Example of a interaction of waves

3.2.3. Example of a interaction of waves

Now, we are interested in the interaction of elementary waves for the generalized Riemann problem associated to the Suliciu relaxation system (1). In this sense, we consider (1) with initial data

x

Figure 3-2: The solution for the generalized Riemann problem.

where ρ_l, ρ_r, u_i, v_i, i = l, m or r, are constants and ρ_m(x) = e^x.

Moreover, we assume that the initial data satisfies the conditions H1 and H2.

Thereby, the solution on the left, right and middle states is given by ρl(t, x) = ρl,

3.2 Generalized Riemann problem 47

and

e

x₁(t) = u_mt + ln(e^b− st), xe₂(t) =

u_r+ u_m

2 − sv_r− vm

2

 t + b, e

x3(t) =



ur+ s ρr

 t + b.

x

t x1(t) x2(t)

x3(t)

Figure 3-3: The curves xi = xi(t) in the plane t − x.

x t

e

x¹(t) xe2(t)

e x3(t)

Figure 3-4: The curves exi = exi(t) in the plane t − x.

Now, we observe that the curves x3 = x3(t) and ex1 = ex1(t) intersect at point (t1, x1) defined

by 





t1 = ^eb_2s⁻¹, x1 = ume^b−1

2s + ln

e^b+1 2

= umt0+ ln

e^b+1 2

. (3-58)

Following the way of getting the interaction of elementary waves, we propose to solve the following Riemann problem associated to the Suliciu relaxation system (1) with initial data

((ρ_∗∗, u_∗∗, v_∗∗)(t1, x), if x < x1,

(eρ_∗, eu_∗, ev∗)(t1, x), if x > x1, (3-59)

x

For time t > t1, we must find the solution in four new regions until some time t2 as shown in Figure 3-6.

3.2 Generalized Riemann problem 49

In the region B and C, we obtain that 1

Observe that, at time t₂ the curves x₂(t) and bx₁(t) intersect and should be considered a new Riemann problem for the Suliciu relaxation system with inicial data

((ρ_∗, u_∗, v_∗)(t2, x), if x < x2,

(bρ_∗, bu_∗, bv∗)(t2, x), if x > x2. (3-68) In each new intersection, we obtain a Riemann problem which can be solve of natural form.

Finally, note that the Figures 3-3 – 3-6, of previous example, correspond to the initial data

(ρ0, u0, v0)(x) =









(1, 3,⁷₂), if x < 0,

(e^x,⁵₂, 4), if 0 < x < ln(4), (¹₂,⁷₂, 3), if x > ln(4).

4 Delta shock solution for the Suliciu relaxation system

There are nonclassical situations where the Cauchy problem (or Riemann problem) for a system of conservation laws either does not possess a weak solution in L^∞. In order to solve the Cauchy problem in these nonclassical situations, it is necessary to seek solutions in other sense.

In 1977, Korchinski [49] in his PhD thesis considered the Riemann problem for system (ut+ ¹₂u²

x = 0, vt+ ¹₂uv

x= 0. (4-1)

Motivated by his numerical results, he constructed the unique Riemann solution using gene-ralized delta functions to prove singular shocks satisfying (4-1) in the sense of distributions.

In 1994, Tan, Zhang and Zheng in [86] established the existence, uniqueness and stability of delta shock waves to some viscous perturbations in the reduced one-dimensional system

(ut+ (u²)_x= 0,

vt+ (uv)_x = 0. (4-2)

and referred to such solutions as delta shock waves. A delta-shock wave is a generalization of an ordinary shock wave. From the physical point of view, a delta shock waves are interpreted as the process of formation of the galaxies in the universe, or the process of concentration of particles (mass) [93]. Other works in this sense are due to Ercole [36] who obtained a delta shock solution as a limit of smooth solutions by the vanishing viscosity method. Sheng and Zhang [81] discussed the Riemann problem for the pressureless gases system. In 2005, Brenier [8] considered the Riemann problem for the Chaplygin gas system. Other works for the Chaplygin gas system can be found in [40, 24, 48].

Finally, recent work on delta shocks for general hyperbolic conservation laws are due to Dani-lov and Mitrovic [31, 32] where they described delta shock wave generation from continuous initial data by using smooth approximations in the weak sense. An interesting work on new developments of delta shock waves, its applications and historical notes is [93].

In this chapter, we show existence and uniqueness of delta shock wave for the Suliciu relaxa-tion system which is an example of delta shock for a 3 × 3 hyperbolic system of conservarelaxa-tion laws.

4.1. Delta shock solution

Recently, Lu et al. [68] showed existence of solutions for the Cauchy problem associated to the Suliciu relaxation system (1) with initial data (1-20) satisfying the conditions H1 and H2.

Now, we show the existence and uniqueness of solutions for the Riemann problem associated with the Suliciu relaxation system, in which the left and right constant states (ρl, ul, vl) and (ρr, ur, vr) satisfies condition H2, but unlike chapter 2, the states satisfy λ1(ρl, ul, vl) ≥ λ3(ρr, ur, vr), i.e., they do not satisfy condition H1 globally. In this way, we have the following situations:

1. (ρl, ul, vl) and (ρr, ur, vr) satisfy locally the condition H1, i.e., (ρl, ul, vl) satisfy

α1 ≤ u^l− sv^l ≤ α², α3 ≤ u^l− sv^l≤ α⁴ and vl+ 1 ρl

> α5

where αi, i = 1, . . . , 5, are suitable constants satisfying α5 −^α42s^−α1 > 0,

(ρr, ur, vr) satisfy

β1 ≤ u^r− sv^r ≤ β², β3 ≤ u^r− sv^r ≤ β⁴ and vr+ 1 ρr

> β5

where βi, i = 1, . . . , 5, are suitable constants satisfying β₅− ^β4_2s^−β1 > 0.

Let c₁ = m´ın{α1, β₁} and c4 = m´ax{α1, β₁}. Then there are ci, i = 1, . . . , 4 such that

c1 ≤





ul− sv^l ur− sv^r



≤ c² and c3 ≤





ul+ svl

ur+ svr



≤ c⁴, but is not possible to find a constant c5 such that c5− ^c42s^−c1 > 0.

2. Only (ρ_r, u_r, v_r) satisfy locally the condition H1, i.e., (ρ_l, u_l, v_l) satisfy β1 ≤ u^l− sv^l ≤ β², β3 ≤ u^l− sv^l≤ β⁴ and vl+ 1

ρ_l > β5

where βi, i = 1, . . . , 5, are suitable constants satisfying β5− ^β4_2s^−β1 ≤ 0.

3. Only (ρl, ul, vl) satisfy locally the condition H1, i.e., (ρr, ur, vr) satisfy β1 ≤ u^r− sv^r ≤ β², β3 ≤ u^r− sv^r ≤ β⁴ and vr+ 1

ρ_r > β5

where βi, i = 1, . . . , 5, are suitable constants satisfying β5− ^β42s^−β1 ≤ 0.

4.1 Delta shock solution 53

4. Neither (ρl, ul, vl) nor (ρr, ur, vr) satisfy the local condition.

De la cruz et al. [33] studied the first case. The idea used by the authors in [33] can be extended for the other cases.

Denote by BM(R) the space of bounded Borel measures on R, and then the definition of a measure solution of Suliciu relaxation system in BM(R) can be given as follows.

Definition 4.1.1. A triple (ρ, u, v) constitutes a measure solution to the Suliciu relaxation system, if it holds that

a) ρ ∈ L^∞((0, ∞), BM(R)) ∩ C((0, ∞), H^−s(R)),

Definition 4.1.2. A two-dimensional weighted delta function w(s)δLsupported on a smooth curve L parameterized as t = t(s), x = x(s) (c ≤ s ≤ d) is defined by

Definition 4.1.3. A triple distribution (ρ, u, v) is called a delta shock wave if it is represented in the form bounded solutions of the Suliciu relaxation system (1).

We set ^dx_dt = uδ(t) since the concentration in ρ need to travel at the speed of discontinuity.

Hence, we say that a delta shock wave (4-5) is a measure solution to the Suliciu relaxation system (1) if and only if the following relation holds,



Relations (4-6) are called the generalized Rankine-Hugoniot condition. It reflects the exact relationship among the limit states on two sides of the discontinuity, the weight, propagation speed and the location of the discontinuity. In addition, to guarantee uniqueness, the delta shock wave should satisfy the admissibility (entropy) condition

λ3(ρr, ur, vr) ≤ u^δ(t) ≤ λ¹(ρl, ul, vl). (4-7) Now, the generalized Rankine-Hugoniot condition is applied to the Riemann problem (1)–

(2-1) with left and right constant states U₋ = (ρ₋, u₋, v₋) and U+ = (ρ+, u+, v+),

Thereby, the Riemann problem is reduced to solving (4-6) with initial data

t = 0, x(0) = 0, w(0) = 0, g(0) = 0, (4-9) under entropy condition

u++ s

ρ+ ≤ u^δ(t) ≤ u−− s

ρ₋. (4-10)

4.1 Delta shock solution 55

From (4-6) and (4-9), it follows that

w(t) = −[ρ]x(t) + [ρu]t,

w(t)uδ(t) = −[ρu]x(t) + [ρu²+ s²v]t, and w(t)g(t) = −[ρv]x(t) + [ρuv + u]t.

(4-11)

Multiplying the first equation in (4-11) by uδ(t) and then subtracting it from the second one, we obtain that When [ρ] = ρ₋−ρ⁺ = 0, the situation is very simple and one can easily calculate the solution



which obviously satisfies the entropy condition (4-10). From condition (4-8), s²[v]

Similarly, we can deduce that uδ−

because and then we can find



Next, with the help of the entropy condition (4-10), we will choose the admissible solution from (4-18) and (4-19). Observe that by the entropy condition and since the system is strictly hyperbolic, we have that

4.1 Delta shock solution 57

then, for the solution given in (4-18), we have uδ−λ¹(ρ₋, u₋, v₋) =

showing that the solution (4-19) does not satisfy the entropy condition (4-10).

Thus we have proved the following result.

Theorem 4.1.1. Given left and right constant states (ρl, ul, vl) and (ρr, ur, vr), respectively,

Then, the Riemann problem (1)–(2-1) admits a unique entropy solution in the sense of measures. This solution is of the form

(ρ, u, v)(t, x) =

Finally, if in (2-23) we suppose that ρ_∗ tends to infinity, then 1

s (λ3(Ur) − λ¹(Ul)) − (R²(Ur) − R²(Ul)) = 0. (4-21) Simlarly, if ρ_∗∗ tends to infinity, then

1

s (λ₃(U_r) − λ1(U_l)) + (R₂(U_r) − R2(U_l)) = 0. (4-22) Thereby, if the initial data satisfies (4-21) or (4-22), then the Riemann problem (1)–(2-1) admits a delta shock solution.

We began by analyzing the case in (4-21). Assume left and right constant states (ρl, ul, vl) and (ρr, ur, vr), respectively, such that satisfy the condition H2, λ1(ρl, ul, vl) ≥ λ³(ρr, ur, vr) and (4-21).

It is easy to see that if λ1(ρl, ul, vl) = λ3(ρr, ur, vr), then, the inequality (4-8) is trivially satisfied. Suppose that λ1(ρl, ul, vl) > λ3(ρr, ur, vr). Then, λ3(ρr, ur, vr) < λ3(ρl, ul, vl) , and so

λ₃(ρ_r, u_r, v_r) − λ1(ρ_l, u_l, v_l) < 2s ρl

, meaning that (4-8) is satisfied. The analysis is similar for (4-22).

4.2. Numerical test

In this section, we show numerical evidence of delta shock solution for the Suliciu relaxation system using, once again, the Lax-Friedrichs method.

In the numerical test, with s = 1, we consider the initial data given by

(ρ0, u0, v0)(x) =

((9, 5, 14/5), if x < 0,

(1, 3, 2), if x > 0. (4-23)

and the spatial discretization parameter for N = 1780 points and a constant CF L = 0,1969889. The numerical results at final time t = 0,1 are presented in Figures 4-1, 4-2 and 4-3. The exact solution at time t is

(ρ, u, v)(t, x) =









(9, 5, 14/5), if x < u_δt, (Atδ(x − u^δt), uδ, Bt), if x = uδt, (1, 3, 2), if x > uδt,

(4-24)

with uδ = ^21√₄^5−√₅^√37, A = ^2√√37₅ and B = ^√5+29₁₀^√₃₇^√37.

4.2 Numerical test 59

Figure 4-1: Numerical solution of ρ. (a) The initial data ρ0(x) = ρ(0, x). (b) Numerical solution of ρ at time t = 0,1.

Figure 4-2: Numerical solution of u. (a) The initial data u0(x) = u(0, x). (b) Numerical solution of u at time t = 0,1

Figure 4-3: Numerical solution of v. (a) The initial data v₀(x) = v(0, x). (b) Numerical solution of v at time t = 0,1

In this work, we showed uniqueness of solutions for the Riemann problem associated to the Suliciu relaxation system (1). We think it was very important the study of the Riemann pro-blem because we obtained more information of the solution. We note that when the initial data satisfies the conditions H1 and H2 the solution is in L^∞ space, whereas when the con-dition H1 is not satisfied delta shock solution is obtained. To guarantee uniqueness of delta shock solutions, we employ a generalization of the Rankine-Hugoniot condition together with a suitable entropy condition.

Also, we showed the uniqueness of solutions for the Cauchy problem of the Suliciu relaxation system (1)–(1-20) and we gave the explicit solutions. Even with the explicit solutions, it is generally difficult to construct a solution for a particular initial data. For additional infor-mation about the behavior of the solution, we solve the generalized Riemann problem and and show a small example of the interaction of the elementary waves.

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