This paper is concerned with the structure of the weak entropy solutions to twodimension Riemann initialboundary value problem with curved boundary. Firstly, according to the definition of weak entropy solution in the sense of BardosLerouxNedelec (1979), the necessary and sufficient condition of the weak entropy solutions with piecewise smooth is given. The boundary entropy condition and its equivalent formula are proposed. Based on Riemann initial value problem, weak entropy solutions of Riemann initialboundary value problem are constructed, the behaviors of solutions are clarified, and we focus on verifying that the solutions satisfy the boundary entropy condition. For different Riemann initialboundary value data, there are a total of five different behaviors of weak entropy solutions. Finally, a workedout specific example is given.
1. Introduction
Multidimensional conservation laws are a famous hard problem that plays an important role in mechanics and physics [1–3]. For Cauchy problem of multidimensional scalar conservation laws, Conway and Smoller [4] and Kruzkov [1] have proved that weak solution uniquely exists if it also satisfies entropy condition, and it is called weak entropy solutions. In order to further understand qualitative behavior of solutions, it is also important to investigate multidimensional Riemann problems. For twodimensional case, Lindquist [5], Wagner [6], Zhang and Zheng [7] Guckenheimer [8], Zheng [9] among others, have discussed some relating Riemann problems. In a previous discussion, initial value contains several constant states with discontinuity lines so that selfsimilar transformations can be applied to reduce twodimensional problem to onedimensional case. The situation that initial value contains two constant states divided by a curve can not be solved by selfsimilar transformations, and Yang [10] proposed a new approach for construction of shock wave and rarefaction wave solutions; especially, rarefaction wave was got by constructing implicit function instead of the usual selfsimilar method. This approach can be expanded to general dimension. In addition, multidimensional scalar conservation laws with boundary are more common in practical problems. Bardos et al. [2] have proved the existence and uniqueness of the weak entropy solution of initialboundary problems of multidimensional scalar conservation laws. The main difficulty for nonlinear conservation laws with boundary is to have a good formation of the boundary condition. Namely, for a fixed initial value, we really can not impose such a condition at the boundary, and the boundary condition is necessarily linked to the entropy condition. Moreover the behavior of solutions for onedimensional problem with boundary was discussed in [11–18]. However, for multidimensional problem with boundary, the behaviors of solutions are still hard to study.
In this paper, twodimensional case as an example of Yang's multidimensional Riemann problem [10] is expanded to the case with boundary. Considering twodimensional Riemann problem for scalar conservation laws with curved boundary,
where , and are both constants, , , is a smooth manifold and divides into two infinite parts, , and and denote .
In Section 2, weak entropy solution of Riemann initialboundary value problem (1.1) is defined, and the boundary entropy condition is discussed. In Section 3, weak entropy solutions of the corresponding Riemann initial value problem are expressed. In Section 4, using the weak entropy solutions of the corresponding Riemann initial value problem, we construct the weak entropy solutions of Riemann initialboundary value problem, and prove that they satisfy the boundary entropy condition. The weak entropy solutions include a total of five different shock and rarefaction wave solutions based on different Riemann data. Finally, in Section 5, we give a workedout specific example.
2. Preliminaries
According to the definition of the weak entropy solution and the boundary entropy condition to the general initialboundary problems of multidimensional scalar conservation laws which was proposed by Bardos et al. [2] and Pan and Lin [13], we can obtain the following definition and three lemmas for Riemann initialboundary value problem (1.1).
Definition 2.1.
A locally bounded and bounded variation function on is called a weak entropy solution of Riemann initialboundary value problem (1.1) if, for any real constant and for any nonnegative function , it satisfies the following inequality:
where is the outward normal vector of curve .
Lemma 2.2.
If is a weak entropy solution of (1.1), then it satisfies the following boundary: entropy condition
where .
It can be easily proved that , , so (2.2) can be rewritten as
thus one can get or
and one notices that , , then boundary entropy condition (2.2) is equivalent to
The proof for onedimension case of Lemma 2.2 can be found in Pan and Lin's work [13], and the proof for dimension case is totally similar to onedimension case; actually the idea of the proof first appears in Bardos et al.'s work [2], so the proof details for Lemma 2.2 are omitted here.
Lemma 2.3.
A piecewise smooth function with smooth discontinuous surface is a weak entropy solution to the Riemann initialboundary value problem (1.1) in the sense of (2.1) if and only if the following conditions are satisfied.
(i) RankineHugoniot condition: At any point on discontinuity surface of solution , is a unit normal vector to at if
then
where , .
For any constant , ,
or equivalently
(ii) Boundary entropy condition:
(iii) Initial value condition:
For piecewise smooth solution with smooth discontinuous surface, RankineHugoniot condition (2.7), entropy conditions (2.8), (2.9) and initial value condition (2.11) are obviously satisfied, see also the previous famous works in [4, 7–9]. As in Lemma 2.2, boundary entropy condition (2.10) also holds. The proof of the converse in not difficult and is omitted here.
According to Bardos et al.'s work [2], we have the following Lemma.
Lemma 2.4.
If is piecewise smooth weak entropy solution of (1.1) which satisfies the conditions of Lemma 2.3, then is unique.
According to the uniqueness of weak entropy solution, as long as the piecewise smooth function satisfying Lemma 2.3 is constructed, the weak entropy solution of Riemann initialboundary value problem can be obtained.
3. Solution of Riemann Initial Value Problem
First, we study the Riemann initial value problem corresponding to the Riemann initialboundary value problem (1.1) as follows:
Condition For ,
where is a certain interval can be a finite number or .
Condition combines flux functions and curved boundary manifold , providing necessary condition for the convex property of the new flux function which will be constructed in formula (4.5). The convex property clarifies whether the characteristics intersect or not, whether the weak solution satisfied internal entropy conditions (2.8) and (2.9) and boundary entropy condition (2.10), In addition, Condition is easily satisfied, for example, , , then , so Condition holds. Here is a cubic curve on the plane, and it is strictly bending.
Yang's work [10] showed that depending on whether the characteristics intersect or not, the weak entropy solution of (3.1) has two forms as follows.
Lemma 3.1 (see [10]).
Suppose () holds. If , then weak entropy solution of (3.1) is shock wave solution , and
and discontinuity surface is
where , , .
Lemma 3.2 (see [10]).
Suppose that () holds. If , then weak entropy solution of (3.1) is rarefaction wave solution , and
where is the implicit function which satisfies
Theorem 3.3 (see [10]).
Suppose that () holds. Given , then
(i)if , weak entropy solution of (3.1) is and has a form as (3.3);
(ii)if , weak entropy solution of (3.1) is and has a form as (3.5);
(iii)weak entropy solutions formed as (3.3) and (3.5) uniquely exist.
The weak entropy solutions constructed here are piecewise smooth and satisfy conditions (i) and (iii) of Lemma 2.3.
4. Solution of Riemann InitialBoundary Value Problem
Now we restrict the weak entropy solutions of the Riemann initial value problem (3.1) constructed in Section 3 in region , and they still satisfy conditions (i) and (iii) of Lemma 2.3. If they also satisfy the boundary entropy condition (ii) of Lemma 2.3, then they are the weak entropy solutions of Riemann initialboundary value problem (1.1).
Based on different Riemann data of and , the weak entropy solutions of the Riemann initial value problem (3.1) have the following five different behaviors when restricted in region .
If , the solution of (3.1) is shock wave and
is formed by moving along the direction of the vector , and the outward normal vector of curve is equal to . According to the angle between and , the solution restricted in has two behaviors as follows.
Case 1.
If and .
See also Figure 1(a); it shows that the angle between and is an acute angle, the shock wave surface is outside region , and the solution is constant state formed as
Figure 1. Case 1.The constant solutionThe phase plane
Case 2.
If and .
See also Figure 2; it shows that the angle between and is an obtuse angle, the shock wave surface is inside region , and the solution is shock wave formed as
Figure 2. The shock wave solution of Case 2.
If , the solution of (3.1) is rarefaction wave and
is formed by moving along the direction of the vector , is formed by moving along the direction of the vector , and the outward normal vector of curve is equal to .
We construct a new flux function
according to condition (), , is convex, and is monotonically increasing function, so . And also
Thus, . According to the angles between , , and , the solution restricted in has three behaviors as follows.
Case 3.
If and .
See also Figure 3; it shows that the angles between , and are obtuse angles, the rarefaction wave surfaces and are both inside region , and the solution is rarefaction wave formed as
where is the implicit function which satisfies (3.6).
Figure 3. The rarefaction wave solution of Case 3.
Case 4.
If and .
See also Figure 4(a); it shows that the angle between and is an obtuse angle, the angle between and is an acute angles, the rarefaction wave surface is inside region , the rarefaction wave surface is outside region , and the solution is rarefaction wave formed as
where is the implicit function which satisfies (3.6).
Figure 4. Case 4.The rarefaction wave solutionThe phase plane
Case 5.
If and .
See also Figure 5(a); it shows that the angles between , , and are acute angles, the rarefaction wave surfaces and are both outside region , and the solution is constant state formed as
Figure 5. Case 5.The constant solutionThe phase plane
Next, we verify the above five solutions all satisfying the boundary entropy condition (ii) of Lemma 2.3. By noticing the definition of (4.5) and its convex property, the boundary entropy condition (ii) of Lemma 2.3 can be equivalent to the following formula
and thus we verify the above five solutions all satisfying the boundary entropy condition (4.10).
Case 1.
When , , the shock wave solution is formed as (4.2). In this case, since
and , where is the extreme point of . For , according to the convex property of , we have that
and so the boundary entropy condition (4.10) is verified.
Case 2.
When , , the shock wave solution is formed as (4.3). In this case, , so the boundary entropy condition (4.10) is naturally verified.
Case 3.
When , , the rarefaction wave solution is formed as (4.7). In this case, , and so the boundary entropy condition (4.10) is naturally verified.
Case 4.
When , , the rarefaction wave solution is formed as (4.8). In this case, , and (see also Figure 4(b)), namely, . For , according to the convex property of and Lagrange mean value theorem, there exists , satisfying
and so the boundary entropy condition (4.10) is verified.
Case 5.
When , , the rarefaction wave solution is formed as (4.9). In this case, since (see also Figure 5(b)) For , according to the convex property of and Lagrange mean value theorem, there exists , satisfying
and so the boundary entropy condition (4.10) is verified.
In summary, we have the following theorem.
Theorem 4.1.
Suppose that () holds. Given , then
(i)if and , the solution of (1.1) is constant state and has form as (4.2),
(ii)if and , the solution of (1.1) is shock wave , and has form as (4.3),
(iii)if and , the solution of (1.1) is rarefaction wave and has a form as (4.7),
(iv)if and , the solution of (1.1) is rarefaction wave and has a form as (4.8);
(v)if and , the solution of (1.1) is constant state and has a form as (4.9).
In addition the solutions formed as (4.2), (4.3), (4.7), (4.8), and (4.9) uniquely exist.
Corollary 4.2.
Suppose that for . can be finite or , and when ,
(i)if and , the solution of (1.1) is rarefaction wave and has a form as (4.7),
(ii)if and , the solution of (1.1) is rarefaction wave and has a form as (4.8),
(iii)if and , the solution of (1.1) is constant state and has a form as (4.9),
(iv)if and , the solution of (1.1) is constant state and has a form as (4.2),
(v)if and , the solution of (1.1) is shock wave and has a form as (4.3).
Corollary 4.3.
The approach here for twodimensional Riemann initialboundary problem can be expanded to the case of general dimension.
5. An Example
Solve the following Riemann initialboundary problem:
where , , , and it denotes . Since , we easily get , and condition holds.
According to the different data of and , the behavior of the solution to Riemann initialboundary problem (5.1) has a total of five situations; they can be described by the following five cases: (i) , ; (ii) , ; (iii) , ; (iv) , ; (v) , .
For case (i), and
and thus the solution is constant state formed as
For case (ii), and
and thus the solution is shock wave solution formed as
For case (iii), and
namely, , thus the solution is rarefaction wave formed as
Here, we only need to solve , where
To solve the following equation of :
using Cardano formula, we can get the unique solution as
Since is the solution of implicit function, we still need to verify satisfying the following three conditions: (a) ; (b) ; (c) . In fact, according to the next proposition, the above three conditions can be easily verified, and the detail the omitted here.
Proposition 5.1.
For any real number , the following formula holds:
Proof.
Let
then satisfies
where must be one root of (5.13). In fact, . Equation (5.13) at most has one real root; but is its real root, thus , and the proposition holds.
For case (iv), and
namely, , and thus the solution is rarefaction wave formed as
where has the same form as (5.10).
For case (v), and
namely, , and thus the solution is constant state formed as
Acknowledgment
This work is supported by the National Natural Science Foundation of China (10771087, 61078040), the Natural Science Foundation of Guangdong Province (7005948).
References

Kruzkov, SN: Generalized solutions of the Cauchy problem in the large for nonlinear equations of first order. Soviet Mathematics Doklady. 10, 785–788 (1969)

Bardos, C, le Roux, AY, Nédélec, JC: First order quasilinear equations with boundary conditions. Communications in Partial Differential Equations. 4(9), 1017–1034 (1979). Publisher Full Text

Razafimandimby, PA, Sango, M: Weak solutions of a stochastic model for twodimensional second grade fluids. Boundary Value Problems. 2010, (2010)

Conway, E, Smoller, J: Clobal solutions of the Cauchy problem for quasilinear firstorder equations in several space variables. Communications on Pure and Applied Mathematics. 19, 95–105 (1966)

Lindquist, WB: The scalar Riemann problem in two spatial dimensions: piecewise smoothness of solutions and its breakdown. SIAM Journal on Mathematical Analysis. 17(5), 1178–1197 (1986). Publisher Full Text

Wagner, DH: The Riemann problem in two space dimensions for a single conservation law. SIAM Journal on Mathematical Analysis. 14(3), 534–559 (1983). Publisher Full Text

Zhang, T, Zheng, YX: Twodimensional Riemann problem for a single conservation law. Transactions of the American Mathematical Society. 312(2), 589–619 (1989). Publisher Full Text

Guckenheimer, J: Shocks and rarefactions in two space dimensions. Archive for Rational Mechanics and Analysis. 59(3), 281–291 (1975)

Zheng, Y: Systems of Conservation Laws: TwoDimensional Riemann Problems, Progress in Nonlinear Differential Equations and Their Applications,p. xvi+317. Birkhäuser, Boston, Mass, USA (2001)

Yang, X: Multidimensional Riemann problem of scalar conservation law. Acta Mathematica Scientia. 19(2), 190–200 (1999)

Dubois, F, LeFloch, P: Boundary conditions for nonlinear hyperbolic systems of conservation laws. Journal of Differential Equations. 71(1), 93–122 (1988). Publisher Full Text

LeFloch, P, Nédélec, JC: Explicit formula for weighted scalar nonlinear hyperbolic conservation laws. Transactions of the American Mathematical Society. 308(2), 667–683 (1988)

Pan, T, Lin, L: The global solution of the scalar nonconvex conservation law with boundary condition. Journal of Partial Differential Equations. 8(4), 371–383 (1995)

Pan, T, Lin, L: The global solution of the scalar nonconvex conservation law with boundary condition. (Continuation). Journal of Partial Differential Equations. 11(1), 1–8 (1998)

Pan, T, Liu, H, Nishihara, K: Asymptotic stability of the rarefaction wave of a onedimensional model system for compressible viscous gas with boundary. Japan Journal of Industrial and Applied Mathematics. 16(3), 431–441 (1999). Publisher Full Text

Pan, T, Jiu, Q: Asymptotic behaviors of the solutions to scalar viscous conservation laws on bounded interval corresponding to rarefaction waves. Progress in Natural Science. 9(12), 948–952 (1999)

Pan, T, Liu, H: Asymptotic behaviors of the solution to an initialboundary value problem for scalar viscous conservation laws. Applied Mathematics Letters. 15(6), 727–734 (2002). Publisher Full Text

Pan, T, Liu, H, Nishihara, K: Asymptotic behavior of a onedimensional compressible viscous gas with free boundary. SIAM Journal on Mathematical Analysis. 34(2), 273–291 (2002). Publisher Full Text