The Perturbed Riemann Problem with Delta Shock for a Hyperbolic System

In this paper, we study the perturbed Riemann problem with delta shock for a hyperbolic system. The problem is different from the previous perturbed Riemann problems which have no delta shock. The solutions to the problem are obtained constructively. From the solutions, we see that a delta shock in the corresponding Riemann solution may turn into a shock and a contact discontinuity under a perturbation of the Riemann initial data. This shows the instability and the internal mechanism of a delta shock. Furthermore, we find that the Riemann solution of the hyperbolic system is instable under this perturbation, which is also quite different from the previous perturbed Riemann problems.


Introduction
In this paper, we are concerned with the following nonstrictly hyperbolic system of conservation laws in the form Equations ( 1) can be derived from a two-dimensional hyperbolic system of conservation laws   + ( 2 )  + (V)  = 0, (2) The above system is the mathematical simplification of Euler equations of gas dynamics.In 1991 and 1994, Yang and Zhang [1] and Tan and Zhang [2] obtained both numerical and analytical solutions to the Riemann problem of system (2).The form of Dirac delta functions supported on shocks was found necessary and was used as parts in the Riemann solutions.We call it a delta shock.A delta shock is the generalization of an ordinary shock.It is more compressive than an ordinary shock in the sense that more characteristics enter the discontinuity line.Mathematically, the delta shocks are new type singular solutions such that their components contain delta functions and their derivatives.Physically, they are interpreted as the process of formation of the galaxies in the universe, or the process of concentration of particles [3].
To investigate the validity of delta shock, in 1994, Tan et al. [4] considered the Riemann problem for the onedimensional model (1).They found that there exist delta shock as the limit of vanishing viscosity for system (1).As for delta shock, there are numerous excellent papers.We refer readers to [3][4][5][6][7][8][9][10][11][12][13][14] and the references cited therein.There are still many open and complicated problems in the delta shock theories.Study of this area gives a new perspective in the theory of conservation law systems.
In this paper, we are interested in the internal mechanism and instability of a delta shock.For this purpose, we study system (1) with the following initial data: ( − 0 () , V − 0 ()) ,  < 0, ( + 0 () , V + 0 ()) ,  > 0, where  ± 0 () and V ± 0 () are all bounded  1 functions with the following property: Here û± and V± are constants with (û − , V− ) ̸ = (û + , V+ ).The initial value (3) is a perturbation of Riemann initial value (5) at the neighborhood of the origin in the  −  plane.The perturbation on the Riemann initial data is reasonable.For example, error is unavoidable in computation and the error forms a perturbation of the initial data.
We divide our work into two parts according to the presence of delta shock or not.When the delta shock is not involved, the perturbed Riemann problem (1) and ( 3) is classical.More importantly, there is no delta shock in the corresponding Riemann solutions for the previous work on the perturbed Riemann problem.Therefore, we only pay attention to the perturbed Riemann problem (1) and (3) when the delta shock is involved.To overcome the difficulty caused by delta shock, we adopt the method of characteristic analysis and the local existence and uniqueness theorem proposed by Li Ta-tsien and Yu Wen-ci [15].We construct the solution to the perturbed problem (1) and (3) locally in time.
Our result shows that a perturbation of initial data may bring essential change when a delta shock appears in the corresponding Riemann solution.A delta shock may turn into a shock and a contact discontinuity.This shows the instability of the delta shock, which allows us to better investigate the internal mechanism of a delta shock.Furthermore, the previous works [15,16] about the perturbed Riemann problem pay more attention to the stability of the corresponding Riemann solution.In other words, the Riemann solution has a local structure stability with respect to the perturbation of Riemann initial data.A distinctive feature for this paper is that, for some initial data (3), the preceding local structure stability fails.We pay more attention to the differences between Riemann solution and perturbed Riemann solution.
The paper is organized as follows.In Section 2, we present some preliminary knowledge about the hyperbolic system (1).Then, the construction and proof of the solution to the perturbed Riemann problem (1) and ( 3) with delta shock are presented in Section 3.
The eigenvalues of the hyperbolic system (1) are with the corresponding left eigenvectors, and the corresponding right eigenvectors, By a direct calculation, Therefore,  1 is always linearly degenerate;  2 is genuinely nonlinear if  ̸ = 0 and linearly degenerate if  = 0.The Riemann invariants of system (1) along the characteristic fields are Definition 1 (see [3,19]).A pair of (, V) is called a generalized delta shock solution to (1) with the initial data (3) on local time [0, ), if there exists a smooth curve  = {(  (), ) : 0 ≤  < } and a weight (, ) such that  and V are represented in the following form: in which () is the delta function,  ∈  1 (), ,  ∈  ∞ ( × [0, ); ) and satisfy for all the test functions  ∈  ∞ 0 ((−∞, +∞) × [0, )).Here   is the tangential derivative of the curve , and (, )/ stands for the tangential derivative of the function  on the curve .Definition 2. For an  ×  matrix  = (  ), define and

The Perturbed Riemann Problem with Delta Shock
In this section, we construct the perturbed Riemann solutions of hyperbolic system (1) with initial data (3) for local time and investigate the internal mechanism and instability of delta shock.About the perturbed Riemann problem, we have six cases according to the different constructions of the solutions to the corresponding Riemann problem (1) and ( 5) as follows: (1) When û+ < û− < 0, the Riemann solution is ←   + (2) When û+ ≤ 0 ≤ û− , the Riemann solution is delta shock (3) When û− < û+ ≤ 0, the Riemann solution is ←   +  (4) When û− < 0 < û+ , the Riemann solution is (5) When û+ > û− > 0, the Riemann solution is  +  →  (6) When û− > û+ > 0, the Riemann solution is  +  → Here "+" means "followed by"; the capitals , , and  denote shock, contact discontinuity, and rarefaction wave, respectively.Since the perturbed Riemann problem (1) and (3) with no delta shock is classical and well known, it will not be pursued here.In this section, we mainly study the differences between the perturbed Riemann solution and the corresponding Riemann solution.Thus we only consider the perturbed Riemann problem (1) and (3) with delta shock, a.e., û+ ≤ 0 ≤ û− .
It is known from classical theory that the classical solution ( − (, ), V − (, )) and ( + (, ), V + (, )) can be defined in a strip domains  − and  + for local time, respectively (see Figure 1).Here ( − (, ), V − (, )) and ( + (, ), V + (, )) are local smooth solutions to the initial problem (1) with corresponding initial data ( − 0 (), V − 0 ()) and ( + 0 (), V + 0 ()) on both sides of  = 0, respectively.The right boundary of domain  − is a −characteristic :  =  − (); namely, The left boundary of domain  + is −characteristic :  =  + (); namely, Now we turn our attention to the solution of ( 1) and (3) between the right boundary of domain  − and the left boundary of domain  + .We note that the corresponding Riemann solution is a delta shock  with speed û− + û+ separating two states (û − , V− ) and (û + , V+ ) (see Figure 2(a)).If the perturbed Riemann problem (1) and (3) has a solution by using a delta shock  connecting two states ( − , V − ) and ( + , V + ) on local time [0, ), we must choose  and V to be Where, here and below, we use the usual notation [] =  − −  + with  − and  + the values of the function  on the left-hand and right-hand sides of the discontinuity  =   ()(  (0) = 0), etc.; () is the Heaviside function, that is, 0 when  < 0 and 1 when  > 0; (  (), ) and   are the weight and the tangential derivative of curve  ≜ {(  (), ) : 0 ⩽  < }, which can be defined by From (19), one can get that the propagating speed of the delta shock We will prove that the delta shock solution constructed in ( 18) is a solution of the initial value problem (1) and (3) in the sense of distributions on [0, ).Proposition 3. The delta shock solution constructed in (18) satisfies ( 1) and (3) in the sense of distributions on a domain where  > 0 is a finite time.Proof.Let Then the delta shock solution ( 18) can be reduced to We need to check that (, V) satisfies ( 1), which is ( 12) and ( 13).
Furthermore, to guarantee uniqueness of the solution, the delta shock solution constructed in (18) should satisfy the entropy condition (27) on the discontinuity  =   ().Definition 4. The delta shock solution constructed in ( 18) is an admissible solution of the initial value problem (1) and (3) in the sense of distributions on [0, ), if (, V) satisfies Definition 1 and the entropy condition on the discontinuity  =   ().
Theorem 5.In case of û+ < 0 < û− , the perturbed Riemann problem ( 1) and ( 3) has a delta shock solution constructed in (18) for local time.The admissible solution has a structure similar to that of the corresponding Riemann problem ( 1) and ( 5) (see Figure 2).Furthermore, the delta shock curve  =   () possesses the following property: (a) If u − 0 (0) < u + 0 (0), then the curve is convex (b) On the other hand, if u − 0 (0) > u + 0 (0), then the curve is concave Proof.By Proposition 3 and inequality (32), we know that the delta shock solution constructed in ( 18) satisfies ( 1) and (3) in the sense of distributions and the entropy condition (27) for local time, respectively.Obviously, the delta shock curve  =   () retains its form in a neighborhood of the origin (0, 0).Namely, the solution of the perturbed Riemann problem (1) and (3) has a structure similar to the corresponding Riemann solution of (1) and ( 5) for local time.
Theorem 6.In case of û+ < 0 = û− and u − 0 (0) > 0, the solution to the perturbed Riemann problem ( 1) and ( 3) is composed of a backward shock  =   () followed by a contact discontinuity  =   () for local time.The perturbed solution is dramatically different from the corresponding Riemann solution of ( 1) and (5), which is a delta shock (see Figure 3).
Next, we proceed to prove that the perturbed Riemann problem (1) and (3) for this subcase admits a solution that contains a backward shock ←   and a contact discontinuity  near the origin in another way.
Let  =  * () be the upwards right −characteristic from any point (, ) on the shock curve  =   () (see Figure 3(b)).The point ( 0 ,  0 ) is the intersection point of the −characteristic curve  =  * () and the contact discontinuity  =   ().Since the Riemann invariant (, V) = − + V must be a constant along −characteristic, we have Differentiate the above equation with respect to  and let  = 0; then one obtains Substituting into (56), with û * = û+ < 0, we get Moreover, differentiating the last equality in (52) with respect to  and letting  = 0, we obtain In view of ( 58) and (59), it is easy to see that In the following, firstly, we will accumulate the second derivative of the shock at the origin ẍ  (0).Along  =   (), differentiating the above equality (41) with respect to  and letting  = 0, by (44) and ( 52 On the one hand, from (35), we have Noting (57), (62), and û− = 0, one can get that On the other hand, by ( 60) and (44), we have Along the contact discontinuity wave curve  =   (), from (35), (57), and (64), it follows that Then, substituting (63) and ( 65) into (61), we get Secondly, we now have an estimate of the second derivative of the contact discontinuity wave curve at the origin, a.e.
We introduce the change of variables Boundary condition (75) on  =   () then reduces to Boundary condition (77) on  =   () can be written as Hence, the characterizing matrix  of this problem is of the form [15]  = ( According to the local existence and uniqueness theorem, what remains is to prove that ‖‖ min < 1.By Remark 4.4 in the introduction of [15], it is not hard to prove that for this subcase Then the free boundary problem under consideration admits a unique piecewise  1 solution on the fan-shaped domain {(, ) |   () <  <   (), 0 ≤  < } ( > 0 so small).We can conclude the following.
Theorem 7. In case of û+ = 0 < û− and u + 0 (0) > 0, the solution of the perturbed Riemann problem ( 1) and ( 3) is composed of a contact discontinuity  =   () followed by a forward shock  =   () for local time.The perturbed solution is different from the corresponding Riemann solution of ( 1) and (5), which is a delta shock (see Figure 4).
Next, we proceed to prove that the perturbed Riemann problem (1) and (3) for this subcase admits a solution which contains a contact discontinuity  =   () followed by a forward shock  =   () near the origin.Let  =  * () be the upwards right −characteristic from any point (, ) on the contact discontinuity curve  =   () (see Figure 4 Differentiate the above equation with respect to  and let  = 0; then one obtains Substituting In the following, firstly, we will accumulate the second derivative of the contact discontinuity curve at the origin ẍ  (0).Along  =   (), differentiating (74) with respect to  and letting  = 0, it yields Finally, by virtue of   (0) =   (0) = 0, (88), and (101), the perturbed Riemann solution to (1) and (3) in this subcase clearly consists of a contact discontinuity  connecting ( − , V − ) to ( * , V * ), followed by a forward shock  →  connecting ( * , V * ) to ( + , V + ) near the origin.Thus we have completed the construction and proof of the perturbed Riemann solution for this subcase (see Figure 4).

Conclusion
So far, we have locally constructed the solution to the perturbed Riemann problem (1) and ( 3) with delta shock.
From the above discussion, we discover a delta shock in the corresponding Riemann solution may turn into a shock and a contact discontinuity after the perturbation of Riemann initial data.This shows the internal mechanism and instability of a delta shock.Then we can obtain the main theorem of this paper.
Theorem 8.There is a unique local solution to the perturbed Riemann problem ( 1) and ( 3) with delta shock.The Riemann solution of ( 1) and ( 5) is instable under a perturbation of the Riemann initial data, which is quite different from the previous perturbed Riemann problems.