Reports of the Academy of Sciences of the USSR
1963, Volume 149, No. 3

MATHEMATICS

A. Ya. LEPIN

APPLICATION OF THE METHOD OF GRIDS TO A HYPERBOLIC SYSTEM OF QUASILINEAR EQUATIONS IN THE PLANE

(Presented by Academician S. L. Sobolev on 13 VIII 1962)

In the strip ((0 \leq t \leq t^0,\ 0 \leq x \leq x^0,\ -\infty < u_1,\ldots,u_n < \infty)) let us consider the system of equations

[
\frac{\partial u_1(t,x)}{\partial t}
=
\lambda_1(t,x,u)\frac{\partial u_1(t,x)}{\partial x}
+
f_1(t,x,u),
]
[
\frac{\partial u_2(t,x)}{\partial t}
=
\lambda_2(t,x,u)\frac{\partial u_2(t,x)}{\partial x}
+
f_2(t,x,u),
\tag{1}
]

where (u_1, u_2, u=\left|\begin{matrix}u_1\ u_2\end{matrix}\right|), (f_1, f_2) are column matrices; (\lambda_1, \lambda_2) are diagonal matrices, with the diagonal elements of (\lambda_1) nonnegative, and positive for (x=x^0), while the diagonal elements of (\lambda_2) are nonpositive, and negative for (x=0), with initial conditions

[
u(0,x)=\varphi(x)
\tag{2}
]

and boundary conditions

[
u_1(t,x^0)=\alpha_1(t,u_2(t,x^0))+\int_0^t \beta_1(t,\tau,u(\tau,x^0))\,d\tau,
]
[
u_2(t,0)=\alpha_2(t,u_1(t,0))+\int_0^t \beta_2(t,\tau,u(\tau,0))\,d\tau.
\tag{3}
]

Analogous systems of linear equations were studied in papers ((^{1-3})). In studying classical and generalized solutions we shall need compatibility conditions:

[
\varphi_1(x^0)=\alpha_1(0,\varphi_2(x^0)),
]
[
\varphi_2(0)=\alpha_2(0,\varphi_1(0));
\tag{4}
]

[
\lambda_1(0,x^0,\varphi(x^0))\varphi_1'(x^0)+f_1(0,x^0,\varphi(x^0))
=
\left.\frac{\partial \alpha_1}{\partial t}\right|_{t=0,\ u_2=\varphi_2(x^0)}
+
\beta_1(0,0,\varphi(x^0)),
\tag{5}
]

[
\lambda_2(0,0,\varphi(0))\varphi_2'(0)+f_2(0,0,\varphi(0))
=
\left.\frac{\partial \alpha_2}{\partial t}\right|_{t=0,\ u_1=\varphi_1(0)}
+
\beta_2(0,0,\varphi(0)).
]

To replace the corresponding boundary-value problem by a difference one, we construct a grid with step (h) in (x) and (k) in (t). Then the difference system is written as follows:

[
u_1^{i+1\,j}
=
\varkappa \lambda_1^{ij}u_1^{i\,j+1}
+
(\varepsilon_1-\varkappa\lambda_1^{ij})u_1^{ij}
+
k f_1^{ij},
]
[
u_2^{i+1\,j}
=
-\varkappa \lambda_2^{ij}u_2^{i\,j+1}
+
(\varepsilon_2+\varkappa\lambda_2^{ij})u_2^{ij}
+
k f_2^{ij};
\tag{6}
]
[
u_1^{i+1\,\nu}
=
\alpha_1^{i+1}
+
k\sum_{p=0}^{i}\beta_1^{i+1\,p},
\qquad
u_2^{i+1\,0}
=
\alpha_2^{i+1}
+
k\sum_{p=0}^{i}\beta_2^{i+1\,p},
]

where (\varkappa=kh^{-1}); (\varepsilon_1, \varepsilon_2) are unit matrices.

For a matrix (A=|a_{pq}|) the norm (|A|) is equal to (\max_q \sum_{p=1}^{p_0}|a_{pq}|). The matrix

(A(t,x,u)) satisfies the Lipschitz condition ((A\in \mathrm{Lip})), if for every (a>0) there is a (K) such that

[
|A(t,x,u)-A(\bar t,\bar x,\bar u)|\leq
K\bigl(|t-\bar t|+|x-\bar x|+|u-\bar u|\bigr)
]

for (|u|,|\bar u|\leq a). If in the rectangle (\Pi^=\Pi_{t^}(0\leq t\leq t^*,\,0\leq x\leq x^0)) the integral relation

[
\iint_{\Pi^}\left[\left(\frac{\partial v}{\partial t}-\frac{\partial v\lambda}{\partial x}\right)u+vf\right]d\Pi
=
\int_{0}^{x^0}vu\Big|_{0}^{t^}\,dx
-
\int_{0}^{t^*}v\lambda u\Big|_{0}^{x^0}\,dt,
]

holds for (u,\lambda,f\in\mathrm{Lip}), where (v) is any continuously differentiable row matrix, then we shall say that (u) is a generalized solution of system (1).

Theorem 1. If (\lambda,f,\varphi,\alpha,\beta) have first partial derivatives satisfying the Lipschitz condition, and the compatibility conditions (4)—(5) are fulfilled, then there exists a (t^>0) such that in (\Pi^) there exists a classical solution of the boundary-value problem (1)—(3), and moreover (du/dt,\,du/dx\in\mathrm{Lip}).

Theorem 2. If (\lambda,f,\varphi,\alpha,\beta\in\mathrm{Lip}) and the compatibility conditions (4) are fulfilled, then there exists a (t^>0) such that in (\Pi^) there exists a generalized solution of the boundary-value problem (1)—(3), and moreover (u\in\mathrm{Lip}).

The proofs of these theorems are based on the following lemmas.

Lemma 1. If (\lambda,f,\varphi,\alpha,\beta\in\mathrm{Lip}) and the compatibility conditions (4) are fulfilled, then there exist (U_1,U_2) and (t^2>0) such that, if for the mesh the relations (\varkappa|\lambda^{ij}|\leq 1,\;0\leq a\leq \varkappa|\lambda_1^i|^{-1},\;a\leq \varkappa|\lambda_2^i|) are fulfilled in (\Pi_{t^1}), then (|u^{ij}|\leq U_1,\;|\Delta_1u^{ij}|\leq U_2,\;|\Delta_2u^{ij}|\leq U_2) in (\Pi_{t^}), where (t^=\min{t^1,t^2}), (\Delta_1u^{ij}=(u^{i+1,j}-u^{ij})k^{-1}), (\Delta_2u^{ij}=(u^{i,j+1}-u^{ij})h^{-1}).

Lemma 2. If (\lambda,f,\varphi,\alpha,\beta) have first partial derivatives satisfying the Lipschitz condition, and the compatibility conditions (4)—(5) are fulfilled, then there exist (U_3) and (t^3>0) such that, if for the mesh the relations (\varkappa|\lambda^{ij}|\leq 1,\;0