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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 ґ 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
295

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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 в 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 в 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 в

= 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

Lax Shocks in Mixed-Type Systems of Conservation Laws

301

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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A. A. Mailybaev & D. Marchesin

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 в 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,

Lax Shocks in Mixed-Type Systems of Conservation Laws

303

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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A. A. Mailybaev & D. Marchesin

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

Lax Shocks in Mixed-Type Systems of Conservation Laws

305

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 the case n · u u 2 , which is more complicated.) Then, according to asymptotic equation (4.14), the following quantities have the same order of magnitude: u 1/2 . For the characteristic speeds k,k+1 (U- ) with U- = U0 + u we have the same asymptotic relation as in (3.27): k,
k+1

(U- ) = 0 ±

n · u,

(4.21)

which contains the terms of order u 1/2 . The main correction term in expansion (4.16) for U+ is U+ r0 u 1/2 . This term vanishes when we substitute (4.16) into (3.26), because n · r0 = 0. This means that the square ro ot and the linear terms in (3.26) are of the same order u 1/2 . In order to find the asymptotic values for characteristic speeds, one must use the second order approximation for the expression inside the square ro ot in (3.24) of the following form [24, Chap. 2]: l0 Br0 + 1 (l1 Br0 + l0 Br1 )2 - l0 BGBr0 + o( B 2 ). 4 (4.22)

(This expression corresponds to a degenerate type of bifurcation of a double eigenvalue when l0 Br0 B 2 .) Using expansions (3.25) and (4.16) in (4.22) and substituting the result into (3.24), one finds the asymptotic expression

k,k+1

(U+ )

r ((q · r0 )2 +

= 0 + q · r 0 ±

1 3 l0 d3 F (r0 ,r0 ,r0 ) - n · Gd2 F (r0 ,r0 )) 2 + n · r1 + n · u, 2 2 (4.23)

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where we kept only the leading terms of order u 1/2 . By using the quantities 0 , 1 defined in (4.5) and Eq. (4.14), we write (4.23) as k,
k+1

(U+ ) = 0 + q · r0 ±

( - q · r0 )2 + (1 +20 ).

(4.24)

The right state U+ lies in the hyperbolic region if ( - q · r0 )2 + (1 +20 ) > 0. (4.25)

First, consider the left states U- in the hyperbolic region (n · u > 0). Using (4.21) and (4.24) in (2.3) (with = 0 + ), we write the admissibility conditions for the k -sho ck as (4.26) k -sho ck: < - n · u, (1 +20 ) > 0, Comparing (4.25) and (4.26), it is easy to see that the right states U+ of k -sho cks lie in the hyperbolic region. Similarly, for (k + 1)-sho cks we find (k +1)-sho ck: - n · u < < n · u, (1 +20 ) < 0, > q · r0 . (4.27) Conditions (4.26) and (4.27) one end at U- and, possibly, Consider now the left sta and (4.24) in (2.3), we write determine two segments in the Hugoniot curve with a separate segment for k and/or (k +1)-sho cks. tes U- in the elliptic region (n · u > 0). Using (4.21) the admissibility conditions for the k -sho ck as (1 +20 ) > 0. (4.28)

k -sho ck: < 0,

There are no (k + 1)-sho cks in this case. We remark that by using formulae (4.21) and (4.24), one can find segments of Hugoniot curves satisfying sho ck speed conditions different from (2.3), e.g. the segments containing possible states of certain non-Lax sho cks (Si,j sho cks [17]). 5. Riemann Solutions with States in the Elliptic Region Consider the Riemann problem for the system of conservation laws (2.1), i.e. the ~ weak scale-invariant solution U (x, t) = U (x/t) for piecewise constant initial data with a single jump at x = 0: U (x, 0) = L, x < 0; R, x > 0. (5.1)

When both L and R are very near each other and both lie in the hyperbolic region, Riemann solutions are described by the standard Lax diagram [25]. For initial data near the elliptic boundary, Riemann solutions were discussed in [5, 7, 12, 21] for systems of two conservation laws. In this section, we contribute with an example of the Riemann solution for two conservation laws with left initial condition L in the elliptic region near the boundary, and right initial condition R in the hyperbolic region. This example illustrates the asymptotic description of Hugoniot curves given in Sec. 3.

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A typical Riemann solution is a sequence of two waves (rarefactions and/or sho cks) separated by a constant state M . Rarefaction waves are smo oth solutions U () parametrized by the speed = x/t satisfying the equation A(U )U = U , where U = dU /d. The rarefaction wave defines a curve in state space, which is tangent at each point to one of the eigenvectors of the matrix A(U ). Given the left state U- in the hyperbolic region, the rarefaction curve and the sho ck curve (the curve of right states for admissible sho cks) lie at opposite sides of U- , where they are tangent, see e.g. [25]. The structure of rarefaction curves near the elliptic boundary is described in [16]. Consider the left initial condition L in the elliptic region close to the boundary. Then the slower wave in the Riemann solution is a sho ck; its right state is the intermediate state of the Riemann solution M , see Fig. 5. The thick solid line in the figure shows possible states M for fixed L (this is the part of the Hugoniot curve for U- = L corresponding to admissible shocks satisfying (2.3); M0 is the end point of this part corresponding to = Re 1 (U- )). Figure 5 corresponds to L far away from any of exceptional points, see Fig. 1(c). The faster wave may be a sho ck or a rarefaction. Thus, there are two typical structures of Riemann solutions for different right initial conditions. The first Riemann solution consists of a sho ck S1 (from L to M ) followed by a sho ck S2 (from M to R). These Riemann solutions correspond to R lying in the region marked S1 S2 in Fig. 5. The second Riemann solution consists of a sho ck S1 (from L to M ) followed by a rarefaction of the second characteristic family R2 (from M to R). These Riemann solutions correspond to R lying in the region marked S1 R2 in Fig. 5. Lo cal Riemann solutions with the right states R to the right of the thick dashed line do not exist. This line consists of the rarefaction curve R2 from M0 (the part of the curve to the right of M0 in Fig. 5) and right states of S2 shocks from M0 (the part of the curve to the left of M0 in Fig. 5). Consider the Riemann problem for system (2.1) with flux function and initial conditions F (U ) = (U2 )2 U
1

,

L=

0 -0.02

,

R=

0.02 0.16

.

(5.2)

Fig. 5. Diagram of possible right initial states R in Riemann solutions for a left initial state L in the elliptic region.

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The characteristic speeds at U = (U1 , U2 ) are 1 = - 2U2 and 2 = 2U2 . The elliptic region is the lower half-plane U2 < 0, so L is in the elliptic region, and R is in the hyperbolic region. For the left state U- = L, we find all possible right states U+ from the RankineHugoniot equation (2.2) as U+ = By checking the < 0 (the value can be compare boundary point a d U (0.04 + 2 ) 0.02 +
2

.

(5.3) that 1-sho cks correspond to = (0, 0.02), see Fig. 5). This 3) evaluated for the elliptic we find

dmissibility conditions (2.3), we find = 0 corresponds to the end point M0 with the asymptotic expression (3.2 0 = (0, 0). At this p oint 0 = 0, and 0 1 , r1 = 1 0 0 -0.02 , l0 = (1, 0), 0 2

r0 =

l1 = (0, 1), (5.4)

u=

,

n=

.

Substituting (5.4) into (3.23) gives the exact expression (5.3). By solving the equation A(U )U = U for the rarefaction of the second family ( = 2 = 2U2 ) with the right state R given in (5.2), we obtain the states on the rarefaction wave as 0.02 + 3 - 0.128 2 /3 . (5.5) U () = 2 /2 The right state of the sho ck (5.3) must be equal to the left state of the rarefaction (5.5). Numerical computation yields the intermediate state and sho ck speed as = -0.2311, M= -0.02158 0.07340 . (5.6)

The sho ck is admissible since < 0. Thus, we found the Riemann solution with one sho ck and one rarefaction wave. The shock has the speed and right state (5.6), and the rarefaction is given by (5.5) with 0.3831 0.4 2. Numerical simulations were carried out using a linearized CrankNicolson scheme for two viscous conservation laws (2.4) with co efficient = 0.01 of the parabolic term and data (5.2). Figure 6 shows the numerical solution in (U1 ,U2 ), (x, U1 ) and (x, U2 ) spaces at a certain time. This solution represents a traveling wave S1 from L to M , which becomes a sho ck as +0, followed by a rarefaction wave R2 from M to R. The numerical solution agrees very well with the analytic Riemann solution (5.3), (5.5), (5.6). We also observed stability of this solution.

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Fig. 6. Numerical simulations of the Riemann solution with one shock and rarefaction wave (the elliptic region is gray).

6. Discussion The Lax diagram for the structure of the Riemann solution changes for elliptic left states near regular points of the elliptic boundary. Sho ck-sho ck and sho ckrarefaction sequences are preserved, but there are no lo cal solutions in half a region near the left state, out of the elliptic region, because there are no rarefaction waves emanating from this left state. This gap may be filled by non-lo cal solutions. The structure of the Riemann solution near exceptional points of the elliptic boundary is very rich, as already shown in our analysis of the Hugoniot curves that parametrize sho cks. To understand it, it is necessary to study the traveling waves for these sho cks. An initial step using the results of [8] on bifurcations of planar vector fields was taken in [4], where quadratic flux functions were studied. A program for finding Riemann solutions for nonhomogeneous quadratic flux functions under the viscous profile condition for shocks was initiated in [2], and continued in [20], where two examples were solved, and theoretically in [9]. Our work shows that cubic terms in the flux function must be considered. Remark that the effect of the cubic terms vanishes as the elliptic region shrinks to a point. At this point, a double semisimple eigenvalue (characteristic speed) appears, and the statement can be proved by analyzing lo cal behavior of eigenvalues and eigenvectors [24]. Acknowledgments This work was supported by CNPq under Grants 301532/2003-6, 300097/2004-2, 304168/2006-8, 472067/2006-0, 300668/2007-4, FAPERJ under Grant E-26/152.163/ 2002, E-26/152.525/2006, E-26/110.310/2007, CAPES under Grant 0722/2003 (PAEP no. 0143/03-00), FINEP under Grant 01.02.0212.00 and President of Russian Federation Grant No. MK-2012.2006.1. References
[1] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, New York, 1983).

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[2] A. V. Azevedo, Multiple fundamental solutions for hyp erb olic-elliptic systems of conservation laws, Ph.D. thesis, PUC-RIO, Rio de Janeiro (1991). [3] A. V. Azevedo and D. Marchesin, Multiple viscous profile Riemann solutions in mixed elliptic-hyp erb olic models for flow in p orous media, in Nonlinear Evolution Equations that Change Type, The IMA Volumes in Mathematics and Its Applications, Vol. 27 (Springer, New York, 1990), pp. 117. [4] A. V. Azevedo, D. Marchesin, B. Plohr and K. Zumbrun, Capillary instability in models for three-phase flow, Z. Angew. Math. Phys. 53 (2002) 713746. [5] J. B. Bell, J. A. Trangenstein and G. R. Shubin, Conservation laws of mixed typ e describing three-phase flow in p orous media, SIAM J. Appl. Math. 46 (1986) 1000 1017. [6] S. Canic, Nonexistence of Riemann solutions for a quadratic model deriving from p etroleum engineering, Nonlinear Anal. Real World Appl. 4 (2003) 373408. [7] S. Canic, B. L. Keyfitz and E. H. Kim, Mixed hyp erb olic-elliptic systems in selfsimilar flows, Bol. Soc. Brasil. Mat. 32 (2002) 123. [8] F. Dumortier, R. Roussarie and J. Sotomayor, Bifurcation of Planar Vector Fields, Lecture Notes in Mathematics, Vol. 1480 (Springer, Berlin, 1991). [9] C. Eschenazi and C. F. Palmeira, Local top ology of elementary waves for systems of two conservation laws, Mat. Contemp. 15 (1998) 127144. [10] H. Frid, Measure-valued solutions for multiphase flow in p orous media, J. Math. Anal. Appl. 196 (1995) 614627. [11] M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory (Springer, New York, 1985). [12] H. Holden, On the Riemann problem for a prototyp e of a mixed typ e conservation law, Commun. Pure Appl. Math. 40 (1987) 229264. [13] E. L. Isaacson, D. Marchesin, C. F. Palmeira and B. J. Plohr, A global formalism for nonlinear waves in conservation laws, Commun. Math. Phys. 146 (1992) 505552. [14] B. L. Keyfitz, A geometric theory of conservation laws which change typ e, Z. Angew. Math. Mech. 75 (1995) 571581. [15] B. L. Keyfitz, R. Sanders and M. Sever, Lack of hyp erb olicity in the two-fluid model for two-phase incompressible flow, Discrete Contin. Dynam. Syst. Ser. B 3 (2003) 541563. [16] A. A. Mailybaev and D. Marchesin, Hyp erb olicity singularities in rarefaction waves, to app ear in J. Dynam. Differential Equations 20 (2008) 129. [17] A. A. Mailybaev and D. Marchesin, Dual-family viscous shock waves in n conservation laws with application to multi-phase flow in p orous media, Arch. Ration. Mech. Anal. 182 (2006) 124. [18] D. Marchesin and C. F. B. Palmeira, Top ology of elementary waves for mixed-typ e systems of conservation laws, J. Dynam. Differential Equations 6 (1994) 427446. [19] D. Marchesin and B. Plohr, Wave structure in WAG recovery, Soc. Petroleum Eng. J. 6 (2001) 209219. [20] V. M. M. Matos, Riemann problem for two conservation laws of typ e IV with elliptic region, Ph.D. thesis, IMPA, Rio de Janeiro (2004) (in Portuguese). [21] V. Matos and D. Marchesin, High amplitude solutions for small data in pairs of conservation laws that change typ e, in Hyperbolic Problems: Theory, Numerics, Applications (Springer, Berlin, 2008), pp. 711719. [22] H. B. Medeiros, Stable hyp erb olic singularities for three-phase flow models in oil reservoir simulation, Acta Appl. Math. 28 (1992) 135159.

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[23] W. Schulte and A. Falls, Features of three-comp onent, three-phase displacement in p orous media, SPE Reservoir Eng. 7 (1992) 426432. [24] A. P. Seyranian and A. A. Mailybaev, Multiparameter Stability Theory with Mechanical Applications (World Scientific, Singap ore, 2003). [25] J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer, New York, 1983).