DELTA SHOCK WAVES AS A LIMIT OF SHOCK WAVES

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1 st Reading October 0, 200 :2 WSPC/JHDE 002 Journal of Hyperbolic Differential Equations Vol. 4, No. 4 (200) 2 c World Scientific Publishing Company DELTA SHOCK WAVES AS A LIMIT OF SHOCK WAVES DARKO MITROVIĆ University of Montenegro, Podgorica, Montenegro Department of mathematical Sciences Norvegian Institute of Science and Technology Sentralbygg 2, Alfred Getz vei, NO-4 Trondheim, Norway mitrovic@math.ntnu.no MARKO NEDELJKOV Department of Mathematics and Informatics University of Novi Sad. Trg D. Obradovića Novi Sad, Serbia markonne@uns.ns.ac.yu Received Oct Accepted 2 Apr. 200 Communicated by the Editors Abstract. We discus the existence of delta shock waves obtained as a limit of two shock waves. For that purpose we perturb a prototype of weakly hyperbolic 2 2system (sometimes called the generalized pressureless gas dynamics model ) by an additional term (called the generalized vanishing pressure ). The obtained perturbed system is strictly hyperbolic and its Riemann problem is solvable. Since it is genuinely nonlinear, its solution consists of shocks and rarefaction waves combination. As perturbation parameter vanishes, the solution converges in the space of distribution. Specially, a solution consisting of two shocks converge to a delta function. Also, we give a formal definition of approximate solution and prove a kind of entropy argument. The paper finishes by a discussion about delta shock interactions for the original system. Keywords: Delta shock wave; hyperbolic perturbation. Mathematics Subject Classification 2000: L6, L6. Introduction It is well-known (see [4,.. Theorem], for example) that Riemann problem for strictly hyperbolic one-dimensional system of conservation laws has a solution at least for initial data with small variation. Such a result does not exist in the case of weakly hyperbolic systems. So, it is not a surprise that some of them have non-classical solutions. One possible non-classical solution is so called delta shock

2 st Reading October 0, 200 :2 WSPC/JHDE D. Mitrović &M.Nedeljkov 2 wave. Roughly speaking, it is a shock wave with added Dirac delta function supported by a shock front. Our aim is to use a well-established theory for strictly hyperbolic 2 2 systems to verify existence of delta shocks for a class of weakly hyperbolic systems. One can find different methods for delta shock verification in the literature (see the papers cited bellow). Let us mention a vanishing viscosity method (that we use for comparison) where one adds a second order spatial derivative perturbed by a small parameter ε on the right-hand side. If a distributional limit to the perturbed system contains the delta function, then we say that a delta shock solution to the original equation is verified. A solution concept for the non-perturbed system may be different as one can find some examples bellow. The present paper is an attempt to show that such kind of solution may exists as a limit of elementary wave solution of the system with a different perturbation: The flux of the system is perturbed by some additional term multiplied by a small parameter ε. Unlike the vanishing viscosity procedure that produces hyperbolicparabolic or parabolic second order systems, now we will get strictly hyperbolic first order system which can be solved by the means of elementary waves combination. The solution to the perturbed system represents a delta shock solution to the unperturbed system if its distributional limit exists and contains delta function. We give a short review of some results from the literature concerning weakly hyperbolic systems and delta shocks. The system u t + f(u) x =0, v t +(a(u)v) x =0, a(u) =f (u), f convex, was studied in [0]. Author used Volpert s averaged product to prove existence and uniqueness of solution to the above system. The function v is the spatial derivative of bounded variation function w, thus it belongs to the space of Borel measures. Function u is a classical weak solution to the first equation and w is a BV (function with bounded variation) solution to non-conservative equation w t + a(u)w x =0. The sticky particles model (or pressureless gas dynamics) with initial data belonging to BV space ρ t +(ρu) x =0, (ρu) t +(ρu 2 ) x =0. (.) was solved in []. The authors used a smart variables change, so ρ and u become space-derivatives of a density-flux pair of some new scalar conservation law. Again, Volpert product is used and ρ belongs to the space of Borel measures. System (.) is one of investigation subjects in []. The authors introduce so called generalized variational principle and solve the system in the space of Borel measures.

3 st Reading October 0, 200 :2 WSPC/JHDE 002 Delta Shock Waves as a Limit of Shock Waves Author of [] used a local (with respect to phase space) change of coordinates in order to define a weak solution to generalized pressureless gas dynamics model ρ t +(ρg(u)) x =0, (ρu) t +(ρug(u)) x =0, (.2) where g is non-decreasing function. Using that definition, the author proved that there exists a solution to initial value problem for (.2) at least in some strip [0,T). It is important to say that there is no solution after the first delta shock delta shock interaction time. Riemann problem for (.2) has been solved by a direct use of Borel measures in []. That result is then used for proof of global existence of initial value problem when initial data are bounded variation functions. Contrary to the above paper, delta shock interaction with another one (or with a contact discontinuity) always produces a single delta shock in that setting. The author in [8] verified delta shock existence to (.) by vanishing viscosity approximation. A different approach called weak asymptotic method was used for a class of conservation law systems in [6]. Roughly speaking, the method consists of expanding functions in the distribution space with respect to a small parameter ε. That expansion is made to be continuous with respect to the time variable in order to see how singularity is arising from initial data. The same authors deal with delta shock interaction problem in []. Smooth function nets represent weak solutions to conservation law systems in [2]. The equality is substituted by the distributional limit. The goal of that paper was to find conditions that permits a delta shock (and a singular shock a similar type to delta shock) Riemann solution. Two sided ( split ) delta functions has been used in [] to solve delta shock interaction problem for simplified magnetohydrodynamics model from [8]. The idea was to avoid definition of an L function on a discontinuity line (set of Lebesgue measure equals zero). That was done by dividing a half-space by discontinuity lines into sets where one can define multiplication of Radon measure with a continuous function. Finally, we refer to [2] where delta shock existence was verified for system (.). The second equation in the system is perturbed with vanishing pressure term: ρ t +(ρu) x =0, (ρu) t +(ρu 2 + εp(ρ)) x =0, (.) where p(ρ) =κρ γ,forγ (, ). The new system is strictly hyperbolic and has a solution of arbitrary Riemann data. The authors proved that Riemann solution to (.) converge to a solution of (.) in the measure theoretical sense (see already cited papers [] or [8]).

4 st Reading October 0, 200 :2 WSPC/JHDE D. Mitrović &M.Nedeljkov Our main intention is to extend that results to a case of generalized pressureless gas dynamics model (.2). It serves as a prototype of a non-strictly hyperbolic 2 2 system. Our method can be simply described as follows. First, we shall perturb the second equation in system (.2) with generalized vanishing pressure term p = p(ε, ρ). Such a perturbation is called the hyperbolic perturbation. Second, we shall discuss a distributional limit of a weak, bounded variation solution to the perturbed system. The limit is a Radon measure (sometimes called signed Radon measure ), precisely the one given in []. All the results from [2] are recovered. After that, we shall propose a form of approximate solutions to the original system and prove some kind of entropy inequality. Delta shock solution to (.2) can be viewed as a net of piecewise constant functions (ρ ε,u ε ) satisfying initial conditions and the system after distributional limit is taken. Approximated solution resembles nets of smooth function used in [2], but they have two crucial differences: They are piecewise constant functions not smooth functions. Instead of fixed width for each t, delta shocks move along edges of the cone x = c,ε t and x = c 2,ε t. The speeds c and c 2 satisfy lim ε 0 c,ε c 2,ε =0. At the end of the paper, we try to give an answer on a problem of interaction of two delta shocks in system (.2). The result is only partial. Restriction to system (.) does not give complete result, too. Thus we dare to say that the interaction problem is still open. At least there is no verification of any solution after some double delta shock interactions. Looking at previous results, one can find opposite indicators: The solution concept from [] does not allow two delta shocks to interact, but with the solution concept in [] one gets a single delta shock as a result of the interaction. Let us mention that the delta shock interaction problem is solved by using the weak asymptotic method (see []) or split delta measures (see []) for some other, let say simpler systems. There is a result about strictly hyperbolic systems and singular shocks in [4]. In all of these papers, interaction problem with delta shocks is solved (in different solution concepts), but there are some numerical indications that the result of two delta shocks interaction may not produce a delta shock in the sense of hyperbolic perturbation for system (.2) or (.). That question is postponed for future research. 2. BV Solutions We shall first describe a perturbed system and its elementary wave solutions. One can look at standard textbooks about conservation law systems, [4] or [6], for example. The domain Ω = {(ρ, u) (R + {0}) R} will be fixed in the whole paper. Let p and q be fixed functions with the following properties: p C 2 ([0, )), p([0, )) = [0, ), p 0, q 0, p > 0. (2.)

5 st Reading October 0, 200 :2 WSPC/JHDE 002 Delta Shock Waves as a Limit of Shock Waves Consider the following Riemann problem ρ t +(ρg(u)) x =0, (ρu) t +(ρug(u)+εp(ρ)) x =0, { { ρ0, x < 0, u0, x < 0, ρ t=0 = u t=0 = ρ, x > 0, u, x > 0, (2.2) where ε is a perturbation parameter. After a change of variables ρu v, forρ>0 system (2.2) can be written in the evolution form ρ t +(ρg(v/ρ)) x =0, v t +(vg(v/ρ)+εp(ρ)) x =0, { { (2.) ρ0, x < 0, v0, x < 0, ρ t=0 = v t=0 = ρ, x > 0, v, x > 0. Eigenvalues associated to the system are λ = g(v/ρ) εg (v/ρ)p (ρ) and λ 2 = g(v/ρ)+ εg (v/ρ)p (ρ), so it is strictly hyperbolic. The appropriate right-handed characteristic vectors are ( r =, v g (v/ρ) ρ ) εp (ρ) ρ, g (v/ρ) ( r 2 =, v g (v/ρ)+ρ ) εp (ρ) ρ. g (v/ρ) Since λ i r i > 0, i =, 2, both fields are genuinely nonlinear. The first rarefaction curve (R ) for system (2.2) is the integral curve of dρ dσ =, dv dσ = v g (v/ρ) ρ εp (ρ)) ρ, g (v/ρ)) ρ σ=0 = ρ 0, v σ=0 = v 0 = ρ 0 u 0, σ > 0, and the second one (R 2 ) is the integral curve of dρ dσ =, dv dσ = v g (v/ρ)+ρ εp (ρ)) ρ, g (v/ρ)) ρ σ=0 = ρ 0, v σ=0 = v 0, σ > 0. (2.4) (2.) In the sequel, (ρ Ri,v Ri )(or(ρ Ri,u Ri ) if one uses (ρ, u) variables) will denote values of an i rarefaction wave between x = λ i (ρ 0,v 0 )t and x = λ i (ρ,v )t, i =, 2, when the point (ρ,v ) lies on R i.

6 st Reading October 0, 200 :2 WSPC/JHDE D. Mitrović &M.Nedeljkov Hugoniot curve is the set of all (ρ, u) satisfying ( (u u 0 )(g(u) g(u 0 )=ε ) (p(ρ) p(ρ 0 )) (2.6) ρ 0 ρ (it directly follows from Rankine Hugoniot conditions for (2.2)). Lax entropy conditions imply that S (-shock curve) is determined by (2.6) for ρ>ρ 0, while S 2 (2-shock curve) is determined by the same relation for ρ<ρ 0. Let us look at the non-perturbed system (.2) now. Contact discontinuity for it is supported by the integral curve of dρ dσ =, dv dσ = v ρ, ρ t=0 = ρ 0, u t=0 = u 0, σ R. (2.) Solution to system (2.) is given by v/ρ = v 0 /ρ 0 (or u = u 0 ) if one uses the other variables. Observe the following useful facts. Shock curves for (2.2) are below the contact discontinuity line u 0 = u for system (.2). We expect a solution to be -shock followed by 2-shock. One would see that the assumption g (u) > 0 suffice to prove that in the next section. The right-hand sides of systems (2.4) and (2.) tends to the right-hand side of system (2.). By the basic theorem in ODE s theory, one can see that the solutions to (2.4) and (2.) tend to the solution to (2.) as ε 0. Since ρ is positive in the interior of the domain Ω, and p is an increasing function, one can see that v from (2.4) and (2.) satisfies v>ρv 0 /ρ 0 (or u>u 0 ), i.e. the rarefaction curves for system (2.2) are above contact discontinuity curve for system (.2). For a given initial data (ρ 0,u 0 )and(ρ,u ), u >u 0,thereexistsε small 2 enough such that one can always find a solution consisting of two rarefaction waves connected by the vacuum state (ρ = 0). Denote by (0,u m ) the intersection point 2 of the R starting from (ρ 0,u 0 ) with the line ρ =0andby(0,u 2 m ) the intersection point of the inverse R 2 starting from (ρ,u ) with the line ρ = 0 (i.e. (ρ,u ) lies 2 on R 2 that starts from (0,u m ). In the vacuum state variable u takes values between u m and u2 m. 2 In the case when u 0 = u (i.e. v 0 /ρ 0 = v /ρ ), a solution to (2.2) will be a -rarefaction wave followed by a 2-shock if ρ <ρ 0.Ifρ >ρ 0, then -shock is 2 followed by 2-rarefaction wave. The proof of the above assertion one can find in Sec. 4.. Shock Waves The main result of the paper is given in the following theorem for the case u 0 >u. Since there is no vacuum state in a solution to the perturbed system in that case, we can use variable u in computations which are to follow.

7 st Reading October 0, 200 :2 WSPC/JHDE 002 Delta Shock Waves as a Limit of Shock Waves 2 Theorem.. For each pair of initial states (ρ 0,u 0 ), (ρ,u ) satisfying u = v /ρ <u 0 = v 0 /ρ 0, the Riemann problem (2.2) has a unique solution consisting from -shock followed by 2-shock for ε small enough. Let us denote the solution by (ρ ε,u ε ). Its distributional limit is given by where lim ε 0 ρε = G(x ct)+t const δ(x ct), lim ε 0 uε = H(x ct), { ρ0, y < 0 G(y) = ρ, y > 0 { u0, y < 0, and H(y) = u, y > 0. (.) (.2) Proof. Let ε 0 be a positive number such that (ρ 0,u 0 ) can be connected with (ρ,u ) for ε<ε 0. We have to find (ρ ε,u ε ) R + R such that (ρ 0,u 0 ) is connected with (ρ ε,u ε ) by an -shock, while (ρ ε,u ε ) is connected with (ρ 0,u 0 )bya2-shock. Rankine Hugoniot conditions for both shocks imply ( (u u 0 )(g(u) g(u 0 )) = ε ) (p(ρ) p(ρ 0 )), (.) ρ 0 ρ ( (u u )(g(u) g(u )) = ε ) (p(ρ) p(ρ )). (.4) ρ ρ The first task is to prove that there exists a solution (ρ ε,u ε ) (max{ρ 0,ρ }, ) (u,u 0 ) to system of nonlinear equations (..4). Lax entropy conditions can be true only if ρ ε max{ρ 0,ρ } because ρ is increasing along S, while ρ decreasing along S 2. Therefore, the initial choice of the domain of (ρ, u) is appropriate. Immediately it follows that for every fixed u<u 0 there exists a unique solution ρ>ρ 0 to (.). That is consequence of the fact that p and g are increasing functions. Therefore there exists a function ρ c :(u,u 0 ) (ρ 0, ), ρ c (u 0 )=ρ 0, such that (.) holds true for (ρ, u) =(ρ c (u),u). Let check that ρ c is a decreasing function. Denote ( H(u) :=ε ) (p(ρ c (u)) p(ρ 0 )). ρ 0 ρ c (u) Differentiation of the left-hand side of (.) gives H (u) = d du ((u u 0)(g(u) g(u 0 ))) = g(u) g(u 0 )+g (u)(u u 0 ) < 0, (.) 2 2 because g > 0andu<u 0. If we differentiate the right-hand side of the same equality, we have H (u) = ε ( ρ 2 c (u)ρ c (u)(p(ρ c(u)) p(ρ 0 )) + ε ) p (ρ c (u))ρ c ρ 0 ρ c (u) (u) ( ( = ερ p(ρc (u)) p(ρ 0 ) c(u) ρ 2 c (u) + ) ) p (ρ c (u)). ρ 0 ρ c (u)

8 st Reading October 0, 200 :2 WSPC/JHDE D. Mitrović &M.Nedeljkov 2 Using the fact p > 0andρ c (u) >ρ 0 for u (u,u 0 ), we have ( p(ρ c (u)) p(ρ 0 ) ρ 2 c (u) + ) p (ρ c (u)) > 0. ρ 0 ρ c (u) Thus, sign(h (u)) = sign(ρ c (u)) and (.) imply ρ c < 0foru (u,u 0 ). In the same way as above, one can see that there exists an increasing function ρ c2 :(u,u 0 ) (ρ, ), ρ c2 (u )=ρ, implicitly defined by (.4): ( (u u )(g(u) g(u )) = ε ) (p(ρ c2 (u)) p(ρ )). ρ ρ c2 (u) Since the term on the left-hand side of (.) is bounded with respect to ε, wehave ρ c (u ) >ρ 0 for ε small enough. With the same type of reasoning, one can see from (.4) that ρ c2 (u 0 ) >ρ. These properties of functions ρ c and ρ c2 ensures that there exist unique interaction point (ρ ε,u ε ) (max{ρ 0,ρ }, ) (u,u 0 )ofcurves (ρ c (u),u)and(ρ c2 (u),u). The first task is finished. Let us now look at a limiting behavior of (ρ ε,u ε )asε 0. From (.) and (.4), it follows that the value ρ ε as ε 0, while u ε stays bounded. More precisely p(ρ ε ) /ε as ε 0. (.6) In order to obtain the distributional limit of the above solution as ε 0, one has to prove that lim ε 0 u ε and lim ε 0 εp(ρ ε ) do exist. Let us define ( F (ε, ρ, u) :=(u 0 u)(g(u 0 ) g(u)) ε ) (p(ρ) p(ρ 0 )) = 0, ρ 0 ρ ( G(ε, ρ, u) :=(u u)(g(u ) g(u)) ε ) (.) (p(ρ) p(ρ )) = 0. ρ ρ Since F F D s = ρ u G G ρ u ( ( ε ρ 2 (p(ρ) p(ρ 0)) + ρ 0 ρ = ( ( ε ρ 2 (p(ρ) p(ρ )) + ρ ρ ( ( = ε ρ 2 (p(ρ) p(ρ 0)) + ρ 0 ρ ( ( + ε ρ 2 (p(ρ) p(ρ )) + ) ρ ρ ) ) p (ρ) ) ) p (ρ) ) p (ρ) g(u) g(u 0 )+(u u 0 )g (u) g(u) g(u )+(u u )g (u) ) (g(u) g(u )+(u u )g (u)) ) p (ρ) (g(u) g(u 0 )+(u u 0 )g (u)) < 0, on the domain (0,ε 0 ) (max{ρ 0,ρ }, ) (u,u 0 ), the Implicit Function theorem implies that it is possible to define ρ ε := ρ(ε) andu ε := u(ε), ε<ε 0 by (.).

9 st Reading October 0, 200 :2 WSPC/JHDE 002 Delta Shock Waves as a Limit of Shock Waves Immediately, one could see that ρ(ε) 0asε 0because ρ(ε) is strictly greater than zero (ρ(ε) > max{ρ 0,ρ }). Also, from the same theorem, we have F F F F ρ ε = ε u G Ds u, G ε = ρ ε G G Ds. ε u ρ ε Simple calculations yields F ( F ε u ) (p(ρ) p(ρ 0 )) g(u) g(u 0 )+(u u 0 )g (u) ρ 0 ρ G = G ( ε u ) (p(ρ) p(ρ )) g(u) g(u )+(u u )g (u) ρ ρ ( = ) (p(ρ) p(ρ 0 ))(g(u) g(u )+(u u )g (u)) ρ 0 ρ ( ) (p(ρ) p(ρ ))(g(u) g(u 0 )+(u u 0 )g (u)) > 0. ρ ρ Since D s < 0, the above relation implies that ρ(ε) is a decreasing function, i.e. ρ ε as ε 0. Also, F F ρ ε G G ρ ε ( ( ε ρ 2 (p(ρ) p(ρ 0)) + ) ) ( p (ρ) ) (p(ρ) p(ρ 0 )) ρ 0 ρ ρ 0 ρ = ( ( ε ρ 2 (p(ρ) p(ρ )) + ) ) ( p (ρ) ) (p(ρ) p(ρ )) ρ ρ ρ ρ = ε ( ρ 2 (p(ρ) p(ρ 0))(p(ρ) p(ρ )) ) ρ 0 ρ ( + ε )( ) (p(ρ ) p(ρ 0 ))p (ρ). ρ 0 ρ ρ ρ If ρ 0 ρ,then ρ 0 ρ 0andp(ρ ) p(ρ 0 ) 0. In that case, the above determinant is positive for ε small enough. So u(ε) is an increasing function, i.e. u ε decreases as ε 0. On the other hand, if ρ 0 >ρ,then ρ 0 ρ > 0, p(ρ ) p(ρ 0 ) < 0, and u(ε) is a decreasing function, i.e. u ε increases as ε 0. In both cases the sequence u ε is a monotone. Since it is bounded, there exists lim ε 0 u ε =: u s. Now, it easy to see that (.) implies lim εp(ρ ε)=ρ 0 (u 0 u s )(g(u 0 ) g(u s )), ε 0

10 st Reading October 0, 200 :2 WSPC/JHDE D. Mitrović &M.Nedeljkov or that (.4) implies lim εp(ρ ε)=ρ (u u s )(g(u ) g(u s )). ε 0 Thus, u s is a unique solution to the equation f(u s ):=ρ 0 (u 0 u s )(g(u 0 ) g(u s )) ρ (u u s )(g(u ) g(u s )) = 0. (.8) The uniqueness follows from the fact that f (u s )=ρ 0 (g(u s ) g(u 0 )+(u s u 0 )g (u s )) + ρ (g(u ) g(u s )+(u u s )g (u s )) < 0. Now we have to show that these shocks are admissible (in Lax sense).the speeds of the shocks constructed above are given by c,ε = ρ εg(u ε ) ρ 0 g(u 0 ) and c 2,ε = ρ g(u ) ρ ε g(u ε ). (.) ρ ε ρ 0 ρ ρ ε The first shock is admissible if g(u 0 ) εg (u 0 )p (ρ 0 ) > ρ εg(u ε ) ρ 0 g(u 0 ) >g(u ε ) εg ρ ε ρ (u ε )p (ρ ε ). 0 These inequalities imply ρ ε (g(u ε ) g(u 0 )) < εg ρ ε ρ (u 0 )p (u 0 ) (.0) 0 and ρ 0 (g(u ε ) g(u 0 )) > εg ρ ε ρ (u ε )p (ρ ε ). (.) 0 The left-hand side of (.0) tends to g(u s ) g(u 0 ) < 0, while the term on the righthand side tends to 0 as ε 0. Thus, (.0) is satisfied for ε small enough. Relation (.6) permits us to substitute ε by /p(ρ ε ) in (.). Since ρ ε as ε 0, it is enough to see that p(ρ) ρ 2 That proves (.). p (ρ) ρ p (ρ) as ρ (L Hospital rule). The second shock is admissible if g(u ε )+ εg (u ε )p (ρ ε ) > ρ g(u ) ρ ε g(u ε ) >g(u ε )+ εg ρ ρ (u )p (ρ ). ε These inequalities imply ρ (g(u ) g(u ε )) < εg ρ ρ (u ε )p (u ε ) (.2) ε and ρ ε (g(u ) g(u ε )) > εg ρ ρ (u )p (ρ ). (.) ε The left-hand side of (.) tends to (g(u ) g(u s )) > 0, while the term on the right-hand side tends to 0 as ε 0. Thus, (.) is satisfied for ε small enough. Proof of inequality (.2) can be done in the same way.

11 st Reading October 0, 200 :2 WSPC/JHDE 002 Delta Shock Waves as a Limit of Shock Waves By (.), we have c 2,ε c,ε = (ρ 0g(u 0 ) ρ g(u )+g(u ε )(ρ ρ 0 )) 0. ρ ε Since (c 2,ε c,ε )tρ ε ρ 0 g(u 0 ) ρ g(u )+g(u s )(ρ ρ 0 ), we have ρ(x, t) G(x ct)+t(ρ 0 g(u 0 ) ρ g(u )+g(u s )(ρ ρ 0 ))δ(x ct), where c = g(u s ) = lim ε 0 c,ε = lim ε 0 c 2,ε. This completes the proof. 4. Other Elementary Wave Combinations In the previous section, we were dealing with the initial data satisfying u 0 >u. Let us now examine the case u 0 u. Theorem 4.. (a) For u 0 = u Riemann problem (2.2) has a solution consisting of -rarefaction wave followed by 2-shock when ε is small enough. Its distributional limit is ρ 0, x < g(u 0 ), ρ(x, t) = ρ, x > g(u 0 ), (4.) u(x, t) =u 0, which is a contact discontinuity solution to (.2). (b) If u 0 < u, then Riemann problem (2.2) has a solution consisting of -rarefaction followed by 2-rarefaction wave for ε small enough. Its distributional limit is the combination of two contact discontinuities joined by the vacuum state, ρ 0, x < g(u 0 )t, ρ(x, t) = 0, g(u 0 )t<x<g(u )t, ρ, x > g(u )t, (4.2) u 0, x < g(u 0 )t, u(x, t) = const, g(u 0 )t<x<g(u )t, u, x > g(u )t. Proof. (a) The initial conditions for system (2.2) are { ρ0, x < 0, ρ t=0 = ρ, x > 0, u t=0 = u 0. (4.) Let us consider two different cases: ρ <ρ 0 and ρ 0 <ρ (the case ρ 0 = ρ is trivial: u u 0, ρ ρ 0 is the solution). Thefirstcaseu < u 0. Let us prove that the states (u 0,ρ )and(u 0,ρ 0 )canbe connected by a -rarefaction wave followed by 2-shock. The left-hand state (ρ, u)

12 st Reading October 0, 200 :2 WSPC/JHDE D. Mitrović &M.Nedeljkov can be connected with the end state (ρ,u 0 )by2-shockif ( (u u 0 )(g(u) g(u 0 )) = ε ) (p(ρ) p(ρ )). (4.4) ρ ρ One has to prove that for every fixed ρ>ρ there exists a unique u>u 0 such that (4.4) holds. Define G(u) :=(u u 0 )(g(u) g(u 0 )). Since G(u 0 )=0andgis increasing we have G(u) as u,andg([u 0, )) = [0, ). The right-hand side of (4.4) is positive for every ρ>ρ,sothereexistsa u ρ >u 0 such that ( G(u ρ )=ε ) (p(ρ) p(ρ )). ρ ρ Since g is increasing and u>u 0,wehave G (u) =g(u) g(u 0 )+(u u 0 )g (u) > 0. So, the solution u ρ is unique. Denote by u S2 = u S2 (ρ) the function defined by the implicit relation ( (u S2 (ρ) u 0 )(g(u S2 (ρ)) g(u 0 )) = ε ) (p(ρ) p(ρ )). (4.) ρ ρ We have just proved that the function u S2 is well-defined, u S2 :[ρ,ρ 0 ) [u 0, + ) and u S2 (ρ )=u 0. Also, it is increasing in the interval (ρ,ρ 0 ): Differentiation of (4.) gives u S 2 (ρ)(g(u S2 (ρ)) g(u 0 )+(u S2 (ρ) u 0 )g (u S2 (ρ))) = ε ( ρ 2 (p(ρ) p(ρ 0)) + ε ) p (ρ). ρ ρ 2 Since p and g are increasing functions, u>u 0 and ρ>ρ,wehaveu S 2 (ρ) > 0and u S2 (ρ 0 ) >u 0. The function u R :(ρ,ρ 0 ) (u 0, + ) is the solution to du R dρ = εp (ρ) ρ g (u R (ρ)), u R (ρ 0 )=u 0. (4.6) 2 (The aboveequationis derivedfrom (2.4) when v is substituted by ρu.) Since p andg are increasing function, one can see that u R is decreasing. Therefore, u R (ρ ) >u 0.

13 st Reading October 0, 200 :2 WSPC/JHDE 002 Delta Shock Waves as a Limit of Shock Waves From the above we conclude that there exists ρ m,ε (ρ,ρ 0 ) such that u m,ε = u S2 (ρ m,ε )=u R (ρ m,ε ). Therefore, the solution to problem (2.2), (4.) is given by: ρ 0, x < λ (u 0,ρ )t, ρ ρ ε R (x/t), λ (u 0,ρ )t<x<λ (u m,ε,ρ m,ε )t, (x, t) = ρ m,ε, λ (u m,ε,ρ m,ε )t<x<ct, ρ, x > ct, (4.) u 0, x < λ (u 0,ρ )t u ε u R (x/t), λ (u 0,ρ )t<x<λ (u m,ε,ρ m,ε )t, (x, t) = u m,ε, λ (u m,ε,ρ m,ε )t<x<ct, u, x > ct. Here, c = g(u)ρ g(um,ε)ρm,ε ρ ρ m,ε is the speed of 2-shock. The second case ρ >ρ 0. This case can be done in a similar way. Now, solution consists from -shock followed by 2-rarefaction wave and it is given by ρ 0, x < ct, ρ m,ε, ct < x < λ 2 (u m,ε,ρ m,ε )t, ρ ε (x, t) = ρ R2 (x/t), λ 2 (u m,ε,ρ m,ε )t<x<λ 2 (u 0,ρ )t, ρ, x > λ 2 (u 0,ρ )t, (4.8) u 0, x < ct, u m,ε, ct < x < λ 2 (u m,ε,ρ m,ε ), u ε (x, t) = u R2 (x/t), λ 2 (u m,ε,ρ m,ε )t<x<λ 2 (u 0,ρ ), u, x > λ 2 (u 0,ρ )t. Here, c = g(um,ε)ρm,ε g(u0)ρ0 ρ m,ε ρ 0 is the speed -shock. Since we have a solution (ρ ε,u ε ) given by (4.) and (4.8) in both cases, it remains to find their limits as ε 0. We have proved above that ρ is always between ρ 0 and ρ, thus bounded with respect to ε. Relation (4.4) implies that u m,ε and ρ m,ε satisfy and ( (u m,ε u 0 )(g(u m,ε ) g(u 0 )) = ε ) (p(ρ m,ε ) p(ρ )) ρ ρ m,ε lim ε 0 (u m,ε u 0 )(g(u m,ε ) g(u 0 )) = 0,

14 st Reading October 0, 200 :2 WSPC/JHDE D. Mitrović &M.Nedeljkov 2 because ρ m,ε is bounded. Since g is increasing, we then have lim ε 0 u m,ε = u 0 and λ i (ρ m,ε,u m,ε ) g(u 0 ), i =, 2, as ε 0. Thus, the solution (u ε,ρ ε ) given by (4.) or (4.8) converge to to (4.) as ε 0. (b) Now, the rarefaction curves for problem (2.2) are given by the following formulas. The curve R starting from (ρ 0,u 0 ) is given by (4.6), so u R (ρ) = ρ p (s) ε ρ 0 s g (u) ds + u 0, ρ [0,ρ 0 ]. Since g > 0, we have u R (ρ) u 0, ε 0, ρ [0,ρ 0 ]. (4.) Similarly, the curve R 2 with the starting point (ρ, u) and the end point (ρ,u ) is a solution to du R2 dρ = εp (ρ) ρ g (u), u R 2 (ρ )=u, ρ < ρ, given by u R2 (ρ) = ρ p (s) ε ρ s g (u) ds + u. Since ρ lies in [0,ρ ]andg > 0, we have u R2 (ρ) u, ε 0, ρ [0,ρ 0 ]. (4.0) Assumption u >u 0, relations (4.) and (4.0) imply that u R (0) := u m,ε >u R2 (0) := u 2 m,ε (diminishing ε if necessary). Finally, the solution is given as a combination of two rarefaction waves connected by the vacuum state: ρ 0, x < λ (u 0,ρ 0 ), ρ R (x/t), λ (u 0,ρ 0 )t x<λ (u m,ε, 0)t, ρ ε (x, t) = 0, λ (u m,ε, 0)t x<λ 2 (u 2 m,ε, 0)t, ρ R2 (x/t), λ 2 (u 2 m,ε, 0)t x<λ 2 (u,ρ )t, ρ,λ 2 (u,ρ )t, u 0, x < λ (u 0,ρ 0 ), u R (x/t), u ε (x, t) = const, u R2 (x/t), u,λ 2 (u,ρ )t. λ (u 0,ρ 0 )t x<λ (u m,ε, 0)t, λ (u m,ε, 0)t x<λ 2 (u 2 m,ε, 0)t, λ 2 (u 2 m,ε, 0)t x<λ 2(u,ρ )t, (4.)

15 st Reading October 0, 200 :2 WSPC/JHDE 002 Delta Shock Waves as a Limit of Shock Waves Limits in (4.) and (4.0) show that u m,ε = u R (0) u 0 and lim ε 0 u 2 m,ε = u. Therefore, in the case of u 0 <u the solution to (2.2) given by (4.) tends to (4.2). 2. Approximated Solutions Let (ρ ε,u ε ), ε (0,ε 0 ), be a net of piecewise constant functions. We say that it is a solution of (.2) in the approximated sense if ρ ε ϕ t + ρ ε g(u ε )ϕ x dxdt + ϕ(x, 0)G(x)dx 0 R 0 R (.) ρ ε u ε ψ t + ρ ε u ε g(u ε )ψ x dxdt + ψ(x, 0)H(x)dx 0, as ε 0, R 0 for every pair of test functions ϕ and ψ in C0 (R2 ). Here, G and H are functions defined in (.2). For the purposes in the present paper, it suffice to verify that the initial data are satisfied in the sense of pointwise convergence almost everywhere and take ϕ and ψ with supports in half-plane t>0. In the third section we havefound that functions u 0, x < c,ε t, ρ 0, x < c,ε t, u ε (x, t) = u ε, c,ε t<x<c 2,ε t, and ρ ε (x, t) = ρ ε, c,ε t<x<c 2,ε t, u, x > c 2,ε t, ρ, x > c 2,ε t, (.2) solve perturbed system (2.2) if u <u 0. One will find a proof that (.2) is an approximate solution to non-perturbed system (.2) in the following theorem. In the next section, we will show that it satisfies a kind of entropy inequality. Theorem.. Let (ρ ε,u ε ) (min{ρ 0,ρ }, ) (u,u 0 ) betheuniquesolutionto system (..4). Letc,ε and c 2,ε be given by (.). Then, the functions ρ ε and u ε given by (.2) are approximate solution to (.2). Proof. We use the following notation from distribution theory. A distributional derivative of a piecewise constant function is a sum of delta function: If w(x, t) = G(x ct) (G is defined in (.2), say), then R 2 where w t = c(ρ ρ 0 )δ x ct and w x =(ρ ρ 0 )δ x ct, δ x ct,ϕ := 0 ϕ(ct, t)dt.

16 st Reading October 0, 200 :2 WSPC/JHDE D. Mitrović &M.Nedeljkov Let us denote by Γ,Γ 2 and Γ the sets {x = c,ε t}, {x = c 2,ε t} and {x = ct}, respectively. To prove (.), we substitute functions defined by (.2) into the first equation of (.2): A ε := c,ε (ρ ε ρ 0 )δ Γ c 2,ε (ρ ρ ε )δ Γ2 +(ρ ε g(u ε ) ρ 0 g(u 0 ))δ Γ +(ρ g(u ) ρ ε g(u ε ))δ Γ2 = (ρ ε g(u ε ) ρ 0 g(u 0 ))δ Γ (ρ g(u ) ρ ε g(u ε ))δ Γ2 +(ρ ε g(u ε ) ρ 0 g(u 0 ))δ Γ +(ρ g(u ) ρ ε g(u ε ))δ Γ2 0, where we have used exact values for c,ε and c 2,ε found in (.). Substitution of the same functions into the second equation gives B ε := c,ε (ρ ε u ε ρ 0 u 0 )δ Γ c 2,ε (ρ u ρ ε u ε )δ Γ2 +(ρ ε u ε g(u ε ) ρ 0 u 0 g(u 0 ))δ Γ +(ρ u g(u ) ρ ε u ε g(u ε ))δ Γ2 = (ρ εg(u ε ) ρ 0 g(u 0 ))(ρ ε u ε ρ 0 u 0 )+(ρ ε ρ 0 )(ρ ε u ε g(u ε ) ρ 0 u 0 g(u 0 )) δ Γ ρ ε ρ 0 + (ρ εg(u ε ) ρ g(u ))(ρ u ρ ε u ε )+(ρ ρ ε )(ρ u g(u ) ρ ε u ε g(u ε )) δ Γ2 ρ ρ ε = ρ ερ 0 u ε g(u 0 )+ρ ε ρ 0 u 0 g(u ε ) ρ ε ρ 0 u 0 g(u 0 ) ρ ε ρ 0 u ε g(u ε ) δ Γ ρ ε ρ 0 + ρ ερ u g(u ε )+ρ ε ρ u ε g(u ) ρ ε ρ u ε g(u ε ) ρ ε ρ u g(u ) ρ ρ ε δ Γ2. In the proof of Theorem. we saw that ρ ε and u ε u s (u,u 0 )asε 0. Therefore, the coefficients of δ Γ and δ Γ2 in B ε converge as ε 0. Since c i,ε g(u s ) (see the last part of Theorem. s proof), we have that Γ and Γ 2 tend to Γ as ε 0. Thus, the distributional limit of B ε equals lim ε 0 B ε =(ρ 0 u s g(u 0 )+ρ 0 u 0 g(u s ) ρ 0 u 0 g(u 0 ) ρ 0 u s g(u s ))δ Γ (ρ u g(u s )+ρ u s g(u ) ρ u s g(u s ) ρ u g(u ))δ Γ =(ρ 0 (u s u 0 )(g(u 0 ) g(u s )) ρ (u u s )(g(u s ) g(u )))δ Γ. Taking the limit of (.) as ε 0, one gets lim εp(ρ ε )=(u s u 0 )(g(u s ) g(u 0 )). ρ 0 ε 0 Repeating the procedure for (.4), we have lim εp(ρ ε )=(u u s )(g(u ) g(u s )), ρ ε 0 because u ε u s as ε 0. Since lim ε 0 εp(ρ ε ) exists (see (.6) in the proof of Theorem.), we have ρ 0 (u s u 0 )(g(u 0 ) g(u s )) = ρ (u u s )(g(u s ) g(u )), and lim ε 0 B ε =0.

17 st Reading October 0, 200 :2 WSPC/JHDE Entropy and Entropy Flux Pairs Delta Shock Waves as a Limit of Shock Waves System (.2) s only weakly hyperbolic, so it does not admit a convex entropy function in general. But one can still construct semi-convex entropy function η (D 2 η is positive semi-definite) for it. We shall look for entropy and entropy-flux functions in the region where ρ>0. Using (ρ, v) variables and (2.), one can find that an entropy function is of the form η(ρ, v) =θ(v/ρ)ρ + θ(v/ρ). Semi-convex condition on η implies that θ > 0and θ 0. The appropriate entropy-flux function is q(ρ, v) = θ(v/ρ)g(v/ρ)ρ. Now, we shall use (ρ, u) variables because the calculations have a simpler form. Recall, an approximate solution (ρ ε,u ε ) to (.2) is admissible (entropy one) if η(ρ ε,u ε )ϕ t + q(ρ ε,u ε )ϕ x dxdt + η(ρ ε 0(x, 0),u ε 0(x, 0)ϕ(x, 0)dx 0 R 0 for every nonnegative ϕ C0 (R2 ). The initial data are given by ρ ε 0 = G and u ε 0 = H, whereg and H are given in (.2). Since lim t 0 η(ρε,u ε ) η(ρ ε 0,uε 0 ), it is enough to check η(ρ ε,u ε ) t + q(ρ ε,u ε ) x 0 in the distributional sense (the test function ϕ has a support in upper half-plane). See [6], for example. As above, denote by Γ,Γ 2 and Γ the sets {x = c,ε t}, {x = c 2,ε t} and {x = ct}, resp. Here, c = g(u s ) = lim ε 0 c i,ε, i =, 2. 2 Substitution of (.2) into the expression η t + q x gives η t + q x = c,ε (θ(u ε )ρ ε θ(u 0 )ρ 0 ) δ Γ c,ε (θ(u )ρ θ(u ε )ρ ε ) δ Γ2 R 2 +(θ(u ε )g (u ε ) ρ ε θ(u 0 )g (u 0 ) ρ 0 ) δ Γ +(θ(u )g (u ) ρ θ(u ε )g (u ε ) ρ ε ) δ Γ2. Using the values of the speeds from (.), one gets c,ε θ(u ε )ρ ε + θ(u ε )g (u ε ) ρ ε ( = ρ ε θ(u ε ) g(u ε ) ρ ) εg(u ε ) ρ 0 g(u 0 ) ρ ε ρ 0 = ρ 0 ρ ε θ(u ε ) g(u 0) g(u ε ) ρ 0 θ(u s )(g(u 0 ) g(u s )),c 2,ε θ(u ε )ρ ε θ(u ε )g (u ε ) ρ ε ρ ε ρ ( 0 ) ρ g(u ) ρ ε g(u ε ) = ρ ε θ(u ε ) g(u ε ) ρ ρ ε = ρ ρ ε θ(u ε ) g(u ε) g(u ) ρ ε ρ ρ θ(u s )(g(u s ) g(u )), c,ε θ(u 0 )ρ 0 θ(u 0 )g (u 0 ) ρ 0 ρ 0 θ(u 0 )(g(u 0 ) g(u s )), c 2,ε θ(u )ρ + θ(u )g (u ) ρ ρ θ(u )(g(u s ) g(u )), ε 0.

18 st Reading October 0, 200 :2 WSPC/JHDE D. Mitrović &M.Nedeljkov Since all of these limits are finite, we have ( η t + q x ρ 0 θ(u s )(g(u 0 ) g(u s )) + ρ θ(u s )(g(u s ) g(u )) ) ρ 0 θ(u 0 )(g(u 0 ) g(u s )) ρ θ(u )(g(u s ) g(u )) δ Γ, ε 0. Thus the entropy condition reduces to ρ 0 (θ(u s ) θ(u 0 ))(g(u 0 ) g(u s )) + ρ (θ(u s ) θ(u ))(g(u s ) g(u )) 0. It is satisfied if ρ 0 (g(u 0 ) g(u s ))θ(u 0 )+ρ (g(u s ) g(u ))θ(u ) (ρ 0 (g(u 0 ) g(u s )) + ρ (g(u s ) g(u )))θ(u s ), i.e. if θ(u s ) ρ 0 (g(u 0 ) g(u s )) ρ 0 (g(u 0 ) g(u s )) + ρ (g(u s ) g(u )) θ(u 0) + ρ (g(u s ) g(u )) ρ 0 (g(u 0 ) g(u s )) + ρ (g(u s ) g(u )) θ(u ). But, since ρ 0 (u 0 u s )(g(u 0 ) g(u s )) ρ (u u s )(g(u ) g(u s )) = 0 (by the definition of u s, see (.8)) is equivalent to u s = ρ 0 (g(u 0 ) g(u s )) ρ 0 (g(u 0 ) g(u s )) + ρ (g(u s ) g(u )) u 0 + ρ (g(u s ) g(u )) ρ 0 (g(u 0 ) g(u s )) + ρ (g(u s ) g(u )) u, the entropy condition for delta shocks follows directly from the fact that θ is a convex function. Remark. As shown in [], the entropy condition alone is not good enough to single out a physically relevant solution to pressureless gas dynamics model (.) and we need an extra condition. Our choice is overcompressibility condition which now means u 0 >u for system (.2). 2. Interaction of δ Shocks A natural question which now arises is how to handle a delta shock interaction. We restrict our investigation to double delta shock interaction. That could give some insight about possible directions in further research. As one could see in this section, the problem is still open.

19 st Reading October 0, 200 :2 WSPC/JHDE 002 Delta Shock Waves as a Limit of Shock Waves Consider system (2.2) with the initial condition: ρ 0, x < a, u 0, x < a, ρ t=0 = ρ, a <x<a 2, v t=0 = u, a <x<a 2, ρ 2, x > a 2, u 2, x > a 2. where u 0 >u >u 2 and a <a 2. According to the results of the previous section, the states (u 0,ρ 0 )and(u,ρ )aswellas(u,ρ )and(u 2,ρ 2 ) are connected by a -shock followed by a 2-shock with intermediate states (u 0,ε,ρ0,ε )and(u0 2,ε,ρ0 2,ε ), respectively. Also, p(ρ 0 i,ε )=O(/ε) andu0 i,ε u0 i,s as ε 0, i =, 2, for some u,s (u,u 0 )andu 2,s (u 2,u ) determined by (.8). The speeds of -shock connecting (u 0,ρ 0 )and(u 0,ε,ρ0,ε ) and 2-shock connecting (u 0,ε,ρ 0,ε) and(u,ρ )are c,ε = ρ 0g(u 0 ) ρ 0,ε g(u0,ε ) ρ 0 ρ 0,ε and c 2,ε = ρ0,ε g(u0,ε ) ρ g(u ) ρ 0,ε ρ, respectively. The speeds of -shock connecting (u,ρ )and(u 0 2,ε,ρ 0 2,ε) and 2-shock connecting (u 0 2,ε,ρ 0 2,ε) and(u 2,ρ 2 )are c 2,ε = ρ0 2,εg(u 0 2,ε) ρ g(u ) ρ 0 2,ε ρ and c 22,ε = ρ 2g(u 2 ) ρ 0 2,εg(u 0 2,ε) ρ 2 ρ 0, respectively. 2,ε Lax admissibility conditions imply c 2,ε λ 2 (ρ 0,ε,u 0,ε) =g(u 0,ε)+ εg (u 0,ε )p (ρ 0,ε ) and c 2,ε λ (ρ 0 2,ε,u 0 2,ε) =g(u 0 2,ε) εg(u 0 2,ε )p (ρ 0 2,ε ). Recall, g is increasing function and u 0,ε >u0 2,ε. Therefore, c 2,ε >c 2,ε and there exists a moment t ε < when the first interaction occurs. That is interaction between the 2-shock connecting (u 0,ε,ρ0,ε )and(u,ρ ), and the -shock connecting 2 (u 0,ε,ρ0,ε )and(u,ρ ). The place of that interaction can be calculated from x ε := c 2,ε t ε + a = c 2,ε t ε + a 2, 2 so the interaction coordinates are t ε = a 2 a c 2,ε c 2,ε, x ε = a 2c 2,ε a c 2,ε c 2,ε c 2,ε.

20 st Reading October 0, 200 :2 WSPC/JHDE D. Mitrović &M.Nedeljkov Now, the initial data at the moment t = t ε are given by: ρ 0, x<c,ε t ε + a, ρ ε (x, t ε )= ρ 0,ε, c,εt ε + a <x<c 2,ε t ε + a, ρ 0 2,ε, c 2,εt ε + a 2 <x<c 22,ε t ε + a 2, ρ 2, x>c 22,ε t ε + a 2, u 0, x<c,ε t ε + a, u ε (x, t ε )= u 0,ε, c,εt ε + a <x<c 2,ε t ε + a, u 0 2,ε, c 2,εt ε + a 2 <x<c 22,ε t ε + a 2, u 2, x>c 22,ε t ε + a 2. The normal procedure would be to track all wave interactions after that time t ε. At least isentropic gas dynamics system (.) has a bounded global solution (see [] and []). But the existence of a solution was not enough for us to conclude something about its limit as ε 0. So, the general result of delta shock interaction is not yet known. There are some numerical results which suggest that such a result would not be (always) a single delta shock as expected in sticky particle model (see [] or []) and in paper []. We propose a different procedure described bellow. In the moment t ε we stop the clock and re-approximate the solution. According to the results of the previous section, distributional limits of the functions ρ ε (x, t ε ) and u ε (x, t ε )aregivenby ρ ε (x, t ε) G 0 (x a c t )+t (ρ 0 g(u 0 )+g(u s )(ρ ρ 0 ) ρ 2 g(u 2 ), + g(u s2 )(ρ 2 ρ )) δ(x a c t ), u ε (x, t ε) H 0 (x a c t ), where (.) { ρ0 G 0, x < 0, (x) = ρ, x > 0, { u0 H 0, x < 0, (x) = u, x > 0, c = lim c,ε = lim c 2,ε = g(u s ),u s (u,u 0 ), ε 0 ε 0 c 2 = lim c 2,ε = lim c 22,ε = g(u s2 ),u s2 (u 2,u ), ε 0 ε 0 t = lim t ε = a a 2. ε 0 c c 2 The pair (u s, u s2 ) (u,u 0 ) (u 2,u ) is the solution to the system ρ 0 (u 0 u s )(g(u 0 ) g(u s )) ρ (u u s )(g(u ) g(u s )=0, ρ (u u 2s )(g(u ) g(u 2s )) ρ 2 (u 2 u 2s )(g(u 2 ) g(u 2s )=0.

21 st Reading October 0, 200 :2 WSPC/JHDE 002 Delta Shock Waves as a Limit of Shock Waves 2 Fig.. That allows us to approximate distributional limits in (.) by piecewise constant families of functions: ρ 0, x x ε < a(ε), ρ ε (x, t )= ρ ε, a(ε) <x x ε <a(ε), ρ 2, x x ε >a(ε), (.2) u 0, x x ε < a(ε), ũ ε (x, t )= u ε, a(ε) <x x ε <a(ε), u 2, x x ε >a(ε). The idea is to solve (2.2) in the region t>t with the new initial data (.2) using the elementary waves and to glue such a solution to the already obtained in the region t<t. (The glued solution will be only approximated solution to (.2)). In order to get such connection, it is necessary to ensure that ρ and ρu are continuous across the line t = t in the distributional sense. That is, one has to take care only about the terms with t-derivative in (.2) across the line. Denote by Γ := {(x, t ): c,ε t ε + a <x<c 2,ε t ε + a }, Γ 2 := {(x, t ):c 2,ε t ε + a <x<c 22,ε t ε + a 2 } and Γ := {(x, t ): a(ε) <x x ε <a(ε)}. Distributional continuity of ρ and ρu holds if ρ ε Γ (ρ 0,ε Γ + ρ 0 2,ε Γ 2 ) 0, ρ ε Γ u ε (ρ0,ε Γ u 0,ε + ρ0 2,ε Γ 2 u 0 2,ε ) 0as ε 0. Here, A denotes one dimensional Lebesgue measure of a set A. In fact, the values α := ρ 0,ε Γ and α 2 := ρ 0 2,ε Γ 2 are strengths of the incoming delta shocks at the

22 st Reading October 0, 200 :2 WSPC/JHDE D. Mitrović &M.Nedeljkov time t = t ε. From the above relations, it follows that u ε u 0,ε and u0 2,ε : is convex combination of u ε = α α + α 2 u 0,ε + α 2 α + α 2 u 0 2,ε, achoiceofρ ε is still free providing Γ 0asε 0. The situation is similar when delta shock interacts with a rarefaction wave or a contact discontinuity. One just has to split the rarefaction wave in a fan of non-entropy shocks and apply a similar procedure. We need the following assumption to obtain a solution to delta shock interaction problem. Assumption.. There exist ρ ε > 0andu ε (u 2,u 0 ) such that ( (u ε u 0)(g(u ε ) g(u 0)) = ε ρ 0 ρ ε ( (u ε u 2 )(g(u ε) g(u 2 )) = ε ρ 2 ρ ε ) (p(ρ ε ) p(ρ 0) ), ) (p(ρ ε) p(ρ 2 ) ), for ε small enough. The solution (ρ ε,u ε) to the above system may not exist, obviously. One possibility to ensure that is to add an additional condition on (ρ 2,u 2 ) in the beginning. A set of such points is called the second delta locus in [, ]. In principle such a set could provide a possibility for adding some other wave on a side of the delta shock wave. There is no such a result in the present paper. With Assumption., the states (u 0,ρ 0 )and(u ε,ρ ε ) are connected by -shock of the speed c,ε = ρ εg(u ε) ρ 0 g(u 0 ) ρ, ε ρ 0 and the states (u ε,ρ ε )and(u 2,ρ 2 ) are connected by 2-shock of the speed c 2,ε = ρ ε g(u ε ) ρ 2g(u 2 ) ρ ε ρ. 2 In order to obtain a continuity of distributional limits from both sides of the line t = t ε, put ( a(ε) = t ε ρ0 g(u 0 )+g(u ε)(ρ ε ρ 0 ) ρ 2 g(u 2 )+g(u 2 ε)(ρ 2 ρ ε) ) (.) for a(ε) defined in (.2). Then 2ρ ε ρ ε (x, t ) G 02 (x a c t )+t ( ρ 0 g(u 0 )+g(u s )(ρ ρ 0 ) ρ 2 g(u 2 ), + g(u s2 )(ρ 2 ρ ) ) δ(x a c t ), ũ ε (x, t ε ) H02 (x a c t ),

23 st Reading October 0, 200 :2 WSPC/JHDE 002 Delta Shock Waves as a Limit of Shock Waves 2 where c := lim ε 0 c,ε = lim ε 0 c 2,ε and { { ρ0 G 02, x < 0, u0 (x) = H 02, x < 0, (x) = ρ 2, x > 0, u 2, x > 0. Therefore, the solution to (2.2), (.) for t>t is given by ρ 0, x < c,εt a(ε), ρ ε (x, t) = ρ ε, c,εt a(ε) <x<c 2,εt + a(ε), ρ 2, x > c 2,ε t + a(ε), u 0, x < c,ε t a(ε), ũ ε (x, t) = u ε, c,ε t a(ε) <x<c 2,ε t + a(ε), u 2, x > c 2,ε t + a(ε). Also, it is not difficult to see that for t>t,wehave ρ ε (x, t) G 02 (x a c t) + ( t (ρ 0 g(u 0 )+g(u s )(ρ ρ 0 ) ρ 2 g(u 2 )+g(u s2 )(ρ 2 ρ )) +(t t )(ρ 0 g(u 0 ) ρ 2 g(u 2 )+g(u s )(ρ 2 ρ 0 )) ) δ(x a c t), ũ ε (x, t) H 02 (x a c t). The constant u s (u 2,u 0 ) is the unique solution of the equation ρ 0 (u 0 u s)(g(u 0 ) g(u s)) ρ 2 (u 2 u s)(g(u 2 ) g(u s)) = 0. Now, the global approximate solution (ρ ε G,uε G ) to (2.2), (.) is given by { ρ ε ρ ε (x, t), t<t ε G(x, t) =, { u ε ρ ε (x, t), t>t ε, and u ε (x, t), t<t ε G(x, t) =, ũ ε (x, t), t>t ε. (.4) It represents a solution to delta shock interaction problem in that special case... Admissibility of re-approximated solution Let us first observe that on both sides of the line t = t ε, the solution (.4) satisfies the entropy condition η t + q x 0, η(ρ, u) =θ(u)ρ, q(ρ, u) =θ(u)f(u)ρ, ρ > 0, where θ is a convex function, θ > 0. If one proves that η t 0 across the line γ : t = t ε, then the solution (.4) satisfies the complete entropy condition. Thus we need η t,ϕ = ηϕ t ηϕ t = t<t ε t>t ε η t=t ε 0ϕdx + η t=t ε +0ϕdx 0. (.) R R

24 st Reading October 0, 200 :2 WSPC/JHDE D. Mitrović &M.Nedeljkov to hold on the line γ. We keep the same definitions for Γ, Γ and Γ 2 as before. The integrals in (.) over γ\(γ Γ Γ 2 ) from upper and lower side annihilates, and (.) rduces to 2a(ε)ρ εθ(u ε) Γ ρ 0,εθ(u 0 ε) Γ 2 ρ 0 2,εθ(u 0 2,ε) 0. One knows that 2a(ε)ρ ε = Γ ρ 0,ε + Γ 2 ρ 0 2,ε (a(ε) was chosen to satisfy this relation from (.)). Also, A := Γ ρ 0,ε t (ρ 0 g(u 0 )+g(u s )(ρ ρ 0 ) ρ g(u )), A 2 := Γ 2 ρ 0 2,ε t (ρ g(u )+g(u s2 )(ρ 2 ρ ) ρ 2 g(u 2 )). With such a notation, entropy inequality reads θ(u s) A A + A 2 θ(u s )+ A 2 A + A 2 θ(u s2 ). (.6) Since u s = A A +A 2 u s + A2 A +A 2 u s2 and θ is convex, (.6) always holds if Assumption. is satisfied. Acknowledgments The research of the first author is partially supported by Norvegian Science Fondation. The research of the second author is partially supported by grant No. 4406, MNZŽS RS References [] Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal. (6) (8) [2] G.-Q. Chen and H. Liu, Formation of δ-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal. 4(4) (200) 2 8. [] G.-Q. Chen, Convergence of the Lax-Fridrichs scheme for isentropic gas dynamics (III), Acta Math. Sci. 6 (86) 20. [4] M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics (Springer, 2000). [] V. G. Danilov and V. M. Shelkovich, Dynamics of propagation and interaction of δ-shock waves in conservation laws system, J. Differential Equations 2 (200) 8. [6] V. G. Danilov and V. M. Shelkovich, Delta-shock wave type solution of hyperbolic systems of conservation laws, Q. Appl. Math. 2 (200) [] W. E., Y. G. Rykov and Ya. G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Commun. Math. Phys. (2) (6) [8] B. T. Hayes and P. LeFloch, Measure solutions to a strictly hyperbolic system of conservation laws, Nonlinearity (6) 4 6. [] F. Huang, Weak solution to pressureless type system, Commun. Partial Differential Equations 0( ) (200)

25 st Reading October 0, 200 :2 WSPC/JHDE 002 Delta Shock Waves as a Limit of Shock Waves 2 [0] P. LeFloch, An existence and uniqueness result for two nonstrictly hyperbolic systems, in: Nonlinear Evolution Equations that Change Type, eds. B.L. Keyfitz, M. Shearer (EDS), The ZMA Volumes in Mathmatics and its Applications, Vol. 2 (Springer Verlag, 0), pp [] P.-L. Lions, B. Perthame and P. Soughanidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Commun. Pure Appl. Math. 4 (6) 68. [2] M. Nedeljkov, Delta and singular delta locus for one-dimensional systems of conservation laws, Math. Meth. Appl. Sci. 2 (2004). [] M. Nedeljkov, Conservation law systems and piecewise constant functions shadow waves, Preprint. [4] M. Nedeljkov, Singular shock waves in interactions, to appear in Q. Appl. Math. [] M. Nedeljkov and M. Oberguggenberger, Delta shock wave and interactions in a simple model case, preprint. [6] D. Serre, Systems of Conservation Laws I (), II (2000) (Cambridge University Press). [] M. Sever, A class of nonlinear, nonhyperbolic systems of conservation laws with wellposed initial value problems, J. Differential Equations 80() (2002) [8] H. Yang, Riemann problems for a class of coupled hyperbolic systems of conservation laws, J. Differential Equations ()