside the elliptic region, the other state is always in the hyperbolic region. 3.3. Admissible shocks near the el liptic boundary Let us check the Lax conditions (2.3) for different points of the Hugoniot curve. In the case when A0 has double eigenvalue 0 with 2 â 2 Jordan blo ck, the asymptotic expression for the eigenvalues of the perturbed matrix A0 + B is given by [24] k,
k+1

= 0 ±

l0 Br0 + o( B )+

1 (l1 Br0 + l0 Br1 )+ o( B ). 2

(3.24)

Hereafter, k corresponds to the minus sign, and k+1 corresponds to the plus sign of the square ro ot. Taking B = A(U ) - A0 (so that A0 + B = A(U ) = dF /dU ) evaluated at U = U0 +U , we have Ba = d2 F (U, a)+ 13 d F (U, U, a)+ o( U 2
2

),

a Rm .

(3.25)

Using the vectors n and q defined in (3.9), we obtain k, For U
- k+1

(U0 +U ) = 0 ±

n · U + o( U )+ q · U + o( U ).

(3.26)

and U+ given by expressions (3.23), we find the asymptotic relations k,
k+1

(U- ) = 0 ±

n · u,

k,

k+1

(U+ ) = 0 ±

22 - n · u.

(3.27)

In these expressions, we kept only the lowest order (square ro ot) terms. First, consider the left states U- in the hyperbolic region (n · u > 0). If (n · u)/2, then the second square root in (3.27) is real, so U+ belongs || > to the hyperbolic region. Then, by using (3.27) in (2.3), we find that the admis sible k -shocks correspond to < - n · u, and (k + 1)-sho cks correspond to (n · u)/2 < < n · u. For U+ in the elliptic region (|| < (n · u)/2), the square ro ot term in the second expression in (3.27) is purely imaginary (the expression inside the square ro ot is negative). Then Re k = Re k+1 = 0 + o( u 1/2 ), and we have (k +1)-sho cks for > o( u 1/2 ). Thus, right states of (k +1)-sho cks correspond to the segment between the point U- and the opposite point of the lo op U+ in Fig. 1(b). Therefore, we found that the sho cks satisfying the extended Lax conditions are given by k -sho ck: < - n · u, (3.28) (k +1)-shock: 0 < < n · u,


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with the terms of order u neglected. These sho cks are represented by thick segments Sk and Sk+1 in Fig. 1(b). If we interchange the left and right states U- and U+ , the other two (thin) segments correspond to admissible sho cks. Now consider the left states U- in the elliptic region (n · u < 0). In this case Re k (U- ) = Re k+1 (U- ) = 0 + o( u 1/2 ), so only k -sho cks are possible. By checking conditions (2.3), one finds with the accuracy o( u 1/2 ): k -sho ck: < 0. (3.29) These sho cks are represented by the thick segment Sk in Fig. 1(c). If we interchange the left and right states U- and U+ , the other (thin) segment corresponds to admissible (k +1)-sho cks. A similar analysis for the left states U- at the elliptic boundary (u = 0) yields existence of k -shocks only. These sho cks are given by condition (3.29) (which is exact), and are represented by the thick segment Sk in Fig. 1(a). The other (thin) segment of the Hugoniot curve in Fig. 1(a) corresponds to (k +1)-sho cks from U+ to U- (with the left and right states interchanged). 4. Exceptional Points of Elliptic Boundary Let us consider a point of the elliptic boundary U0 , where the nondegeneracy condition (3.12) is violated, i.e. the eigenvector r0 is tangent to the elliptic boundary: n · r0 = 0. (4.1) Such a point was called exceptional in [16]. Generically, a set of exceptional points is a co dimension 2 manifold in state space. For example, in mo dels of petroleum engineering [5, 19], exceptional points typically exist on the boundaries of elliptic regions. Condition (4.1) implies that we cannot apply the implicit function theorem for solving the equation (, e, , u) = 0 with respect to e and . The standard way to study solutions of such an equation is provided by singularity theory, see [11]. It consists of (i) Liapunov­Schmidt reduction of the system (, e, , u) = 0 to a single scalar equation g (, , u) = 0 (with a one-to-one correspondence between the solutions (, u), e(, u) and (, u)), (ii) analysis of the reduced scalar equation, and (iii) interpretation of the results in terms of the original system variables. In this section, we will use the direction vector e normalized as (e - r0 ) · r0 = 0 (4.2)

instead of the condition e = 1 used above. The reason is that condition (4.2) facilitates the Liapunov­Schmidt reduction. 4.1. Liapunov­Schmidt reduction Theorem 4.1. For the exceptional point U0 at the el liptic boundary, Eq. (3.7) can be solved for and e in the neighborhood of (, e, , u) = (0,r0 , 0, 0) in the form 1 e(, u) = r0 - Gd2 F (r0 ,r0 ) + r1 - GE d2 F (u, r0 )+ o(, u ), (4.3) 2


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where (, u) is a solution of the scalar equation 0 2 + 1 - 2 + n · u + gu u +2q · u 1 + guu (u, u)+ o( 2 ,2 , u 2 ) = 0 2 with coefficients 1 l0 d3 F (r0 ,r0 ,r0 ) - 6 1 1 = n · r1 + q · r0 , 2 1 gu u = l0 d3 F (u, r0 ,r0 ) - 2 0 = 1 n · Gd2 F (r0 ,r0 ), 2

(4.4)

1 l0 d2 F (u, Gd2 F (r0 ,r0 )) 2

- l0 d2 F (r0 ,GE d2 F (u, r0 )), g
uu

(u, u) = l0 d3 F (u, u, r0 ) - 2l0 d2 F (u, GE d2 F (u, r0 )).

(4.5)

The matrix G is given in (3.15), and E the matrix E = I - r1 l0 . Pro of. Let us intro duce the vector y = to (4.2),
h

R

m+1

, where h = e - r0 . According (4.6)

h · r0 = 0. Then we can write the equation (, e, , same letter is not confusing as the number The (m + 1)-dimensional vector y belongs by (4.6); also, y = 0 when = 0 and e = formulae for the derivatives of (y, , u) at Ldy y dy =

u) = 0 as (y, , u) = 0 (keeping the and type of the arguments is different). to the m-dimensional subspace defined r0 . Using (3.16), we find the following (y, , u) = (0, 0, 0):

12 d F (r0 ,r0 ) d +(A0 - 0 I ) dh, 2 1 yy (dy1 ,dy2 ) = d2 F (dh2 ,r0 )d1 + d2 F (dh1 ,r0 )d2 + d3 F (r0 ,r0 ,r0 ) d1 d2 , 3 u du = d2 F (du, r0 ), yu (dy , du) = = -r0 ,
uu

(du, du) = d3 F (du, du, r0 ),

13 d F (du, r0 ,r0 )d + d2 F (du, dh), 2
u

y dy = -dh,

= 0,





= 0.

(4.7)

Here subscripts denote derivatives with respect to the corresponding variable. The linear operator L has one-dimensional null-space for vectors y satisfying condition (4.6). It is easy to see that the vector v0 ker L can be taken as 1 v0 = 1 (4.8) , 2 - Gd F (r0 ,r0 ) 2


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where the matrix G is defined in (3.15). Indeed, by using the degeneracy condition (3.9), (4.1) and the last equality in (3.21), one can check that the vector - 1 Gd2 F (r0 ,r0 ) is orthogonal to r0 , i.e. condition (4.6) is satisfied. Thus, using 2 (4.6) and (4.7), Lv0 = 12 d F (r0 ,r0 )+ G- 2
1

1 - Gd2 F (r0 ,r0 ) 2

= 0.

(4.9)

The left null-space of L is given by the left eigenvector l0 , so that l0 L = 0. The matrix E = I - r1 l0 defines the pro jection operator from Rn onto range L; this property follows from relations (3.4) and (3.5). Finally, we define the inverse operator L-1 on range L as L
-1

a

0 Ga

.

(4.10)

According to the Liapunov­Schmidt reduction pro cedure, the solution of equation (y, , u) in the neighborho o d of (0, 0, 0) can be given as y = v0 + W (, , u), (4.11) where is a solution of a scalar equation g (, , u) = 0, and W belongs to range L-1 , i.e. its first component is zero. The first component of relation (4.11) yields = . The smo oth functions g (, , u) and W (, , u) can be found as a Taylor expansion. Their derivatives taken at (0, 0, 0) are given by the formulae [11, Chap. 1]: g = 0, g


g



= l0 yy (v0 ,v0 ), +2l0
y

g = l 0 ,

g



= l0

y v0

+ l0 yy (v0 ,W ),
-1

= l0



W + l0 yy (W ,W ),

W = 0,

W = -L

E , (4.12)

where subscripts denote derivatives with respect to the corresponding variables, and stands for or u. By using (4.7), (4.8), (4.10) in (4.12), we find 1 l0 d3 F (r0 ,r0 ,r0 ) - n · Gd2 F (r0 ,r0 ), 3 1 g = -2, g = n · r1 + q · r0 , gu du = n · du, gu du = 2q · du, 2 1 1 gu du = l0 d3 F (du, r0 ,r0 ) - l0 d2 F (du, Gd2 F (r0 ,r0 )) 2 2 g = 0, g


=

g = 0,

- l0 d2 F (r0 ,GE d2 F (du, r0 )), guu (du, du) = l0 d3 F (du, du, r0 ) - 2l0 d2 F (du, GE d2 F (du, r0 )), W = 0, W = 0 r1 , Wu du = 0 -GE d F (du, r0 )
2

.

(4.13)

Here we also used (3.4), (3.5), (3.9), (3.21) and the relations GE r0 = Gr0 = r1 , l0 GE = l1 E = l1 . Then the expressions in the theorem follow directly from (4.11) and (4.13).


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4.2. Singularities of Hugoniot curves Let us consider Eq. (4.4) for left states U- = U0 + u such that n · u u . Formally, this corresponds to perturbations u along a line passing through U0 transversal to the elliptic boundary (the other situation, when n · u u , will be considered later). In the case n · u u , we can neglect higher order terms (the terms in the second row) in (4.4) to obtain the following asymptotic equation 0 2 + 1 - 2 + n · u = 0. (4.14) This equation defines a curve in the (, )-plane. If n · u > 0, this curve intersects the -axis at the points (4.15) = 0, = ± n · u. Remark that Eq. (4.14) with u = 0 corresponds to the normal form 2 ± 2 = 0 whose universal unfolding is 2 ± 2 + = 0, see [11]. This means that Eq. (4.14) describes the generic unfolding of a singularity in the neighborho o d of (, , u) = (0, 0, 0). Using asymptotic expression (4.3) in (3.6), the Hugoniot curve for U+ = U0 + u + e is given by U+ = U0 + u + r0 - 1 Gd2 F (r0 ,r0 ) + r1 , 2 (4.16)

where again we kept only the essential lowest order terms (according to (4.14), the u and geometry of the Hugoniot curve is described by in the interval ). The points with = 0 in (4.15) correspond to the self-intersection point of the Hugoniot curve at U+ = U- = U0 + u. Naturally, the self-intersection exists only for n · u > 0, when U- belongs to the hyperbolic region. There are two cases distinguished by the sign of the quantity D = 2 +40 . 1 (4.17)

If D < 0, then for u = 0 (i.e. for the left sho ck state at the exceptional point U- = U0 ), Eq. (4.14) has only the trivial solution = = 0. For left states U- lying in the hyperbolic region (n · u > 0), the solution of (4.14) represents an ellipse in the (, )-plane, see Fig. 2(a). This ellipse determines an eight-shaped curve in state space, see Fig. 2(b). According to (4.16), this curve is elongated in the direction of r0 , i.e. parallel to the elliptic boundary (recall that r0 is tangent to the elliptic boundary at the exceptional point, see (4.1)). As U- U0 in the hyperbolic region, the eight-shaped Hugoniot curve shrinks to a point. For left states U- lying in the elliptic region (n · u < 0), Eq. (4.14) has no solutions, so the Hugoniot curve do es not exist (at least lo cally). For D > 0, the solution of (4.14) for u = 0 is given by two lines (4.18) = 1 ± D /2 intersecting at the origin, see Fig. 3(a). These lines determine two curves in state space passing through the exceptional point U0 tangent to the elliptic boundary


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(a)

(b)

Fig. 2. Hugoniot curves near the exceptional point for D < 0. The elliptic region lies below the gray surface.

along the eigenvector r0 , Fig. 3(b). A detailed analysis using inequality (4.25) below shows that either both curves lie in the hyperbolic region, or one curve lies in the hyperbolic region and the other in the elliptic region. Figure 3 corresponds to the case when both curves lie in the hyperbolic region. For left states U- lying in the hyperbolic region (n · u > 0), the solution of (4.14) is given by two branches of a hyperbola in the (, )-plane lying in the left and right quadrants. In state space, these branches intersect at U- , Fig. 3(c). Finally, for left states U- lying in the elliptic region (n · u < 0), the solution is given by two branches of a hyperbola lying in the upper and lower quadrants. Then there are two separated Hugoniot branches which do not pass through U- , Fig. 3(d). Now consider the case n · u u , when the left sho ck state is much closer to the elliptic boundary than to the exceptional point (this happens for perturbations u along a curve tangent to the elliptic boundary at U0 ). We are interested in the situation when n · u u 2 . In this case all terms in Eq. (4.4) are of the same order of magnitude. The solution of Eq. (4.4) in the (, )-plane is an ellipse (or the empty set) if D < 0, and it is a hyperbola if D > 0. Unlike the previous case, these curves (ellipse or hyperbola) are not centered at the origin. The curves intersect the -axis for U- in the hyperbolic region, they are tangent to the -axis for U- at the elliptic boundary, and have no intersection with the -axis for U- in the elliptic region. This follows from the result in classical theory that there are two transversal Hugoniot curves passing through U- (recall that this corresponds to = 0) for U- in the hyperbolic region. In state space, the Hugoniot curves are described by the formula U+ = U0 + u + e with e given by expression (4.3). If the position of the ellipse or hyperbola relative to the -axis is the same as in Fig. 2(a) or 3(a), then the Hugoniot curves in state space have qualitatively the same form as described above. The other four possibilities, when the ellipse is tangent or do es not intersect the -axis, and when the hyperbolae are tangent or intersect the -axis, are shown in Fig. 4. These cases represent transitions from singular structures of shock curves near exceptional points to regular structures described in Sec. 3.


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(a)

(b)

(c)

(d)

Fig. 3. Hugoniot curves near the exceptional point for D > 0. The elliptic region lies below the gray surface.

Remark 4.2. For a system of two conservation laws with a quadratic flux function, T T we have r0 = l1 , r1 = l0 and G = r0 l0 + r1 l1 . Then, at the exceptional point (n · r0 = 0), we find 1 1 0 = - n · Gd2 F (r0 ,r0 ) = - (n · r1 )l1 d2 F (r0 ,r0 ) 2 2 1 = -(n · r1 ) q · r0 - n · r1 . 2 Using 1 from (4.5) yields D = 2 +40 = 1 q · r0 - 3 n · r1 2
2

(4.19)

0.

(4.20)

This proves that the singularity of type shown in Fig. 2 (D < 0) do es not arise in systems of two conservation laws with quadratic flux functions. 4.3. Admissible shocks Let us check the Lax conditions (2.3) along the Hugoniot curve. We consider the cases n · u u shown in Figs. 2 and 3. (In this paper, we do not study the Lax


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Fig. 4. Hugoniot curves near the exceptional point for n · u u

2

(the elliptic region is gray).

conditions in