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ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Delta and Singular Delta Locus for One Dimensional Systems of Conservation Laws M. Nedeljkov Vienna, Preprint ESI 837 (000) February 9, 000 Supported by Federal Ministry of Science and Transport, Austria Available via http://www.esi.ac.at

DELTA AND SINGULAR DELTA LOCUS FOR ONE DIMENSIONAL SYSTEMS OF CONSERVATION LAWS Marko Nedeljkov Abstract. A condition for existence of singular and delta shock waves for systems of conservation laws is given in the paper. The systems considered here have uxes which are linear in one of the dependent variables. The condition obtained here is analogous to the one for the standard Hugoniot locus. Three dierent solution concept are used in the paper: associated solution in Colombeau sense, limits of nets of smooth functions together with Rankin-Hugoniot conditions and a kind of a measure valued solutions. 1. Introduction The aim of this paper is to give some criterion for which certain classes of function nets, called delta and singular shock waves, satisfy the following system of conservation laws (1) () u t + (f 1 (u)v + f (u)) x = 0 v t + (g 1 (u)v + g (u)) x = 0; in an approximated sense. Here, f i, g i, i = 1; are smooth functions, polynomially bounded together with all their derivatives, and u; v : R R +! R satisfy (3) u(x; 0) = u 0 ; x < 0; u(x; 0) = u 1 ; x > 0 v(x; 0) = v 0 ; x < 0; v(x; 0) = v 1 ; x > 0: A pair of nets (u " ; v " ) D 0, " (0; 1), is a solution in an approximated sense if hu "t + (f 1 (u " )v " + f (u " )) x ; i! 0 hv "t + (g 1 (u " )v " + g (u " )) x ; i! 0 as "! 0, for every D(R R + ). In order to have a well dened composition of functions we use families of smooth function which can be viewed as representatives of Colombeau functions ([1], [9]) or families of piecewise constant functions which are called box approximations. For the rst case we will use the space of generalized functions, G g, and the concept of a approximate solution dened in [10]. In the second case we use the standard Rankin-Hugoniot conditions for systems of conservation laws like in [6]. Roughly speaking, delta and singular shock wave are pairs of nets (u " ; v " ) which converge to sums of step and delta functions in D 0. They will be precisely dened bellow. Let us remark that singular shock waves have similarities to innite narrow 1 Typeset by AMS-TEX

M. NEDELJKOV solitons which are introduced by Maslov and Omel'anov [8]. Further results in this direction for systems of conservation laws can be found in the paper [3] from Danilov, Maslov and Shelkovich. For a given point (u 0 ; v 0 ) R, a delta (singular delta) locus is a subset of R consists of points (u 1 ; v 1 ) for which there exists a delta (singular) shock wave joining (u 0 ; v 0 ) and (u 1 ; v 1 ). This denition is an analogue of the denition of the classical Hugoniot locus in the case of piecewise constant solutions (shock waves) for Riemann problem. Admissibility condition for such shocks could be the overcompressiveness like in [6], [11], [4] and [5], i.e. (u 0 ; v 0 ) > 1 (u 0 ; v 0 ) c (u 1 ; v 1 ) > 1 (u 1 ; v 1 ); where c is a speed of the delta or singular shock wave, 1 ; are the eigenvectors for the system, (u 0 ; v 0 ) and (u 1 ; v 1 ) denotes left and right hand side initial value, respectively. Let us remark that the hyperbolicity condition is not used in the construction of approximate solutions. Like in the classical theory of Hugoniot locus, the (singular) delta locus can be used for construction of solutions containing also a rarefaction wave on the left or on the right hand side of a singular or delta shock. Our investigations are motivated by Keytz and Kranzer ([6]) who found singular shock wave solutions to the system (4) u t + (u? v) x = 0 1 v t + 3 u3? u = 0; r t in the form u " (x; t) = G " (x? ct) + a a t x x? ct " ", v " (x; t) = H " (x? ct) + x? ct ", where G " and H " converge to appropriate step functions dened " by the Riemann initial data (3), " := "?1 (=") converges to the delta distribution and i ", i = 1; 3 converge to zero in D 0 as "! 0. This system is included in our general system (1), (). Our paper shows that their approach is suciently general for solving (1), (). An example of a system with a delta shock wave solution is (5) u t + (u ) x = 0 v t + (uv) x = 0 examined in [11]. The solution is of the form u " (x; t) = G " (x? ct), v " (x; t) = H " (x? ct) + st " (x? ct), where G " and H " are the same as before and " is a delta net. By adding a viscosity term Tan, Zhang and Zheng ([11]) proved that the limit of vanishing viscosity solutions is a solution of this form. Let us notice that there is no additional "singular" term in u ". Here, the delta shock wave will be used to denote that there are no other terms in a solution except approximations of step functions and the delta distribution. Systems which are to follow have the similar form of solutions as (5), that is they do not contain term added to G " and a delta net is a part of v ".

DELTA AND SINGULAR DELTA LOCUS 3 Oberguggenberger ([9]) proved that for (6) u t + (u =) x = 0 v t + (uv) x = 0 the viscosity limit is a delta shock wave. By using Le Flock and Vol'pert denition of the product, Hayes and Le Flock ([5]) found a delta shock wave solution to (7) u t + (u ) x = 0 v t + ((u? 1)v) x = 0: They also proved that the vanishing viscosity limit exists and it is equal to the solution of (7). Finally, Ercole ([4]) proved that the system (8) u t + f(u) x = 0 v t + (g(u)v) x = 0; with some minor assumptions on f and g, has a delta shock solution as a limit of vanishing viscosity solutions. The dierences between system (4) and systems (5)-(8) are obvious. Only system (4) has second dependent variable in the rst ux function and it can not be solved with delta shock waves. In this paper we will not use the viscosity approximation. The existence and convergence of vanishing viscosity solutions is not a trivial task even for some specic choice of a function in the ux. An example of this one can nd in Dafermos- DiPerna paper []. All the systems considered here have a ux which is a linear function in one variable and this will be a general form of a ux for which we nd a delta locus. The interpretation of a limit of " where " is a delta net brings diculties. Colombeau proved in [1] that " denes an element in the space of the generalized functions which is not associated with any classical distribution. In all the cited papers a delta locus is a subset of R with the non-zero Lebesgue measure. In general, this set is just a curve, but for systems (5)-(8) our procedure gives the same locus as it was already obtained. Singular delta locus is an set with non-zero Lebesgue measure in general. In the sequel we shall consider only solutions which equals to initial data for every " (also after regularization when we use smooth nets) at the initial time. This means that the \value" of the Heaviside function in zero does not have any inuence, for example. The paper is organised as follows. In Section the algebra of generalized functions, delta and singular shock wave solution are precisely dened. In Section 3 we give a form of approximate solutions which will be used in nding delta locus's for smooth functions f i, g i, i = 1;. Roughly speaking, the main property of approximations is a distribution of a \mass" of singular part on the left and right-hand side from zero. Properly placed right and left hand sided masses determines the delta locus. For the use of a delta locus in constructions of admissible solutions one has to remove the points for which the overcompressiveness do not hold.

4 M. NEDELJKOV A box approximation solution is used in Section 4 and it gives the same delta locus as in the case of Colombeau generalized functions. Theorem 1 applied to systems (5)-(8) gives the results previously obtained in the quoted papers. System (4) can not be solved by this procedure. A concept and construction of a solution in a certain measure space is given in Section 5. Approximated solutions constructed in Section 3 converge to this solution. In Section 6 systems with polynomial uxes are examined. There exists a broad class of polynomials for which we can nd singular delta locus of the non-zero Lebesgue measure modifying u " by adding a net which converge to zero but some of its powers do not. In fact we use an idea of [6] adopted for arbitrary polynomials. In contrast to the singular shock wave solution obtained in [6], we have to assume that singular parts of approximations u " and v " have disjoint supports. This can be omitted if f 1 and g 1 are constants, or linear functions in u. There are many open problems concerning the system (1), (). For example one can try to obtain a limit of viscosity self-similar solutions to the system like it was done in Dafermos and DiPerna's paper []. Also, one can try to describe singular shock wave solution formally obtained as a net of approximate solutions by using a weighted measure space as it was done by Keytz and Kranzer in [6].. Definitions We shall briey recall some of the denitions of Colombeau algebra given in [10]. Let R + = R (0; 1), R + = R [0; 1) and Cb 1 () be the algebra of smooth functions on bounded together to all its derivatives. Let C 1 b (R + ) be a set of all (R (0; T )) for every T > 0. Let functions u C 1 (R +) such that ujr(0;t ) Cb 1 us remark that every element of Cb 1(R +) has a smooth extension up to the line ft = 0g, i.e. Cb 1(R +) = Cb 1(R +). This is also true for C 1 b (R +). E M;g (R +) is the set of all maps G : (0; 1) R +! R, ("; x; t) 7! G " (x; t), smooth for every " (0; 1), which satisfy: For every (; ) N 0 and T > 0, there exists N N such that sup j@ x @ t G " (x; t)j = O("?N ); as "! 0: (x;t)r(0;t ) N g (R +) is the set of all G " E M;g (R +) which satisfy: For every (; ) N 0, a R and T > 0 sup j@ x @ t G " (x; t)j = O(" a ); as "! 0: (x;t)r(0;t ) The dierential algebra G g (R +) of generalized functions is the factor algebra G g (R + ) = E M;g(R + )=N g(r + ). By G will be denoted a class in G g which has a representative G " E M;g. Since C 1 b (R +) = C 1 b (R +) one can dene a restriction of a generalized function to ft = 0g as follows. For given G G g (R + ), its restriction Gj t=0 G g (R) is the class which contain a function G " (x; 0). Similarly, G(x? ct) G g (R) is dened with G " (x? ct) E M;g (R). Let us note that if f is a smooth function polynomially bounded together with all its derivatives, then the composition f(g), G G g is well-dened.

DELTA AND SINGULAR DELTA LOCUS 5 Let u D 0 L1(R). Fix an element D(R), (x) 0, x R, R (x)dx = 1, supp [?1; 1]. Let " (x) = "?1 (x="), x R. Then : u! class of u " denes an embedding of D 0 L1(R) into G g (R). Obviously, commutes with derivative. We shall call a generalized function G a step function if it equals (u), where u is a step function in the usual sense. The delta generalized function is given by a representative ". In the sequel we shall omit the subscript. An element G G g () is associated with u D 0 () if for some (and hence every) representative G " of G, G "! u in D 0 () as "! 0. It will be denoted by G u. Two generalized functions G and H are associated, G H if G? H 0. A generalized function G is of the bounded type if sup jg " (x; t)j = O(1) as "! 0; (x;t)r(0;t) for every T > 0. The initial data for the Riemann problem are regularized in the following way. Let u0 ; x < 0 v0 ; x < 0 1 (x) = and (x) = u 1 ; x > 0; v 1 ; x > 0: The initial data are now (G; H) where G " = 1 " and H " = ". This means that a solution to system (1), () satises u " (x; 0) = G " (x); G " (x) = u 0 ; x <?"; G " (x) = u 1 ; x > "; v " (x; 0) = H " (x); H " (x) = v 0 ; x <?"; H " (x) = v 1 ; x > "; where G " and H " are of the bounded type, i.e. they are uniformly bounded with respect of ". Generalized functions of this type are called the generalized step functions. One will see that the results obtained here will be valid for all representatives of step functions which satisfy these conditions. Let us remark that if u " is a function satisfying the above conditions, this will be true for every its power u m, m N, also. Denition 1. (Generalized functions) (U; V ) G(R + ) is a singular shock wave solution to (1), () and (3) if a) (9) (10) (11) U t + (f 1 (U)V + f (U)) x 0 V t + (g 1 (U)V + g (U)) x 0: Uj t=0 = G; V j t=0 = H: b) U(x; t) 1 (x? ct), where c is a constant (speed of the shock) and V (x; t) (x? ct) + s(t)(x? ct), where c is the same constant, s C 1 ([0; 1)), s(0) = 0 and is Dirac delta function. If in addition U is of the bounded type, the solution is called delta shock wave.

6 M. NEDELJKOV Denition. (Box approximations) A pair of nets (u " ; v " ) consisting of piecewise smooth functions is called an approximated singular shock wave solution to (1), () and (3) if a) limhu " ; t i + hf 1 (u " )v " + f (u " ); x i = 0 "!0 lim hv "; t i + hg 1 (u " )v " + g (u " ); x i = 0: "!0 b) (u " ; v " ) satisfy (3) for " small enough. c) u " converge in the distributional sense to 1 (x?ct), where c is a constant (speed of the shock) and 1 a step function. v " converge in the distributional sense to (x? ct) + s(t)(x? ct), where c is the same constant, s C 1 ([0; 1)), s(0) = 0, ) is a step function and is Dirac delta function. If in addition u " is bounded independently on ", the solution is called delta shock wave. 3. Generalized functions and delta locus Theorem 1. a) Let f 1 6 const. Then a delta shock wave solution to (9), (10) and (11) exists if u 0 6= u 1, f 1 (u 0 ) 6= f 1 (u 1 ) and (1) c = f 1(u 1 )v 1 + f (u 1 )? f 1 (u 0 )v 0? f (u 0 ) u 1? u 0 = g 1(u 0 )f 1 (u 1 )? g 1 (u 1 )f 1 (u 0 ) ; f 1 (u 1 )? f 1 (u 0 ) where c is the velocity of the delta shock. The set of all pairs of right hand states (u 1 ; v 1 ) such that (1) holds is called the delta locus for (1), () and (3). b) If f 1 (u 0 ) = f 1 (u 1 ) = 0 (specially, if f 1 0) and g 1 6 const, then the delta locus is the set of all points (u 1 ; v 1 ) such that g 1 (u 0 ) 6= g 1 (u 1 ). c) If f 1 0 and g 1 b R, then the delta locus is the set of all points (u 1 ; v 1 ) such that b(u 1? u 0 ) = f (u 1 )? f (u 0 ). Proof. Let C 1 0 (R), supp [?1; 1], R (x)dx = 1, 0 and let D " = D? " A?1 x + A+1 " A?1! + D+ " A?1 x? A+1 " A?1 where D? + D + = 1 Let s 1 (t); s (t) be smooth functions for t 0, such that s 1 (0) = s (0) = 0 and A > 0. Since in the general we do not have any information on the behaviour of f 1, f, g 1 and g we shall try to nd delta shock wave solution in the form (13) u " (x; t) = G " (x? ct); v " (x; t) = H " (x? ct) + s(t)d " (x? ct); where D " is dened as above and s is from Denition 1. Let us note that D " converge to delta function as "! 0. After a substitution of (13) into (1) we have (14) G "t + (f 1 (G " )H " + f (G " )) x + s(t)(f 1 (G " )D " ) x 0:! ;

DELTA AND SINGULAR DELTA LOCUS 7 A missing argument of a function means that it equals x? ct. [?X; X] [?T; T ], for some X; T > 0. Then ZZ Let supp (@ t G " (x? ct) + @ x (f 1 (G " (x? ct))h " (x? ct) + f (G " (x? ct))) (x; t)dxdt ZZ = (cg " (x?ct)? f 1 (G " (x?ct))h " (x?ct)? f (G " (x?ct)))@ x (x; t)dxdt = = = Z T Z ct?"?t?x Z T Z ct+"?t ct?" Z T Z X?T ct+" =I 1 + I + I 3 : At rst let us notice that (cg " (x?ct)? f 1 (G " (x?ct))h " (x?ct)? f (G " (x?ct)))@ x (x; t)dxdt (cg " (x?ct)? f 1 (G " (x?ct))h " (x?ct)? f (G " (x?ct)))@ x (x; t)dxdt (cg " (x?ct)? f 1 (G " (x?ct))h " (x?ct)? f (G " (x?ct)))@ x (x; t)dxdt ji j Z T?T "CC dt! 0; as "! 0; since all of G ", H " and are bounded independently on ". Next, I 1 + I 3 = + Z T Z ct?"?t?x Z T Z ct?"?t?x (cu 0? f 1 (u 0 )v 0? f (u 0 ))@ x (x; t)dxdt (cu 1? f 1 (u 1 )v 0? f (u 1 ))@ x (x; t)dxdt = (cu 0? f 1 (u 0 )v 0? f (u 0 ))? (cu 1? f 1 (u 1 )v 1? f (u 1 )) Z T?T Z T?T (ct? "; t)dt (ct + "; t)dt!?(c[u]? [f 1 (u)v + f (u)])j x=ct ; as "! 0: This means that the sum of rst two members of (14) is associated to (x? ct) multiplied by a constant, i.e. G "t + (f 1 (G " )H " + f (G " )) x?(c[u]? [f 1 (u)v + f (u)]): In the sequel we shall omit this type of calculations. The function (f 1 (G " )D " ) x is a derivative of a product of approximations for step and delta function. This product is associated with the delta function multiplied by a constant, and the constant depend of a representative of the delta function. This constant as well as (c[u]? [f 1 (u)v + f (u)]) has to be zero. (c[u]? [f 1 (u)v + f (u)]) = 0 gives the Rankine-Hugoniot conditions for G " and H " of the rst equation, i.e. c = f 1(u 1 )v 1 + f (u 1 )? f 1 (u 0 )v 0? f (u 0 ) u 1? u 0 :

8 M. NEDELJKOV Now we shall nd a representative of the delta function such that f 1 (G " )D " 0. Let us note that supp D [?A;?1] [ [1; A], A is a real number greater than 1, R D(y)dy = 1 and D" (y) = "?n D(y="). The condition on supports ensures that the product of a step function and will not depend on the regularization of the initial data. It will depend only on values of a step function away of zero, R 0?1 D(y)dy R?1 and D(y)dy. Thus, we have the following system 0 Z?1?A f 1 (u 0 ) The solution is given by D? := Z?1?a D(y)dy = D(y)dy + Z?1?A Z A 1 D(y)dy = 1 D(y)dy + f 1 (u 1 ) Z A f 1 (u 1 ) f 1 (u 1 )? f 1 (u 0 ) ; D+ := 1 Z b 1 D(y)dy = 0: D(y)dy =?f 1 (u 0 ) f 1 (u 1 )? f 1 (u 0 ) ; if f 1 (u 1 ) 6= f 1 (u 0 ). In the case when f 1 const 6= 0, we shall nd singular shock solutions of system (1), () but for some special forms of f, g 1 and g. There are no delta shock solutions of this form (only classical shocks, i.e. s = 0). If f 1 0, then D? and D + can be choosen such that cd 0 " (g 1 (G " )D " ) x : The only neccesary condition is that g 1 (u 0 ) 6= g 1 (u 1 ). If g 1 is also a constant then c equals to this constant. The set of all u 1 for which this is true determines the delta locus. When we substitute u " and v " into (), H "t + s 0 (t)d "? cs(t)d "t + (g 1 (G " )H " + g (G " ) + g 1 (G " )s(t)d " ) x =H "t + (g 1 (G " )H " + g (G " )) x + s 0 (t)d "? cs(t)d "t + s(t)(g 1 (G " )D " ) x : Since G " and H " are representatives of classical step functions, it follows that H "t + (g 1 (G " )H " + g (G " )) x?; where is Rankine-Hugoniot decit ([6]) given by = c(v 1? v 0 )? (g 1 (u 1 )v 1 + g (u 1 )? g 1 (u 0 )v 0? g (u 0 )): By putting = 0 one can see that (u 1 ; v 1 ) has to belong to the standard Rankine- Hugoniot locus (then s = 0, since only the term s 0 (t)d " is associated with the delta function multiplied by a constant and all others are associated with the derivative of the delta function multiplied by some constant). In the case 6= 0 the condition s 0 (t) = should be satised, which implies that s(t) = t and This implies that cd 0 " (g 1 (G " )D " ) x : (15) c = g 0(u 1 )f 1 (u 1 )? g 1 (u 1 )f 1 (u 0 ) f 1 (u 1 )? f 1 (u 0 ) (since the representative of delta function D " (i.e. D? and D + ) is already determinated). That means that the delta locus is dened as the set of all (u 1 ; v 1 ) such that (15) holds. Remark. In the case when f 1 const 6= 0 the delta locus is the empty set.

DELTA AND SINGULAR DELTA LOCUS 9 4. Box approximations and delta locus Theorem 1 is also true for the box approximations. In this case D " (y) consists of two boxes, D " (y) = 8 >< >: D? " D + ; y [?"; 0] ; y [0; "] " 0; otherwise: After the substitution of u and v into (1), by using the standard Rankine- Hugoniot conditions for step functions, we have (?c[g] + [f 1 (G)H + f (G)]) 0 + s(t)([f 1 (G)D " ]?"?" +[f 1 (G)D " ] 0 0 + [f 1 (G)D " ] " " ) = 0; where h a ; i = (a) and [] denotes a jump in a point. Again the same condition for the speed holds, c = [f 1(G)H + f (G)] ; [G] and h[f 1 (G)D " ]?"?" ; i = f 1 (u 0 )D? "?1 (?") = f 1 (u 0 )D? "?1 ( (0)? " 0 (y " )) = f 1 (u 0 )D? "?1 (0)? f 1 (u 0 )D? 0 (y " ); where y " is some number in [?"; 0]. Similarly, ~y " [0; "]. Finally, Then h[f 1 (G)D " ] " " ; i =?f 1 (u 1 )D + "?1 (0)? f 1 (u 1 )D + 0 (~y " ); h[f 1 (G)D " ] 0 0 ; i = (?f 1 (u 0 )D? + f 1 (u 1 )D + )"?1 (0): h[f 1 (G)D " ]?"?" + [f 1 (G)D " ] 0 0 + [f 1 (G)D " ] " " ; i =? f 1 (u 0 )D? 0 (y " ) + f 1 (u 1 )D + 0 (~y " )! (?f 1 (u 0 )D?? f 1 (u 1 )D + ) 0 (0) =(?f 1 (u 0 )D?? f 1 (u 1 )D + )h 0 ; i; as "! 0. This gives the following two conditions D? + D + = 1 f 1 (u 0 )D? + f 1 (u 1 )D + = 0: These conditions are the same as in the previous case. In the similar way as above one can obtain the third condition which denes the delta locus g 1 (u 0 )D? + g 1 (u 1 )D + = c:

10 M. NEDELJKOV The analysis of possible cases is the same as for smooth aproximations. In both cases it seems that one splits delta function in two parts, left delta,?, and right one, +, such that its product with piecewise continuous and bounded functions is dened by (y)f(y) = lim f(y); where is? or + : y!0 In the case when system (1), () is hyperbolic, admissible part of the delta locus is the set of (u 1 ; v 1 ) such that (u 1 ; v 1 ) c 1 (u 0 ; v 0 ) where 1 < are the eigenvalues of (1), (). Remark. Eigenvalues of (1), () equals 1; (u; v) = f0 1(u)v + f(u) 0 + g 1 (u) p (f 0 1 (u)v + f0(u)? g 1(u)) + 4f 1 (u)(g 0 (u)v + 1 g0 (u)) If one uses the box approximation then one can see that the hyperbolicity of the system means 1 6= R, for u = u i, v = v i, i = 1; and for v equals some number depending on " which tends to innity as "! 0. The last will be true if f 0 1(u i ) 6= 0, i = 1;. If one uses smooth approximation, then the question of hyperbolicity is much more dicult and highly depends on functions in the uxes. Overcompressivity condition can be written in the following explicit form f1(u 0 0 )v 0 + f(u 0 0 ) + g 1 (u 0 ) p (f 0? 1 (u 0)v 0 + f 0 (u 0)? g 1 (u 0 )) + 4f 1 (u 0 )(g 0 1 (u 0)v 0 + g 0 (u 0)) g 1(u 0 )f 1 (u 1 )? g 1 (u 1 )f 1 (u 0 ) f 1 (u 1 )? f 1 (u 0 ) f0 1(u 1 )v 1 + f(u 0 1 ) + g 1 (u 1 ) p (f 0 + 1 (u 1)v 1 + f 0 (u 1)? g 1 (u 1 )) + 4f 1 (u 1 )(g 0 1 (u 1)v 1 + g 0 (u 1)) 5. Measure valued solutions In order to explain more simply the concept of solution we use the self similar solutions to (1) and (), i.e. u = u(x=t), v = v(x=t). In that case the system is (16) (17)?yu 0 (y) + (f 1 (u(y))v(y) + f (u(y))) 0 = 0?yv 0 (y) + (g 1 (u(y))v(y) + g (u(y))) 0 = 0; (18) u(?1) = u 0 ; u(1) = u 1 ; v(?1) = v 0 ; v(1) = v 1 : Our aim is to nd a solution (u; v), where u is an element of the space of piecewise continuous functions having the right and left hand limits in all points in R, P L (R), while v = (v 0 : : : :; v m+1 ) belonges to the space y0;:::;ymm(r) = M((?1; y 0 ])

DELTA AND SINGULAR DELTA LOCUS 11 M([y 0 ; y 1 ]) M([y m ; 1)), where y i, i = 0; : : :; m are some real numbers. In this case the products f 1 (u)v and g 1 (u)v exist and belong to the space y0;:::;ymm(r). Denote by M F (R) the union of all y0;:::;ymm(r), where fy 1 ; : : :; y m g is a nite set of real numbers dened above. Let w = (w 0 ; : : :; w m+1 ) y0;:::;ym M(R), w i = w i + i?1 + (? y i?1 ) + i? (? y i ); i = 1; : : :; m; w 0 = w 0 + 0? (? y 0 ); w m+1 = w m+1 + m + (? y m ); where w i are the measures with the zero mass in the points fy 0 ; : : :; y m g. The mapping : w 7! (w) = W D 0 (R) is dened in the following way. (w) := W + ( 0 + 0 )(? y 0 ) + + ( m + m )(? y m ); where W is the image of w = w 0 + +w m+1 M(R) by the usual injection of the space of measures into the space of distributions. Thus, (w) is a sum of images of two measures. The rst one is a measure which has zero mass at points y 0 ; : : :; y m and equals w 0 + + w m+1 when restricted to Rn fy 0 ; : : :; y m g. The second one is a linear combination of delta measures in the points fy 0 ; : : :; y m g. We say that (u; v) P L (R) M F (R) is a solution to (16), (17) if?y(u) 0 + (f 1 (u)v + f (u)) 0 = 0?y(v) 0 + ((g 1 (u)v + g (u))) 0 = 0 in D 0. With such interpretation, for given (u 0 ; v 0 ), the delta locus will be the set of all pairs (u 1 ; v 1 ) R such that there exists a solution in the above sense which is constant except in one point. Now we will nd the delta locus (with the above denition) for system (16), (17) and (18). Suppose that y 0 = c is the point of discontinuity and let Then u(y) = u0 ; y < c u 1 ; y > c ; v = w +? + + ; where w(y) = f 1 (u)v + f (u) = f 1 (u)w + f (u) + f 1 (u)(? + + ) v0 ; y < c v 1 ; y > c : = f 1 (u)w + f (u) + f 1 (u 0 )? + f 1 (u 1 ) + : After the injection of the above term in D 0, equation (16) becomes?yu 0 + (f 1 (u)w + f (u)) 0 + (f 1 (u 0 ) + f 1 (u 1 )) 0 = 0: The term?yu 0 + (f 1 (u)w + f (u)) 0 = const determines the shock wave speed c as usual, i.e. const = 0. So, the term (f 1 (u 0 ) + f 1 (u 1 )) 0 has to be zero, i.e. we have (19) c = f 1(u 1 )v 1 + f (u 1 )? f 1 (u 0 )v 0? f (u 0 ) u 1? u 0

1 M. NEDELJKOV and (0) f 1 (u 0 ) + f 1 (u 1 ) = 0: Equation (19) denes the speed of the shock and (0) gives the rst condition on and. Equation (17) is transformed in the following way. The rst term, yv = yw + y? +y +, is mapped into yw+y(+) D 0. The second term, g 1 (u)v+g (u) = g 1 (u)w+g (u)+g 1 (u 0 )? +g 1 (u 1 ) +, is mapped into g 1 (u)w+g (u)+(g 1 (u 0 ) + g 1 (u 1 )) D 0. After the substitution of these elements of D 0 into (17) we obtain?yw 0 + (g 1 (u)w + g ) 0? c( + ) 0 (y? c) + ( + )(y? c) +(g 1 (u 0 ) + g 1 (u 1 )) 0 (y? c) = 0: Since?yw 0 + (g 1 (u)w + g ) 0 =?(y? c), where is Rankine-Hugoniot decit, = c(v 1? v 0 )? (g 1 (u 1 )v 1 + g (u 1 )? g 1 (u 0 )v 0? g (u 0 )), and satisfy (1) () + = g 1 (u 0 ) + g 1 (u 1 ) = c: The solution to system (0), (1) and () in respect of and exists if f 1 (u 0 ) f 1 (u 1 ) 0 1 1? g 1 (u 0 ) g 1 (u 1 )?c 6= 0: The result is the same as in the case when the approximations are used. One can see that s = +, D? = =s, and D + = =s. 6. Singular shocks In the case when the uxes f 1, f, g 1 and g are polynomials it is often possible to nd a singular shock solution to system (1)-(3) for a larger set of the initial data. This will be done by adding some terms in u " beside G ". When all functions in the ux are polynomials we can control behaviour of these additional terms in u ". Suppose that the maximal degree of all polynomials in the uxes equals m. Let f 1 (y) = mx i=0 a i y i ; f (y) = mx i=0 a i y i ; g 1 (y) = We will try to nd solutions in the form mx i=0 b i y i ; g (y) = mx i=0 b i y i : (3) u " (x; t) = G " (x? ct) + s 1 (t)d " (x? ct) v " (x; t) = H " (x? ct) + s (t)d " (x? ct); where s i, u = 1; and D " are of the same form as before. A net d " is a representative of the generalized function associated with zero, and it will be described in the proof of the following theorem.

DELTA AND SINGULAR DELTA LOCUS 13 Theorem. A constant c = f 1(u 1 )v 1 + f (u 1 )? f 1 (u 0 )v 0? f (u 0 ) u 1? u 0 = [f 1(u)v + f (u)] [u] will be called a speed of the singular shock, and = c(v 1? v 0 )? (g 1 (u 1 )v 1 + g (u 1 )? g 1 (u 0 )v 0? g (u 0 )) = c[v]? [g 1 (u)v + g ] will be called Rankin-Hugoniot decit. A singular shock wave solution to (9)-(11) in the form (3) exists in the following cases. a) Let m be an even number. Then there must be possible to nd > 0 such that there exists a solution (D? ; d? ) R R + to the system (3) [f 1 (u)]d? + [v]a m d? = a m + f 1 (u 1 ) + v 1 a m [g 1 (u)]d? + [v]b m d? = b m + g 1 (u 1 ) + v 1 b m? c: b) Let m be an odd number. One of the two following conditions has to hold. (i) There exists Rn f0g such that there exists a solution (D? ; d? ) R to system (3). (ii) There exists > 0 such that there exists a solution (D? ; d? ) R R + to the system (4) [f 1 (u)]d? + (a m?1 [u] + ma m [uv] + ma m [u])d? =(a m v 1 + ma m u 1 v 1 + ma m u 1 ) + f 1 (u 1 ) + a m?1 [g 1 (u)]d? + (b m?1 [u] + mb m [uv] + mb m [u])d? =(b m v 1 + mb m u 1 v 1 + mb m u 1 ) + g 1 (u 1 ) + b m?1? c: The set of all points (u 1 ; v 1 ) R for which there exists a singular shock wave solution is called the singular delta locus. d " = d? " b?a Proof. a) Let C 1 0 (R), supp [?1; 1], R (x)dx = 1, 0 and let x + a+b " b?a!! 1=m + d + " b?a x? a+b " b?a!! 1=m ; where (d? ) m + (d + ) m = 1. Let s 1 (t); s (t) be smooth functions for t 0, s 1 (0) = s (0) = 0, and let D " be of the same form as in Theorem 1. If G " is a representative of a generalized function of the bounded type (some power of a step function, for example), then Z G " (x? ct)d j "(x? ct) (x; t)dxdt C GC Z " 1?j=m 1=m (y)dydt The last term converge to zero as "! 0, for j < m. Since the multiplication with a smooth function preserves association, there holds (G " + s(t)d " ) j G j "; j < m and (G " + s(t)d " ) m G m " + s m (t)d m " :

14 M. NEDELJKOV Let G " (y) be a generalized step function which equals u 0 for y <?" and equals u 1 for y > " One can see that G " d m " (u 0 d? + u 1 d + )(x? ct) from the construction of d ". Also, the P construction of D " i d " implies that D " d " 0. m If?(y) = i=0 iy i, then?(g " + s(t)d " )?(G " ) + s m (t) m d m " : After substitution of U " = G " +s 1 (t)d " and V " = H " +s (t)d " in system (9)-(11), one can see that from the rst equation that c = [f1(u)v+f(u)] (Rankin-Hugoniot [u] condition) and from the second that s (t) = t, where = c[v]? [g 1 (u)v + g ] is a Rankin-Hugoniot decit. Further on, after association procedure like in the proof of Theorem 1 and grouping coecients in respect of and 0 one nds that s 1 (t) = ~st 1=m and (recall that D? + D + = (d? ) m + (d + ) m = 1) [f 1 (u)]d? + [v]a m ~s m d? = a m ~s m + f 1 (u 1 ) + v 1 a m ~s m [g 1 (u)]d? + [v]b m ~s m d? = b m ~s m + g 1 (u 1 ) + v 1 b m ~s m? c: This proves the rst part of the theorem. b) (i) The proof of the assertion is the same as the one of a). The only dierence is that ~s m has not to be positive. (ii) The construction of d " in the second case is a slightly dierent. Let be as above and d? 1=(m?1) x + (a + 3b)=4 d? 1=(m?1) x + (3a + b)=4 d " = "(b? a)? "(b? a)=4 "(b? a) "(b? a)=4 d + x? 1=(m?1) (3a + b)=4 d + x? 1=(m?1) (a + 3b)=4 + "(b? a) + "(b? a)=4 "(b? a) ; "(b? a)=4 where (d? ) m?1 + (d + ) m?1 = 1. Like in the previous case, one can see that (G " + s(t)d " ) j G j "; j < m? 1 and (G " + s(t)d " ) m?1 G m?1 " + s m?1 (t)d m?1 " for every generalized function G " of the bounded type. Because of the specic denition of d " one can see that G " d m " 0 if G " is a generalized step function. The above means that?(g " + s 1 (t)d " )?(G " ) + m?1 s m?1 1 (t)d m?1 " + m m s m?1 1 (t)g " d m?1 " : Substituting U " = G " + s 1 (t)d " and V " = H " + s (t)d " in system (9)-(11) one can see that c = [f1(u)v+f(u)], s [u] (t) = t, where = c[v]? [g 1 (u)v + g ] like in a). Also, one nds that s 1 (t) = ~st 1=(m?1) and [f 1 (u)]d? + (a m?1 [u] + ma m [uv] + ma m [u])~s m?1 d? =~s m?1 (a m v 1 + ma m u 1 v 1 + ma m u 1 ) + f 1 (u 1 ) + a m?1 ~s m?1 [g 1 (u)]d? + (b m?1 [u] + mb m [uv] + mb m [u])~s m?1 d? =~s m?1 (b m v 1 + mb m u 1 v 1 + mb m u 1 ) + g 1 (u 1 ) + b m?1 ~s m?1? c: This proves the theorem. In the following corollaries we shall describe singular delta locus in the case f 1 a where a is a nonzero constant. This is the case when the delta locus is the empty set (see Theorem 1).

DELTA AND SINGULAR DELTA LOCUS 15 Corollary 1. Let f 1 a, a 6= 0 is a constant, and let g 1 be non constant function. a) If m is an even number, then (u 1 ; v 1 ) is in a delta singular locus if a m 6= 0 (i.e. deg f = m), there exists > 0 such that sgn([v]a m ) = sgn(a + v 1 a m ) and [g 1 (u)] 6= 0. b) If m is an odd number and a m 6= 0, then the delta singular locus is the set R n fu 1 R: [g 1 (u)] = 0g. c) Let m be an odd number, a m = 0 and a m?1 6= 0 (i.e. deg f = m? 1). Then (u 1 ; v 1 ) is in a delta singular locus if a=a m?1 < 0 and [g 1 (u)] 6= 0. Let us note that in the case deg f < m? 1 the delta singular locus is the empty set ( = 0 is the condition for Rankin-Hugoniot locus). Proof. In all of these cases, the condition [g 1 (u)] 6= 0 ensures that there always exists D? if appropriate d? exists. a) The proof easily follows from (4) since D? is an arbitrary real number. b) The proof is an immediate consequence of b) (i) in the previous theorem. c) One can see from the rst equation in (4) that =?a=a m?1 > 0. Corollary. Let f 1 a and g 1 b, where a 6= 0 and b are some constants. a) Let a m and b m be dierent from zero. Then = ((b? c)a m? ab m ) a m?1 b m? a m b m?1 has to be greater than zero, and (u 1 ; v 1 ) is in delta singular locus if d? = ma mu 1 + a + a m?1 ma m [u] [0; 1] for dened above. b) Let a m 6= 0 and b m = 0. Then = c? b b m?1 has to be positive, and (u 1 ; v 1 ) is in a delta locus if d? = ma mu 1 + a + a m?1 ma m [u] [0; 1] for dened above. c) Let a m = 0 and b m 6= 0. Then =?a a m?1 has to be positive, and (u 1 ; v 1 ) is in a delta locus if d? = mb mu 1 + b + b m?1? c ma m [u] [0; 1] for dened above.

16 M. NEDELJKOV d) Let a m = b m = 0. Then a delta singular locus exist, and it is R, only if c? b b m?1 =?a a m?1 > 0: Proof. In all of these cases we are using the second denition of d " and the assertions follow by solving the system of equations (4) with the unknowns and d?. Remark 1. As in the rst section all the calculation can be made for box approximations. Remark. In [6] the singular parts of functions u and v have nonempty intersection (even d " = D " ). This is possible because in the system (4) there are no multiplication of d " and D ", i.e. f 1 =?1, g 1 = 0 are the constants. It is also possible to choose d " and D " with non-disjoint supports if ux functions are linear in u. In general case, problems with terms D " d j " can appear. The solution of Keytz and Kranzer is the same as in the case c) of Corrolary for m = 3, a =?1, b = 0, a m?1 = 1, b m = 1=3 and b m?1 = 0. When all points for which the overcompressibility condition does not hold are extracted, one obtain the same area as them (it is denoted by Q 7 in their paper). Acknowledgement The author is grateful to Barbara Keytz, Michael Oberguggenberger and Stevan Pilipovic for their great help in the preparation of this paper. He would like to thank to Austrian Academic Exchange Service ( OAD) for nancing his visit to Austria during which the paper was written. Also he is thankful to Institut f. Geometrie u. Mathematik, Univeristat Innsbruck and E. Schrodinger Institute, Wien, for their great hospitality. References 1. J. F. Colombeau, Elementary Introduction in New Generalized Functions, North Holland, 1985.. C. M. Dafermos, R. J. DiPerna, The Riemann problem for certain classes of hyperbolic systems of conservation laws, J. Dierential Equations 0 (1976), 90-114. 3. V. G. Danilov, V. P. Maslov, V. M. Shelkovich, Algebras of singularities of singular solutions to rst-order quasilinear strictly hyperbolic systems, Preprint (1998). 4. G. Ercole, Delta shock waves as self-similar viscosity limits, Preprint (1997). 5. B. T. Hayes, P. G. Le Flock, Measure solutions to a strictly hyperbolic system of conservation laws, Nonlinearity 9 (1996), 1547-1563. 6. B. L. Keytz, H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Di. Eq. 118, (1995), 40-451. 7. P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia, 1973. 8. V. P. Maslov, G. A. Omel'anov, Asymptotic solitonlike solutions of equations with small dispersion, Russian Math. Surveys 36 (1981), 73-119. 9. M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Dierential Equations, Pitman Res. Not. Math. 59, Longman Sci. Techn., Essex, 199. 10. M. Oberguggenberger, Y.-G. Wang, Generalized solutions to conservation laws, Zeitschr. Anal. Anw. 13 (1994), 7-18. 11. D. Tan, T. Zhang, Y. Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Di. Eq. 11 (1994), 1-3. 1. Y. Zheng, Systems of conservation laws with incomplete sets of eigenvectors everywhere, in A Volume Dedicated to Professor Xiaqi Ding's 70th Birthay (to appear).