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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c0208201.png" />-dimensional surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c0208202.png" />, oriented in a certain way, may serve as the initial surface in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c0208203.png" /> of independent variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c0208204.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c0208205.png" /> is a space-like surface then the [[Cauchy problem|Cauchy problem]] (with initial data on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c0208206.png" />) 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.
+
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 (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c0208207.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c0208208.png" />)
+
For example, the characteristic Cauchy problem for the equation ($n=1$, $x_1=x$)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c0208209.png" /></td> </tr></table>
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$$u_{xt}=0$$
  
with data on the characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082010.png" />:
+
with data on the characteristic $t=0$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082011.png" /></td> </tr></table>
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$$u(x,0)=\tau(x),\quad u_t(x,0)=\nu(x),$$
  
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: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082012.png" />, i.e. the characteristic Cauchy problem may be solvable only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082013.png" />. In that case, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082015.png" />, a solution indeed exists and is given by
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082016.png" /></td> </tr></table>
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$$u(x,t)=\tau(x)+\alpha t+\rho(t),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082017.png" /> is any function of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082019.png" />, satisfying the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082020.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082021.png" />.
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\begin{equation}\sum_{i=1}^nu_{x_ix_i}-u_{tt}=0,\label{1}\end{equation}
  
when the characteristic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082023.png" /> is the cone
+
when the characteristic surface $S$ is the cone
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082025.png" /> of equation (1) which is regular within the cone (2) and takes prescribed values on the cone (2).
+
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 (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082027.png" />), the cone (2) is a pair of straight lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082028.png" /> passing through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082029.png" />. These straight lines divide the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082030.png" /> of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082031.png" /> into four angles. Let the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082032.png" /> be one of these angles. Then the characteristic problem is customarily known as the Goursat problem: Determine a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082033.png" /> of the equation
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082034.png" /></td> </tr></table>
+
$$u_{xx}-u_{tt}=0$$
  
which is regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082035.png" /> and satisfies the conditions
+
which is regular in $\Omega$ and satisfies the conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082036.png" /></td> </tr></table>
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$$u=\phi\quad\text{if }x-x_0=t-t_0,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082037.png" /></td> </tr></table>
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$$u=\psi\quad\text{if }x-x_0=t_0-t,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082038.png" /></td> </tr></table>
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$$\phi(x_0,t_0)=\psi(x_0,t_0).$$
  
If the characteristic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082039.png" /> is at the same time a surface of degenerate type or order, the characteristic Cauchy problem may prove to be well-posed.
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
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\begin{equation}y^mu_{yy}-u_{xx}+au_x+bu_y+cu=f,\label{3}\end{equation}
  
which is hyperbolic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082041.png" />, the curve of degeneracy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082042.png" /> is a characteristic. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082043.png" />, the Cauchy problem
+
which is hyperbolic for $y>0$, the curve of degeneracy $y=0$ is a characteristic. If $0<m<1$, the Cauchy problem
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$u(x,0)=\tau(x),\quad u_y(x,0)=\nu(x)\label{4}$$
  
for equation (3) is well-posed, but if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082045.png" /> it becomes ill-posed. In that case it is natural to investigate the problem either with modified initial data:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082046.png" /></td> </tr></table>
+
$$\lim_{y\to0}\alpha(x,y)u(x,y)=\tau(x),$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082047.png" /></td> </tr></table>
+
$$\lim_{y\to0}\beta(x,y)u_y(x,y)=\nu(x),$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020820/c02082048.png" /></td> </tr></table>
+
$$\lim_{y\to0}\alpha(x,y)=0,\quad\lim_{y\to0}\beta(x,y)=0,$$
  
or with incomplete initial data, i.e. dropping one of the conditions (4).
+
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)
How to Cite This Entry:
Cauchy characteristic problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_characteristic_problem&oldid=43536
This article was adapted from an original article by V.A. Eleev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article