Reports of the Academy of Sciences of the USSR
1962. Volume 144, No. 1

MATHEMATICS

B. L. ROZHDESTVENSKII

CONSTRUCTION OF DISCONTINUOUS SOLUTIONS OF A SYSTEM OF TWO QUASILINEAR EQUATIONS

(Presented by Academician S. L. Sobolev on 26 XII 1961)

For a system of two quasilinear equations of hyperbolic type

\[ \frac{\partial u_i}{\partial t}+\frac{\partial \varphi_i(u)}{\partial x}=0;\qquad u=\{u_1,u_2\}\quad (i=1,2) \tag{1} \]

a method is indicated for constructing generalized discontinuous solutions. Under certain restrictions on the class of systems (1), this method leads to the construction of a generalized discontinuous solution; the uniqueness of the latter is established.

1. Restrictions on the class of systems (1). Let the functions \(\varphi_i(u)\) possess continuous second derivatives with respect to their arguments. As is known \((^{1,2})\), system (1) can be reduced to the form

\[ \frac{\partial r_k}{\partial t}+\xi_k(r)\frac{\partial r_k}{\partial x}=0 \quad (k=1,2), \tag{2} \]

where \(r_k=r_k(u)\) are the Riemann invariants. We shall assume that \(\xi_1(r)<\xi_2(r)\). We shall require of system (1) that

\[ \frac{\partial \xi_k(r)}{\partial r_k}>0 \quad (k=1,2). \tag{3} \]

Let \(u^0=\{u_1^0,u_2^0\}\) and \(u=\{u_1,u_2\}\) be the values of the solution on different sides of the discontinuity line \(x=Dt\). Then, as is known \((^{1,2})\), these quantities are related by the Hugoniot conditions

\[ D(u_i-u_i^0)=\varphi_i(u)-\varphi_i(u^0) \quad (i=1,2). \tag{4} \]

Fig. 1

Eliminating the quantity \(D\) from the two equations (4), we obtain one equation relating the quantities \(u,u^0\). We shall regard the point \(u^0\) as fixed. Then, as is known \((^{1-3})\), the equation obtained determines two smooth curves passing through the point \(u^0\) and tangent at this point to the lines \(r_2=r_2^0\), \(r_1=r_1^0\) (Fig. 1).

We shall assume that the line tangent to the straight line \(r_2=r_2^0\) is projected one-to-one onto the straight lines \(r_2=\mathrm{const}\), and the line tangent to the straight line \(r_1=r_1^0\) is projected one-to-one onto the straight lines \(r_1=\mathrm{const}\). Thus, our second assumption means that equations (4) can be written in the form

\[ r_2=R_2(r_1,r_1^0,r_2^0);\qquad D=D_1(r_1,r_1^0,r_2^0); \tag{5} \]

\[ r_1=R_1(r_2,r_1^0,r_2^0);\qquad D=D_2(r_2,r_1^0,r_2^0); \tag{6} \]

moreover, the functions \(R_2,D_1,R_1,D_2\) are single-valued and possess continuous first derivatives with respect to their variables.

The following properties of these functions are known \((^{2-4})\):

\[ R_2(r_1^0,r_1^0,r_2^0)=r_2^0;\qquad D_1(r_1^0,r_1^0,r_2^0)=\xi_1(r_1^0,r_2^0), \tag{7} \]

\[ R_1(r_2^0,r_1^0,r_2^0)=r_1^0;\qquad D_2(r_2^0,r_1^0,r_2^0)=\xi_2(r_1^0,r_2^0). \]

We shall require that:

\[ \text{for } r_1<r_1^0\quad \xi_1\bigl(r_1,R_2(r_1,r_1^0,r_2^0)\bigr) < D_1(r_1,r_1^0,r_2^0) < \xi_1(r_1^0,r_2^0), \tag{8} \]

for \(r_2>r_2^0\)
\[ \xi_2\bigl(R_1(r_2,r_1^0,r_2^0),r_2\bigr)>D_2(r_2,r_1^0,r_2^0)>\xi_2(r_1^0,r_2^0), \tag{9} \]
\[ \left|\partial R_2(r_1,r_1^0,r_2^0)/\partial r_1\right|<1;\qquad \left|\partial R_1(r_2,r_1^0,r_2^0)/\partial r_2\right|<1. \]

The listed conditions are satisfied by systems of type (A), considered in papers \((^{1,4})\), by systems of type V in paper \((^3)\), and, in particular, by the systems of equations
\[ \frac{\partial \rho}{\partial t}+\frac{\partial \rho v}{\partial x}=0;\qquad \frac{\partial \rho v}{\partial t}+\frac{\partial}{\partial x}\,[p(\rho)+\rho v^2]=0; \]
\[ \frac{\partial \eta}{\partial t}-\frac{\partial v}{\partial x}=0;\qquad \frac{\partial v}{\partial t}+\frac{\partial}{\partial x}\,p\left(\frac{1}{\eta}\right)=0, \tag{10} \]
which are important for applications, in the case \(p'>0,\ p''\geq 0\).

2. The method of characteristics. The construction of a classical solution of the Cauchy problem for systems of quasilinear equations by the method of characteristics is presented in papers \((^{5,6})\). In constructing discontinuous solutions of system (1), we use an improved method of characteristics which makes it possible, in particular, to prove the existence of Lipschitz-continuous solutions of (1) under weaker restrictions on the initial functions.

Suppose that for system (2) the Cauchy problem is posed
\[ r(0,x)=r^0(x);\qquad |x|\leq a, \tag{11} \]
where the vector function \(r^0(x)\) satisfies the Lipschitz condition with respect to the variable \(x\) with bounded constant \(C\) (this Cauchy problem is equivalent to the Cauchy problem for system (1)). We shall call \(r(t,x)\) a weak solution of the Cauchy problem (2), (11) if \(r_k(t,x)\) are constant along the integral curves of the equations
\[ \partial x_k/\partial t=\xi_k(r_1(t,x_k),r_2(t,x_k))\qquad (k=1,2), \]
issuing from the segment \(|x|\leq a\) of the initial axis.

Theorem 1. For \(0\leq t\leq T(C)\) there exists a unique weak solution of problem (2), (11), satisfying the Lipschitz condition with respect to the variable \(x\) with constant \(2C\). The theorem is valid under the single condition that the functions \(\varphi_i(u)\) have second derivatives and \(\xi_k(r)\) have first derivatives with respect to their arguments.

We indicate here the method of successive approximations by means of which the solution of problem (2), (11) is constructed.

Suppose that an approximation
\[ {}^{(n)}r(t,x)=\{\,{}^{(n)}r_1(t,x);\ {}^{(n)}r_2(t,x)\,\} \]
is known; moreover
\[ {}^{(n)}r(0,x)=r^0(x). \]
Define
\[ {}^{(n+1)}r(t,x) \]
as the solution of two Cauchy problems:
\[ \frac{\partial\,{}^{(n+1)}r_1}{\partial t} +\xi_1\bigl({}^{(n+1)}r_1,{}^{(n)}r_2(t,x)\bigr) \frac{\partial\,{}^{(n+1)}r_1}{\partial x}=0;\qquad {}^{(n+1)}r_1(0,x)=r_1^0(x); \tag{12} \]
\[ \frac{\partial\,{}^{(n+1)}r_2}{\partial t} +\xi_2\bigl({}^{(n)}r_1(t,x),{}^{(n+1)}r_2\bigr) \frac{\partial\,{}^{(n+1)}r_2}{\partial x}=0;\qquad {}^{(n+1)}r_2(0,x)=r_2^0(x). \tag{13} \]
The functions
\[ {}^{(n+1)}r_1(t,x)\quad\text{and}\quad{}^{(n+1)}r_2(t,x) \]
are constant along the integral curves of the equations
\[ \frac{d\,{}^{(n+1)}x_1}{dt} =\xi_1\bigl({}^{(n+1)}r_1,{}^{(n)}r_2(t,{}^{(n+1)}x_1)\bigr);\qquad \frac{d\,{}^{(n+1)}x_2}{dt} =\xi_2\bigl({}^{(n)}r_1(t,{}^{(n+1)}x_2),{}^{(n+1)}r_2\bigr). \tag{14} \]
If
\[ {}^{(n)}r(t,x) \]
satisfies the Lipschitz condition, then through every point for \(0\leq t\leq T(C)\) there passes a unique integral curve of each of equations (14).

Lemma. If
\[ {}^{(n)}r(t,x) \]
satisfies the Lipschitz condition with constant equal to \(2C\), then
\[ {}^{(n+1)}r(t,x) \]
for \(0\leq t\leq T(C)\) also satisfies the Lipschitz condition with constant \(2C\).

The proof of the lemma and the definition of \(T(C)\) are omitted here.

We must specify the first approximation satisfying the conditions of the lemma: set \(\overset{(0)}{r}(t,x)=r^0(x)\). The proof of convergence of the method of successive approximations is based on the estimate

\[ \left|\overset{(n+1)}{r}(t,x)-\overset{(n)}{r}(t,x)\right| \leq 4MCt\max_{\tau,\xi}\left|\overset{(n)}{r}(\tau,\xi)-\overset{(n-1)}{r}(\tau,\xi)\right|, \tag{15} \]

where \(M>|\partial \xi_k/\partial r_j|\), from which follows the uniform convergence of the successive approximations to a weak solution \(r(t,x)\). The uniqueness of the weak solution of problem (2), (11) also follows from estimate (15).

3. Construction of discontinuous solutions

The method described above for constructing Lipschitz-continuous solutions of system (1) can be applied up to the moment at which a singularity is formed in the solution. For problem (1), (11), such singularities are the occurrence of a jump in the solution, or the more frequent case—the unboundedness of the ratio \(\Delta r_k/\Delta x\) as \(\Delta x\to 0\). It is characteristic, however, that in all these cases the Riemann invariant \(r_k(t,x)\) possessing the singularity is, in a neighborhood of the point containing the singularity, monotonically decreasing.

Therefore, let us consider for system (1) the Cauchy problem (11), assuming that the Riemann invariants \(r_k^0(x)\) at the point \(x=0\) have an isolated singularity.

Theorem 2. Let the initial data (11), given for \(|x|\leq a\), have at the point \(x=0\) an isolated singularity of one of the two types:

1) \(r^0(x)\) is a discontinuous function; moreover \(r_k^0(-0)>r_k^0(+0)\), and everywhere except the point \(x=0\) it satisfies the Lipschitz condition;

2)

\[ \frac{r_k^0(\Delta x)-r_k^0(0)}{\Delta x}\to -\infty \quad \text{as } \Delta x\to 0; \]

moreover \(r_k^0(x)\) decreases monotonically for \(|x|\leq a\). Then there exists \(T>0\) such that for \(0\leq t\leq T\) there exists a unique generalized discontinuous solution of problem (1), (11).

Fig. 2

We shall not dwell on the definition of the quantity \(T\) and the proof of the theorem, restricting ourselves to a description of the method for constructing a generalized discontinuous solution of problem (1), (11) in the case where the initial function \(r^0(x)\) is discontinuous.

By the method indicated above we solve the Cauchy problem (1), (11) separately on the intervals \(-a\leq x\leq 0\) and \(0\leq x\leq a\). The solution \(r(t,x)\) will be defined in the regions to the left of \(OL_1^{-}\) and to the right of \(OL_2^{+}\) (Fig. 2). The construction differs somewhat in the cases

\[ R_2\bigl(r_1^0(+0),\,r_1^0(-0),\,r_2^0(-0)\bigr)>r_2^0(+0) \]

and

\[ R_2\bigl(r_1^0(+0),\,r_1^0(-0),\,r_2^0(-0)\bigr)\leq r_2^0(+0). \]

We consider only the latter. We solve for system (2) the problem:

\[ r_1\big|_{OL_2^+}=r_1(t,x)\big|_{OL_2^+},\quad r_2(0,0)=r_2^0(+0)+ \]

\[ +\alpha\left[ R_2\bigl(r_1^0(+0),\,r_1^0(-0),\,r_2^0(-0)\bigr)-r_2^0(+0) \right], \quad 0\leq \alpha\leq 1. \tag{16} \]

Such a problem reduces to that considered in § 2. Its solution will be defined in the zone \(L_2^{-}OL_2^{+}\), with \(OL_2^{-}\) lying between \(OL_1^{-}\) and \(OL_2^{+}\) (Fig. 2). The solution of this problem in the zone \(L_2^{-}OL_2^{+}\) will also be denoted by \(r(t,x)\). It is easy to establish that on the line \(OL_2^{-}\) \(r_2(t,x)\) satisfies the Lipschitz condition. Suppose that in the zone \(L_2^{-}ONM\) a discontinuous function \(\overset{(n)}{r}(t,x)\) is given, having a single line of discontinuity \(OL_{D_1}^{(n)}\), and satisfying outside this line of discontinuity—

* The magnitude \(T\) is limited by the possibility that a new singularity may appear.

the Lipschitz condition with constant \(C\) (to the left of \(OL_{D_1}^{(n)}\), \(r^{(n)}(t,x)=r(t,x)\)).

We indicate a method for constructing the approximation \(r^{(n+1)}(t,x)\). Let us solve the Cauchy problems*:

\[ \frac{\partial \widetilde r_1^{(n+1)}}{\partial t} +\xi_1\bigl(\widetilde r_1^{(n+1)},\, r_2^{(n)}(t,x)\bigr) \frac{\partial \widetilde r_1^{(n+1)}}{\partial x}=0; \qquad \left.\widetilde r_1^{(n+1)}\right|_{OL_2^-} = \left.r_1(t,x)\right|_{OL_2^-}. \tag{17} \]

Let the solution of this problem be known in the zone \(L_2^-OL_1^{(n+1)}\), containing within itself the line \(OL_{D_1}^{(n)}\). It is established that \(\widetilde r^{(n+1)}(t,x)\) satisfies the Lipschitz condition with constant \(C\). In the zone \(L_1^{(n+1)}OL_1^-\) consider the ordinary differential equation:

\[ \frac{d x_{D_1}^{(n+1)}}{dt} = D_1\bigl(\widetilde r^{(n+1)}(t,x_{D_1}^{(n+1)}),\, r_1(t,x_D^{(n+1)}),\, r_2(t,x_D^{(n+1)})\bigr); \qquad x_{D_1}^{(n+1)}(0)=0. \tag{18} \]

Since the right-hand side of (18) satisfies the Lipschitz condition and conditions (8) are fulfilled, there exists a unique integral curve of equation (18) issuing from the point \((0,0)\). We denote it by \(OL_{D_1}^{(n+1)}\). After this we define \(r_2^{(n+1)}(t,x)\):

\[ \frac{\partial \widetilde r_2^{(n+1)}}{\partial t} + \xi_2\bigl(\widetilde r_1^{(n+1)}(t,x),\, \widetilde r_2^{(n+1)}\bigr) \frac{\partial \widetilde r_2^{(n+1)}}{\partial x} =0; \]

\[ \left.\widetilde r_2^{(n+1)}\right|_{OL_{D_1}^{(n+1)}} = \left. R_2\bigl(\widetilde r_1^{(n+1)}(t,x),\, r_1(t,x),\, r_2(t,x)\bigr) \right|_{OL_{D_1}^{(n+1)}}. \tag{19} \]

The solution of problem (19) will satisfy the Lipschitz condition and is defined in the zone \(L_{D_1}^{(n+1)}OL_2^-\). After this we define the approximation \(r^{(n+1)}(t,x)\):

\[ r^{(n+1)}(t,x)= \begin{cases} r(t,x), & \text{to the left of } OL_{D_1}^{(n+1)},\\ \widetilde r^{(n+1)}(t,x), & \text{in the zone } L_{D_1}^{(n+1)}OL_2^-,\\ r(t,x), & \text{to the right of } OL_2^-. \end{cases} \tag{20} \]

The zeroth approximation is chosen so as to satisfy the assumptions made.

One proves the uniform convergence of the lines \(OL_{D_1}^{(n)}\) to the discontinuity line \(OL_{D_1}\), and the uniform convergence of the approximations \(\widetilde r^{(n)}(t,x)\) in any common part of the domains of definition of \(\widetilde r^{(n)}(t,x)\). Thus, there exists a limiting line \(OL_{D_1}\), which is a line of strong discontinuity of \(r(t,x)\). In view of the uniform convergence of the sequence \(r^{(n)}(t,x)\), on the line \(OL_{D_1}\) the Hugoniot conditions (4) are fulfilled, and outside the discontinuity line the limit \(r(t,x)\) is a weak solution of system (1).

The remaining possible cases are considered in an analogous way.

Received
19 XII 1962

CITED LITERATURE

1. B. L. Rozhdestvenskii, UMN, 15, no. 6 (96), 59 (1960).
2. P. D. Lax, Comm. Pure and Appl. Math., 10, 4, 537 (1957).
3. N. N. Kuznetsov, DAN, 131, no. 3, 503 (1960).
4. B. L. Rozhdestvenskii, DAN, 122, no. 5, 762 (1958).
5. R. Courant, P. D. Lax, Comm. Pure and Appl. Math., 2, 255 (1949).
6. P. Hartman, A. Wintner, Am. J. Math., 74, 834 (1952).

* On the discontinuity lines of the coefficients of equation (17), the continuity condition is imposed on \(r_1^{(n+1)}(t,x)\).