UDC 517.945

MATHEMATICS

V. A. BOROVIKOV

ON THE PROBLEM OF THE DECAY OF A DISCONTINUITY FOR A SYSTEM OF TWO QUASILINEAR EQUATIONS

(Presented by Academician M. V. Keldysh on 15 VII 1968)

1. In this note an example is constructed satisfying the Lax condition (see below) of a hyperbolic quasilinear system

\[ \frac{\partial u}{\partial t}+\frac{\partial f(u,v)}{\partial x}=0, \qquad \frac{\partial v}{\partial t}+\frac{\partial \varphi(u,v)}{\partial x}=0, \tag{1} \]

for which the shock adiabat \(A(u_1,v_1)\), i.e. the set of points \(u_2, v_2\) satisfying, for some \(\omega\), the relation

\[ \omega(u_1-u_2)=f(u_1,v_1)-f(u_2,v_2);\quad \omega(v_1-v_2)= \]
\[ =\varphi(u_1,v_1)-\varphi(u_2,v_2), \tag{2} \]

is a bounded set in the plane \(u_2, v_2\). Therefore it is easy to specify such values \(u_-, v_-\) and \(u_+, v_+\) that system (1) has no self-similar solution of the problem of the decay of a discontinuity, i.e. no solution of the form \(u=u(x/t)\), \(v=v(x/t)\), satisfying the Cauchy data

\[ u(x,0)= \begin{cases} u_+, & x>0,\\ u_-, & x<0; \end{cases} \qquad v(x,0)= \begin{cases} v_+, & x>0,\\ v_-, & x<0. \end{cases} \tag{3} \]

Theorem. There exists a constant \(\eta_0>1\) such that, if \(u_+>\eta_0^2u_-\), then for system (1), where

\[ f=3\ln u+v,\qquad \varphi=2/u, \tag{4} \]

the problem of the decay of a discontinuity (3) has no self-similar solution \(u=u(x/t)\), \(v=v(x/t)\).

The proof of this theorem is based on the following lemma.

Lemma. There exist constants \(\eta_0>1\) and \(C\) such that, for system (1), (4), the shock adiabat \(A(u_1,v_1)\), i.e. the set of points \(u_2, v_2\) satisfying (2), is contained in the rectangle
\[ u_1/\eta_0<u_2<\eta_0u_1,\qquad |v_1-v_2|<C. \]

2. Let us prove the lemma. Eliminating \(\omega\) from (2), we obtain that \(A(u_1,v_1)\) coincides with the zeros of the function

\[ K(u_2,v_2)=(u_2-v_1)\bigl(\varphi(u_2,v_2)-\varphi(u_1,v_1)\bigr)- \]
\[ -(v_2-v_1)\bigl(f(u_2,v_2)-f(u_1,v_1)\bigr)=0. \tag{5} \]

If we use (4) and then solve (5) with respect to \(v_2\), we obtain the equation

\[ v_2=v_1+\frac{3}{2}\ln\eta\pm\sqrt{\Delta}, \]

where \(\eta=u_2/u_1\), \(\Delta=\frac{9}{4}\ln^2\eta-2(\eta-1)^2/\eta\). But \(\Delta\ge0\) only when \(1/\eta_0<\eta<\eta_0\), where \(\eta_0>1\) is the root of the equation \(\Delta=0\). Therefore (5) has real roots \(u_2, v_2\) only when \(u_1/\eta_0<u_2<\eta_0u_1\), and it is easily verified that, for these values of \(u_2\), the difference \(|v_1-v_2|\) is bounded in modulus.

3. Before proving the theorem, let us recall how a self-similar solution of the problem of the decay of a discontinuity is defined. According to \((^{1,2})\), a self-similar solution of the problem of the decay of a discontinuity is understood to mean piecewise continuous functions \(u(\xi)\), \(v(\xi)\) (where \(\xi=x/t\)), having piecewise continuous derivatives and satisfying the conditions:

\(1^\circ.\) \(u(\pm\infty)=u_{\pm};\ v(\pm\infty)=v_{\pm}.\)

\(2^\circ.\) If at a point \(\xi\) the functions \(u(\xi)\), \(v(\xi)\) are continuous and have nonzero derivatives, then they satisfy the equations

\[ \xi u'_{\xi}=f'_u u'_{\xi}+f'_v v'_{\xi}, \]
\[ \xi v'_{\xi}=\varphi'_u u'_{\xi}+\varphi'_v v'_{\xi}. \tag{6} \]

Formula (6) is obtained if one substitutes into (1) the functions \(u(x/t)\), \(v(x/t)\) and then sets \(x/t=\xi\). From (6) it is clear that the tangent \(u'(\xi)\), \(v'(\xi)\) to the curve \(u=u(\xi)\), \(v=v(\xi)\), considered in the \(u,v\)-plane, is at each point an eigenvector of the matrix

\[ \begin{pmatrix} f'_u & f'_v\\ \varphi'_u & \varphi'_v \end{pmatrix}, \tag{7} \]

and \(\xi\) is the corresponding eigenvalue. Curves in the \(u,v\)-plane touching at each of their points an eigenvector of the matrix (7) at that point are called characteristics. The naturally arising Lax condition consists in the requirement that along each characteristic the corresponding eigenvalue vary strictly monotonically. If this condition is fulfilled, then on any characteristic \(u(\xi)\), \(v(\xi)\) are single-valued functions of \(\xi\), and therefore \(u(x/t)\), \(v(x/t)\) is a single-valued and continuous solution of the system (1).

\(3^\circ.\) If at \(\xi=\omega\) the functions \(u(\xi)\), \(v(\xi)\) have a discontinuity, with

\[ u(\omega+0)=u_2;\quad u(\omega-0)=u_1; \]
\[ v(\omega+0)=v_2;\quad v(\omega-0)=v_1, \]

then \(u_1,v_1,u_2,v_2\) and \(\omega\) are related by relation (2), i.e., the point \(u_2,v_2\) lies on the shock adiabat \(A(u_1,v_1)\).

Equation (2) is obtained if the system (1) is written in the \(x,t\)-plane in the form

\[ \operatorname{div}(u,f(u,v))=0;\qquad \operatorname{div}(v,\varphi(u,v))=0 \]

and these expressions are integrated over a neighborhood of the straight line \(x/t=\omega\). If the solution \(u(x/t)\), \(v(x/t)\) has a discontinuity at \(x/t=\omega\), then we say that the solution contains at \(x/t=\omega\) a shock wave.

\(4^\circ.\) Only stable shock waves are allowed.

The stability condition consists in requiring that in the \(x,t\)-plane three characteristics of the system (1) impinge on the discontinuity line \(x/t=\omega\): either two from the left and one from the right, in which case \(\omega<\lambda_1,\ \mu_1;\ \mu_2<\omega<\lambda_2\), or one from the left and two from the right, in which case \(\mu_1<\omega<\lambda_1;\ \omega>\lambda_2,\ \mu_2\). Here \(\mu_1,\lambda_1\) and \(\mu_2,\lambda_2\) \((\mu_1<\lambda_1,\ \mu_2<\lambda_2)\) are respectively the eigenvalues of the matrix (7) at the points \((u_1,v_1)\) and \((u_2,v_2)\).

4. Let us outline the proof of the theorem. Simple calculations show that for the system (4) \(\lambda=2/u,\ \mu=1/u\), and the corresponding characteristics have the form \(u=e^{-2(v-s)}\) and \(u=e^{-(v-\sigma)}\), where \(s,\sigma\) \((-\infty<s,\sigma<\infty)\) are parameters specifying the characteristic. In Fig. 1 the characteristics passing through some point \(u_1,v_1\) are shown; the arrows indicate the directions of increase of the corresponding eigenvalues, and the dashed curve is the shock adiabat \(A(u_1,v_1)\).

Let us now consider what conditions \(u_-,v_-\) and \(u_+,v_+\) must satisfy in order that the problem of the decay of a discontinuity have a self-similar solution. To do this, let us trace the change of \(u(\xi)\) as \(\xi\) increases from \(-\infty\) to \(+\infty\).

As \(\xi\) increases on an interval where \(u(\xi)\), \(v(\xi)\) are continuous and have nonzero derivatives \(u'(\xi)\), \(v'(\xi)\), the point \(u(\xi), v(\xi)\) moves along a certain characteristic of system (4) in the direction of increase of the corresponding eigenvalue (equal, according to (6), to \(\xi\)). But since \(\lambda=2/u\), \(\mu=1/u\), under such motion \(u(\xi)\) must decrease. It is easy to show, following (1) or (2), that for any system satisfying Lax’s condition, the solution \(u(\xi), v(\xi)\) cannot contain more than two shock waves. But, according to the lemma, at each shock wave the limit on the right from the discontinuity point \(x/t=\omega\), \(u_2=u(\omega+0)\), can exceed the limit on the left \(u_1=u(\omega-0)\) by no more than a factor of \(\eta_0\). Therefore, as \(\xi\) varies from \(-\infty\) to \(+\infty\), the function \(u(\xi)\) cannot increase by more than a factor of \(\eta_0^2\), which proves the theorem.

Fig. 1

5. Example (4) is a system defined in the half-plane \(u>0\), \(-\infty<v<\infty\). It is easy to construct an example of a system possessing analogous properties but defined on the entire \(u,v\)-plane. For this it is enough to set

\[ f=5\left[uv\sqrt{u^2+1}+u^2+\ln\left(u+v\sqrt{u^2+1}\right)\right]+v, \tag{8} \]

\[ \varphi=-^{16}/_{3}\left[2u^3+3u+2(u^2+1)\sqrt{u^2+1}\right]. \]

The investigation of this system is analogous to the investigation of system (4) carried out above, but differs in more cumbersome calculations.

Received
20 VI 1968

REFERENCES

\(^1\) I. M. Gel'fand, UMN, 14, no. 2, 87 (1959). \(^2\) B. L. Rozhdestvenskii, UMN, 15, no. 6, 59 (1960).