UDC 517.946

MATHEMATICS

V. N. GOLDBERG

DISCONTINUOUS SOLUTIONS OF NONLINEAR MIXED PROBLEMS FOR ALMOST LINEAR HYPERBOLIC SYSTEMS IN THE PLANE

(Presented by Academician I. G. Petrovskii on 6 IV 1970)

1°. In the rectangle \(\bar{\Pi}_T=\{0\le x\le 1,\ 0\le t\le T\}\), \(0<T<\infty\), consider the mixed problem

\[ \partial u/\partial t+\lambda\,\partial u/\partial x=P(x,t,u,q_1,\ldots,q_n); \tag{1} \]

\[ \partial q_i/\partial t-v_i\,\partial q_i/\partial x=Q_i(x,t,u,q_1,\ldots,q_n); \tag{2} \]

\[ u(x,0)=u_0(x),\qquad q_i(x,0)=q_{i,0}(x); \tag{3} \]

\[ q_i(1,t)=0; \tag{4} \]

\[ H(t,u(0,t),q_1(0,t),\ldots,q_n(0,t))=0, \tag{5} \]

where the constants \(\lambda, v_i>0,\ v_i\ne v_j\) for \(i\ne j\), and the functions \(P,Q_i\in C_2(\bar{\Pi}_T\times R^{n+1})\), \(H\in C_2([0,T]\times R^{n+1})\), \(u_0,q_{i,0}\in C_2[0,1]\) satisfy the compatibility conditions necessary for the existence in \(\bar{\Pi}_T\) of a solution of problem (1)—(5) of class \(C_1\). For simplicity, suppose that \(0<T<(\lambda+\max v_i)^{-1}\).*

Following the ideas, developed in works \((^{1-4})\), on unique solvability in the small and continuation in \(t\) of solutions of mixed problems for quasilinear and almost linear hyperbolic systems in the plane, it is not difficult to establish the following proposition.

Theorem 1. Suppose that \(H'_u(0,u_0(0),q_0(0))\ne 0\). Then either A) in \(\bar{\Pi}_T\) there exists a unique solution \((\dot u,\dot q)\in C_1(\bar{\Pi}_T)\) of problem (1)—(5), and
\[ |H'_u(t,\dot u(0,t),\dot q(0,t))|>0 \]
for \(0\le t\le T\); or B) there is a \(0<T^*\le T\) such that in \(\bar{\Pi}_{T^*}=\{0\le x\le 1,\ 0\le t<T^*\}\) there exists a unique solution \((\dot u,\dot q)\in C_1(\Pi_{T^*})\) of problem (1)—(5),
\[ |H'_u(t,\dot u(0,t),\dot q(0,t))|>0 \]
for \(0\le t<T^*\), and at least one of the following equalities holds:**

\[ s\equiv \sup_{\Pi_{T^*}}|\dot u|+\sum_{i=1}^n \sup_{\Pi_{T^*}}|\dot q_i|=\infty,\qquad m\equiv \inf_{0\le t<T^*}|H'_u(t,\dot u(0,t),\dot q(0,t))|=0. \]

Below we investigate the correctness of the formulation of problem (1)—(5) in \(\bar{\Pi}_T\) under the assumption that assertion B) of Theorem 1 holds, but \(T^*<T\), \(s<\infty\), and, consequently, \(m=0\). In this situation, when inequalities (6), (7) are satisfied, problem (1)—(5) has in \(\bar{\Pi}_{T^*+\Delta T}\), for arbitrarily small \(\Delta T>0\), not even a continuous generalized solution (c.g.s.), and increasing the smoothness and compatibility of the functions \(P,Q_i,H,u_0,q_{i,0}\) does not lead to the existence of a c.g.s.

* The inequality \(0<T<(\lambda+\max v_i)^{-1}\) excludes the mutual influence of the boundary conditions (4) and (5) on the properties of the solution of problem (1)—(5) in \(\bar{\Pi}_T\). Therefore all the results of this note hold under more general boundary conditions than (4).

** Examples show that all logical possibilities contained in assertion B) are realized.

In the present note, under the assumption that inequalities (6), (7) and condition b), formulated in 3°, are satisfied, a discontinuous solution (d.s.) of problem (1)—(5) in \(\overline{\Pi}_{\tau}\) for \(\tau>T^*\) is constructed, and theorems are given on the stability of the d.s. under small perturbations of the initial conditions and under perturbation of the function \(H\) by the term

\[ \mu L(u(0,t),q(0,t))\equiv \mu\left[\gamma_1\partial u/\partial t+\gamma_2\partial u/\partial x+ \sum_{k=1}^{n}\left(\rho_1^k\partial q_k/\partial t+\rho_2^k\partial q_k/\partial x\right)\right]_{x=0}, \]

where \(\gamma_i,\rho_i^k\) are constants, and \(\mu\) is a small parameter.

Discontinuous solutions of nonlinear mixed problems for hyperbolic equations of second order were constructed in \((^5)\).

2°. Smoothness \((\dot u,\dot q)\) in \(\overline{\Pi}_{T^*}\). Absence of a d.s. in \(\overline{\Pi}_\tau\) for \(\tau>T^*\).

1. For \(T^*<\tau\le T\), denote by

\[ G_\tau^0=\{(x,t)\in \overline{\Pi}_{\tau},\ 0\le t\le \tau,\ 0\le x\le \lambda t\},\quad K_\tau^0=\overline{\Pi}_{\tau}\setminus G_\tau^0. \]

Let \(\overline{\Pi}_{T^*}'\) be the rectangle \(\overline{\Pi}_{T^*}\) with the “punctured point” \((0,T^*)\), and let \(\mathcal D=\overline{\Pi}_{T^*}'\cap G_\tau^0\).

Below it is assumed throughout that

\[ I\equiv \int_0^{T^*}\left|H'_u(\tau,\dot u(0,\tau),\dot q(0,\tau))\right|^{-1}d\tau<\infty . \tag{6} \]

Theorem 2. The functions \(\dot u\in C(\overline{\Pi}_{T^*})\), \(\dot u\in C_1(\overline{\Pi}_{T^*}')\), \(\dot q\in C_1(\overline{\Pi}_{T^*})\),

\[ \sup_{\mathcal D}|\dot u_x(x,t)|\,\chi(t-x/\lambda)<\infty,\quad \sup_{\mathcal D}|\dot u_t(x,t)|\,\chi(t-x/\lambda)<\infty, \]

where \(\chi(t)=|H'_u(t,\dot u(0,t),\dot q(0,t))|\) for \(0\le t<T^*\).

From Theorem 2 and the equality \(m=0\) it follows that \(H'_u(T^*,u^*,q^*)=0\), where \(u^*=\dot u(0,T^*)\), \(q^*=\dot q(0,T^*)\). Below it is assumed throughout that

\[ H''_{uu}(T^*,u^*,q^*)\ne 0. \tag{7} \]

1. Consider problem (1)—(5) in \(\overline{\Pi}_{\tau}\) for \(\tau>T^*\). Let \(T^*<T_1\le T\) be such that in \(K_{T_1}^0\) there exists a unique solution \((\dot u,\dot q)\in C_1(K_{T_1}^0)\) of problem (1)—(4).*

Definition 1. A vector-function \((u,q)\in C(\overline{\Pi}_{\tau})\), \(T^*<\tau\le T_1\), such that \(u=\dot u,\ q=\dot q\) in \(K_\tau^0\), is called a d.s. of problem (1)—(5) in \(\overline{\Pi}_{\tau}\) if \(H(t,u(0,t),q(0,t))=0\) for \(0\le t\le \tau\), and at each point \((x,t)\in G_\tau^0\)

\[ u(x,t)=u(0,t-x/\lambda)+ \int_{t-x/\lambda}^{t}P(\xi,\tau,u(\xi,\tau),q(\xi,\tau))\big|_{\xi=x(\tau,x,t)}\,d\tau, \tag{8} \]

\[ q_i(x,t)=\dot q_i(\xi_i(x,t),\tau_i(x,t))+ \int_{\tau_i(x,t)}^{t}Q_i(\xi,\tau,u(\xi,\tau),q(\xi,\tau))\big|_{\xi=x_i^+(\tau,x,t)}\,d\tau, \tag{9} \]

where \(x(\tau,x,t)=\lambda\tau+x-\lambda t\), \(x_i^+(\tau,x,t)=-\nu_i\tau+x+\nu_i t\), and \((\xi_i(x,t),\tau_i(x,t))\) is the point of intersection of the characteristics \(\xi=x_i^+(\tau,x,t)\), \(\xi=x(\tau,0,0)\).

Theorem 3. Whatever \(T^*<\tau\le T_1\), there is no d.s. of problem (1)—(5) in \(\overline{\Pi}_{\tau}\).

3°. Construction of a d.s. of problem (1)—(5) in \(\overline{\Pi}_{\tau}\) for \(\tau>T^*\).

1. For \(T^*<\tau\le T_1\), denote by

\[ K_\tau^1=\{(x,t)\in \overline{\Pi}_{\tau},\ T^*\le t\le \tau,\ 0\le x\le \lambda(t-T^*)\},\quad \mathscr L_\tau=G_\tau^0\setminus K_\tau^1, \]

\[ \mathcal D_\tau=\mathscr L_\tau\cup K_\tau^0. \]

* Considering problem (1)—(5) in \(K_T^0\), it is not difficult to establish a theorem of existence and uniqueness of a continuously differentiable solution, analogous to Theorem 1.

Fix an arbitrary \(T^{*}<T_{2}\le T_{1}\) such that in \(\overline{\mathscr L}_{T_{2}}\) there exists a solution \((u,q)\in C(\overline{\mathscr L}_{T_{2}})\) of the system (8), (9), satisfying the equality \(u(0,t)=\mathring u(0,t)\) for \(0\le t\le T^{*}\). By uniqueness in \(\overline{\mathscr L}_{T_{2}}\) of the solution \((u,q)\),

\[ u=\mathring u,\quad q=\mathring q \quad \text{in } \overline{\mathscr D}. \]

In \(\overline{\mathscr D}_{T_{2}}\) define the vector-function \((\widetilde u,\widetilde q)\), setting \((\widetilde u,\widetilde q)=(u,q)\) in \(\overline{\mathscr L}_{T_{2}}\), and \((\widetilde u,\widetilde q)=(\mathring u,\mathring q)\) in \(K_{T_{2}}^{\circ}\).

Theorem 4. The functions \(\widetilde u\in C(\overline{\mathscr D}_{T_{2}})\), \(\widetilde u\in C_{1}(\mathscr D_{T_{2}})\), \(\widetilde q\in C_{1}(\overline{\mathscr D}_{T_{2}})\),

\[ \sup_{\mathscr L_{T_{2}}}|\widetilde u_{x}(x,t)|\,\chi(t-x/\lambda)<\infty, \qquad \sup_{\mathscr L_{T_{2}}}|\widetilde u_{t}(x,t)|\,\chi(t-x/\lambda)<\infty, \]

and the vector-function \((\widetilde u,\widetilde q)\) satisfies equation (1) in \(\mathscr D_{T_{2}}\), and equation (2) in \(\overline{\mathscr D}_{T_{2}}\).

Below the vector-function \((\widetilde u,\widetilde q)\) is denoted by \((\mathring u,\mathring q)\).

2. Suppose

a) \(H'_{u}(t,\mathring u(0,t),\mathring q(0,t))>0\) for \(0\le t<T^{*}\), \(H''_{uu}(T^{*},u^{*},q^{*})<0\)*; (10)

b) the equation \(H(T^{*},u,q^{*})=0\) has a real root \(u^{*}<\overline u^{*}<\infty\) such that \(H(T^{*},u,q^{*})\ne 0\) for \(u^{*}<u<\overline u^{*}\), \(H'_{u}(T^{*},\overline u^{*},q^{*})>0\).

Definition 2. A vector-function \((\overset{1}{u},\overset{1}{q})\in C_{1}(K_{T}^{1})\), \(T^{*}<T\le T_{2}\), such that

\[ \overset{1}{u}(0,T^{*})=\overline u^{*},\qquad \overset{1}{q}(x(t,0,T^{*}),t)=\mathring q(x(t,0,T^{*}),t) \quad \text{for } T^{*}\le t\le T, \]

is called a solution of problem (1)—(5) in \(K_{T}^{1}\), if it satisfies equations (1), (2) in \(K_{T}^{1}\), and for \(T^{*}\le t\le T\)

\[ H(t,\overset{1}{u}(0,t),\overset{1}{q}(0,t))=0, \qquad H'_{u}(t,\overset{1}{u}(0,t),\overset{1}{q}(0,t))>0. \]

Theorem 5. The following alternative holds:

A) in \(K_{T_{2}}^{1}\) there exists a unique solution \((\overset{1}{u},\overset{1}{q})\) of problem (1)—(5);

B) there is a \(T^{*}<\widetilde T\le T_{2}\) such that, for any \(\varepsilon>0\), in \(K_{\widetilde T-\varepsilon}^{1}\) there exists a unique solution \((\overset{1}{u},\overset{1}{q})\) of problem (1)—(5), and at least one of the following relations holds:

\[ \max_{K_{\widetilde T-\varepsilon}^{1}}|\overset{1}{u}|+ \sum_{i=1}^{n}\max_{K_{\widetilde T-\varepsilon}^{1}}|\overset{1}{q}_{i}|\to\infty \quad \text{as } \varepsilon\to 0, \]

\[ \inf_{T^{*}\le t<\widetilde T} H'_{u}(t,\overset{1}{u}(0,t),\overset{1}{q}(0,t))=0. \]

Set \(T^{0}=T_{2}\) in case A), and \(T^{0}=T^{*}+\theta(\widetilde T-T^{*})\) in case B) (\(0<\theta<1\) is an arbitrary fixed number).

Definition 3. The vector-function \((u^{p},q^{p})=(\mathring u,\mathring q)\) in \(\mathscr D_{T^{0}}\), \((u^{p},q^{p})=(\overset{1}{u},\overset{1}{q})\) in \(K_{T^{0}}^{1}\) is called an r.d. of problem (1)—(5) in \(\overline{\Pi}_{T^{0}}\).

4. Stability of the r.d. under small perturbations of the function \(H\). In \(\overline{\Pi}_{T^{0}}\) consider the mixed problem \((1')\)—\((5')\), determined by equations (1)—(4) and the boundary condition

\[ \mu L(u(0,t),q(0,t))+H(t,u(0,t),q(0,t))=0,\quad \mu\ne 0, \tag{5'} \]

We note that \((\mathring u,\mathring q)\) is a solution of problem \((1')\)—\((5')\) in \(K_{T^{0}}^{0}\) for any \(\mu\).

Definition 4. A vector-function \((u,q)\in C_{1}(G_{T^{0}}^{0})\) such that

\[ u(0,0)=u_{0}(0),\qquad q(x(t,0,0),t)=q(x(t,0,0),t) \quad \text{for } 0\le t\le T^{0} \]

is called a solution of problem \((1')\)—\((5')\) in \(G_{T^{0}}^{0}\), if it satisfies equations (1), (2) in \(G_{T^{0}}^{0}\) and equation \((5')\) for \(0\le t\le T^{0}\).

\[ \underline{\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}} \]

* The inequalities (10) do not restrict generality.

Set \(\varepsilon=\mu(\gamma_1-\lambda^{-1}\gamma_2)\). Let \(\varepsilon>0\) when \(\mu\ne0\).

Theorem 6. One can indicate an \(\varepsilon^*>0\) such that, for \(0<\varepsilon<\varepsilon^*\), in \(G_{T^0}\) there exists a unique solution \((u(\mu,x,t),q(\mu,x,t))\) of problem \((1')\)—\((5')\), and

\[ \|u(\mu,\cdot)\|_{C(G_{T^0})}<C<\infty,\qquad \|q_i(\mu,\cdot)\|_{C_1(G_{T^0})}<C<\infty, \]

where \(C\) is a constant independent of \(\mu\).

Theorem 7. Whatever closed domain \(\Omega\subset G_{T^0}\), having no common points with the characteristic \(t=x/\lambda+T^*\), may be,

\[ \|u(\mu,\cdot)-u^p(\cdot)\|_{C(\Omega)} +\|q_i(\mu,\cdot)-q_i^p(\cdot)\|_{C_1(\Omega)} \to0\quad \text{as }\mu\to0 . \]

If \(\Omega\) has no common points with the characteristic \(t=x/\lambda\), then

\[ \|u(\mu,\cdot)-u^p(\cdot)\|_{C_1(\Omega)}\to0\quad \text{as }\mu\to0 . \]

5°. Stability of d.s. under small perturbations of the initial conditions. In \(\overline{\Pi}_{T^0}\) consider problem \((1)\)—\((5)\) with initial conditions \(u(x,0)=u_0(x)\), \(q_i(x,0)=q_{i,0}(x)\), where the functions \(u_0,q_{i,0}\in C_2[0,1]\) satisfy the compatibility conditions necessary for the existence in \(\overline{\Pi}_{T^0}\) of a solution of problem \((1)\)—\((5)\) of class \(C_1\). Denote by

\[ \Delta=\|u_0-\overset{\circ}{u}_0\|_{C_1[0,1]} +\sum_{i=1}^n \|q_{i,0}-\overset{\circ}{q}_{i,0}\|_{C_1[0,1]} . \]

Theorem 8. For every \(\eta>0\) there exists a \(\delta(\eta)>0\) such that if \(\Delta<\delta(\eta)\), then one can indicate a \(\tau^*\), \(|\tau^*-T^*|<\eta\), such that in \(\Pi_{\tau^*}=\{0\le x\le1,\ 0\le t<\tau^*\}\) there exists a unique solution \((\overset{\circ}{u},\overset{\circ}{q})\in C_1(\Pi_{\tau^*})\) of problem \((1)\)—\((5)\), \(H_u'(t,\overset{\circ}{u}(0,t),\overset{\circ}{q}(0,t))>0\) for \(0\le t<\tau^*\), and the quantities \(s<\infty\), \(m=0\), \(I<\infty\) corresponding to the solution \((\overset{\circ}{u},\overset{\circ}{q})\).

By Theorem 2 there exist

\[ \overset{\vee}{u}{}^*=\lim_{t\to\tau^*}\overset{\circ}{u}(0,t),\qquad q^*=\lim_{t\to\tau^*}\overset{\circ}{q}(0,t). \]

Theorem 9. If \(\Delta\) is sufficiently small, then \(H_{uu}''(\tau^*,\overset{\vee}{u}{}^*,q^*)<0\).

Theorem 10. If \(\Delta\) is sufficiently small, then

1) the equation \(H(\tau^*,u,q^*)=0\) has a real root \(\bar u^*<\bar u^*<\infty\) such that \(H(\tau^*,\overset{\vee}{u}{}^*,q^*)\ne0\) for \(\overset{\vee}{u}{}^*<\overset{\vee}{u}<\bar u^*\), and \(H_u'(\tau^*,\bar u^*,q^*)>0\);

2) in \(\overline{\Pi}_{T^0}\) there exists a unique d.s. \((u^p,q^p)\) of problem \((1)\)—\((5)\)*.

Theorem 11. Whatever closed domain \(\Omega\subset\overline{\Pi}_{T^0}\), having no common points with the characteristic \(t=x/\lambda+T^*\), may be,

\[ \|u^p-\mathfrak{u}^p\|_{C_1(\Omega)} +\|q_i^p-\mathfrak{q}_i^p\|_{C(\overline{\Pi}_{T^0})} +\|q_i^p-\mathfrak{q}_i^p\|_{C_1(\Omega)} \to0\quad \text{as }\Delta\to0 . \]

Scientific Research
Radiophysical Institute
Gorky

Received
26 III 1970

CITED LITERATURE

1. R. Courant, Partial Differential Equations, Moscow, 1964.
2. B. L. Rozhdestvenskii, N. N. Yanenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics, Moscow, 1968.
3. V. E. Abolina, A. D. Myshkis, Mat. sbornik, 50 (92), 4, 423 (1960).
4. A. D. Myshkis, Report at the Joint Soviet-American Symposium on Partial Differential Equations, Novosibirsk, 1963.
5. V. N. Gol’dberg, DAN, 182, No. 6, 1257 (1968).

* We do not dwell on the obvious definition of a d.s. of problem \((1)\)—\((5)\) with initial conditions \((u_0,q_0)\).