Difference between revisions of "Cauchy characteristic problem"
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The problem of finding a solution of a partial differential equation (or a system of partial differential equations) which assumes prescribed values on a characteristic manifold. | The problem of finding a solution of a partial differential equation (or a system of partial differential equations) which assumes prescribed values on a characteristic manifold. | ||
− | There is a large class of equations of hyperbolic and parabolic types for which a non-closed | + | There is a large class of equations of hyperbolic and parabolic types for which a non-closed $n$-dimensional surface $S$, oriented in a certain way, may serve as the initial surface in the space $E_{n+1}$ of independent variables $x_1,\dots,x_n,t$. For example, if $S$ is a space-like surface then the [[Cauchy problem|Cauchy problem]] (with initial data on $S$) is always well-posed. In a characteristic Cauchy problem the initial surface is always a characteristic manifold (or a well-defined part thereof). In this case the Cauchy problem may have no solution at all; and if it has a solution it need not be unique. |
− | For example, the characteristic Cauchy problem for the equation ( | + | For example, the characteristic Cauchy problem for the equation ($n=1$, $x_1=x$) |
− | + | $$u_{xt}=0$$ | |
− | with data on the characteristic | + | with data on the characteristic $t=0$: |
− | + | $$u(x,0)=\tau(x),\quad u_t(x,0)=\nu(x),$$ | |
− | is not well-posed. If the characteristic Cauchy problem has a solution, then the equation and the second initial condition imply a necessary condition for its solvability: | + | is not well-posed. If the characteristic Cauchy problem has a solution, then the equation and the second initial condition imply a necessary condition for its solvability: $\nu'(x)=0$, i.e. the characteristic Cauchy problem may be solvable only if $\nu(x)=\mathrm{const}=\alpha$. In that case, if $\tau(x)\in C^2$, $t\geq0$, a solution indeed exists and is given by |
− | + | $$u(x,t)=\tau(x)+\alpha t+\rho(t),$$ | |
− | where | + | where $\rho(t)$ is any function of class $C^2$, $t\geq0$, satisfying the conditions $\rho(0)=\rho'(0)=0$. |
− | A necessary condition for the existence of a solution to a characteristic Cauchy problem for a linear system of hyperbolic equations is that the rank of the augmented matrix of the system equals the rank of the singular matrix along the characteristic surface | + | A necessary condition for the existence of a solution to a characteristic Cauchy problem for a linear system of hyperbolic equations is that the rank of the augmented matrix of the system equals the rank of the singular matrix along the characteristic surface $S$. |
There is a wide class of hyperbolic equations and systems for which one may take a characteristic surface as an initial surface. For example, for the equation | There is a wide class of hyperbolic equations and systems for which one may take a characteristic surface as an initial surface. For example, for the equation | ||
− | + | \begin{equation}\sum_{i=1}^nu_{x_ix_i}-u_{tt}=0,\label{1}\end{equation} | |
− | when the characteristic surface | + | when the characteristic surface $S$ is the cone |
− | + | \begin{equation}\sum_{i=1}^n(x_i-x_i^0)^2-(t-t_0)^2=0,\label{2}\end{equation} | |
− | the characteristic Cauchy problem may be phrased as follows: Find a solution | + | the characteristic Cauchy problem may be phrased as follows: Find a solution $u(x,t)$ of equation \eqref{1} which is regular within the cone \eqref{2} and takes prescribed values on the cone \eqref{2}. |
− | In the case of a space-like variable ( | + | In the case of a space-like variable ($n=1$, $x_1=x$), the cone \eqref{2} is a pair of straight lines $(x-x_0)^2=(t-t_0)^2$ passing through the point $(x_0,t_0)$. These straight lines divide the plane $E_2$ of the variables $x,t$ into four angles. Let the domain $\Omega$ be one of these angles. Then the characteristic problem is customarily known as the Goursat problem: Determine a solution $u(x,t)$ of the equation |
− | + | $$u_{xx}-u_{tt}=0$$ | |
− | which is regular in | + | which is regular in $\Omega$ and satisfies the conditions |
− | + | $$u=\phi\quad\text{if }x-x_0=t-t_0,$$ | |
− | + | $$u=\psi\quad\text{if }x-x_0=t_0-t,$$ | |
− | + | $$\phi(x_0,t_0)=\psi(x_0,t_0).$$ | |
− | If the characteristic surface | + | If the characteristic surface $S$ is at the same time a surface of degenerate type or order, the characteristic Cauchy problem may prove to be well-posed. |
For the equation | For the equation | ||
− | + | \begin{equation}y^mu_{yy}-u_{xx}+au_x+bu_y+cu=f,\label{3}\end{equation} | |
− | which is hyperbolic for | + | which is hyperbolic for $y>0$, the curve of degeneracy $y=0$ is a characteristic. If $0<m<1$, the Cauchy problem |
− | + | $$u(x,0)=\tau(x),\quad u_y(x,0)=\nu(x)\label{4}$$ | |
− | for equation | + | for equation \eqref{3} is well-posed, but if $m\geq1$ it becomes ill-posed. In that case it is natural to investigate the problem either with modified initial data: |
− | + | $$\lim_{y\to0}\alpha(x,y)u(x,y)=\tau(x),$$ | |
− | + | $$\lim_{y\to0}\beta(x,y)u_y(x,y)=\nu(x),$$ | |
− | + | $$\lim_{y\to0}\alpha(x,y)=0,\quad\lim_{y\to0}\beta(x,y)=0,$$ | |
− | or with incomplete initial data, i.e. dropping one of the conditions | + | or with incomplete initial data, i.e. dropping one of the conditions \eqref{4}. |
====References==== | ====References==== |
Revision as of 20:37, 22 December 2018
The problem of finding a solution of a partial differential equation (or a system of partial differential equations) which assumes prescribed values on a characteristic manifold.
There is a large class of equations of hyperbolic and parabolic types for which a non-closed $n$-dimensional surface $S$, oriented in a certain way, may serve as the initial surface in the space $E_{n+1}$ of independent variables $x_1,\dots,x_n,t$. For example, if $S$ is a space-like surface then the Cauchy problem (with initial data on $S$) is always well-posed. In a characteristic Cauchy problem the initial surface is always a characteristic manifold (or a well-defined part thereof). In this case the Cauchy problem may have no solution at all; and if it has a solution it need not be unique.
For example, the characteristic Cauchy problem for the equation ($n=1$, $x_1=x$)
$$u_{xt}=0$$
with data on the characteristic $t=0$:
$$u(x,0)=\tau(x),\quad u_t(x,0)=\nu(x),$$
is not well-posed. If the characteristic Cauchy problem has a solution, then the equation and the second initial condition imply a necessary condition for its solvability: $\nu'(x)=0$, i.e. the characteristic Cauchy problem may be solvable only if $\nu(x)=\mathrm{const}=\alpha$. In that case, if $\tau(x)\in C^2$, $t\geq0$, a solution indeed exists and is given by
$$u(x,t)=\tau(x)+\alpha t+\rho(t),$$
where $\rho(t)$ is any function of class $C^2$, $t\geq0$, satisfying the conditions $\rho(0)=\rho'(0)=0$.
A necessary condition for the existence of a solution to a characteristic Cauchy problem for a linear system of hyperbolic equations is that the rank of the augmented matrix of the system equals the rank of the singular matrix along the characteristic surface $S$.
There is a wide class of hyperbolic equations and systems for which one may take a characteristic surface as an initial surface. For example, for the equation
\begin{equation}\sum_{i=1}^nu_{x_ix_i}-u_{tt}=0,\label{1}\end{equation}
when the characteristic surface $S$ is the cone
\begin{equation}\sum_{i=1}^n(x_i-x_i^0)^2-(t-t_0)^2=0,\label{2}\end{equation}
the characteristic Cauchy problem may be phrased as follows: Find a solution $u(x,t)$ of equation \eqref{1} which is regular within the cone \eqref{2} and takes prescribed values on the cone \eqref{2}.
In the case of a space-like variable ($n=1$, $x_1=x$), the cone \eqref{2} is a pair of straight lines $(x-x_0)^2=(t-t_0)^2$ passing through the point $(x_0,t_0)$. These straight lines divide the plane $E_2$ of the variables $x,t$ into four angles. Let the domain $\Omega$ be one of these angles. Then the characteristic problem is customarily known as the Goursat problem: Determine a solution $u(x,t)$ of the equation
$$u_{xx}-u_{tt}=0$$
which is regular in $\Omega$ and satisfies the conditions
$$u=\phi\quad\text{if }x-x_0=t-t_0,$$
$$u=\psi\quad\text{if }x-x_0=t_0-t,$$
$$\phi(x_0,t_0)=\psi(x_0,t_0).$$
If the characteristic surface $S$ is at the same time a surface of degenerate type or order, the characteristic Cauchy problem may prove to be well-posed.
For the equation
\begin{equation}y^mu_{yy}-u_{xx}+au_x+bu_y+cu=f,\label{3}\end{equation}
which is hyperbolic for $y>0$, the curve of degeneracy $y=0$ is a characteristic. If $0<m<1$, the Cauchy problem
$$u(x,0)=\tau(x),\quad u_y(x,0)=\nu(x)\label{4}$$
for equation \eqref{3} is well-posed, but if $m\geq1$ it becomes ill-posed. In that case it is natural to investigate the problem either with modified initial data:
$$\lim_{y\to0}\alpha(x,y)u(x,y)=\tau(x),$$
$$\lim_{y\to0}\beta(x,y)u_y(x,y)=\nu(x),$$
$$\lim_{y\to0}\alpha(x,y)=0,\quad\lim_{y\to0}\beta(x,y)=0,$$
or with incomplete initial data, i.e. dropping one of the conditions \eqref{4}.
References
[1] | V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) |
[2] | S.K. Godunov, "The equations of mathematical physics" , Moscow (1971) (In Russian) |
[3] | F.G. Tricomi, "Equazioni a derivate parziale" , Cremonese (1957) |
[4] | L. Bers, F. John, M. Schechter, "Partial differential equations" , Interscience (1964) |
[5] | A.V. Bitsadze, D.F. Kalinichenko, "A collection of problems on the equations of mathematical physics" , Moscow (1977) (In Russian) |
[6] | A.V. Bitsadze, "Linear partial differential equations of mixed type" , Proc. 3-rd All-Union Math. Congress , 3 , Moscow (1958) (In Russian) |
Comments
The more common English term for this problem is characteristic Cauchy problem. A general discussion is given in [a1], Sect. 12.8.
References
[a1] | L.V. Hörmander, "The analysis of linear partial differential operators" , 2 , Springer (1983) |
[a2] | A.V. Bitsadse, "Equations of mixed type" , Pergamon (1964) (Translated from Russian) |
[a3] | R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) |
Cauchy characteristic problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_characteristic_problem&oldid=43536