Well-posedness for system of linear PDES

Question

I am studying systems of partial differential equations (PDEs) with smooth coefficients and am trying to understand the general solution. Consider the following system:

\begin{align} u_x &= a_{11}(x,y)u + a_{12}(x,y)v, \\ u_y &= a_{21}(x,y)u + a_{22}(x,y)v, \\ v_x &= a_{31}(x,y)u + a_{32}(x,y)v, \\ v_y &= a_{41}(x,y)u + a_{42}(x,y)v. \end{align}

Q: What types of initial and boundary conditions ensure that this system is well-posed? Are there any standard criteria or a general formula like in the case of linear ODEs?

One complication I encounter is ensuring the compatibility condition $u_{xy}=u_{yx}$ ​To address this issue, let's consider a simpler system that might bypass this complication given by:

• System A: \begin{align} u_x &= b_{11}(x,y)u + b_{12}(x,y)v, \\ v_x &= b_{21}(x,y)u + b_{22}(x,y)v. \end{align}
• System B: \begin{align} u_x &= c_{11}(x,y)u + c_{12}(x,y)v, \\ v_y &= c_{21}(x,y)u + c_{22}(x,y)v. \end{align}

For these simpler systems, what are the criteria for existence of the solution? How does a general solution to these systems look like?

Any help is appreciated. Thanks in advance!

Answer 1

System A

\begin{align} u_x &= b_{11}(x,y)u + b_{12}(x,y)v, \\ v_x &= b_{21}(x,y)u + b_{22}(x,y)v. \end{align}

This system involves derivatives with respect to (x) only. It can be treated as a system of ordinary differential equations (ODEs) in $x$ with $y$ as a parameter.

Initial Conditions: To ensure the well-posedness of this system, we require initial conditions at $x = x_0$: $$ u(x_0, y) = f(y), \quad v(x_0, y) = g(y). $$

The Solution: The system can be written in matrix form as:

$$\frac{d}{dt} X = A X,$$

where $$X=\begin{pmatrix} u \\ v \end{pmatrix} \text{ and } A= \begin{pmatrix} b_{11}(x,y) & b_{12}(x,y) \\ b_{21}(x,y) & b_{22}(x,y) \end{pmatrix},$$ namely

$$ \frac{d}{dx}\begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} b_{11}(x,y) & b_{12}(x,y) \\ b_{21}(x,y) & b_{22}(x,y) \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix}. $$

As a general explination, let's assume that the matrix $A= A(x,y)$ is diagonalizable, i.e. $A = P^{-1} B P$, then we have

\begin{align} \frac{d}{dt} X = AX = P^{-1} D P \Rightarrow X= C e^{P^{-1}D P}= C P^{-1}e^{D}P \end{align}

where $C$ is a constant which can be determine by the initial values. Therefore, in this case the system is well-posed in a domain when the matrix $A$ is diagonalizable. You can find a characterization for all the cases too, i.e when the matrix $A$ has a generalized eigenvalues, or when the solution behave like a center manifold.

By standard theory for systems of linear ODEs, if the coefficients $b_{ij}(x,y)$, $i,j=1,2$ are continuous in $x$ and $y$, the existence and uniqueness of the solution are guaranteed by the Picard-Lindelöf theorem using a sutable norm space.

System B

\begin{align} u_x &= c_{11}(x,y)u + c_{12}(x,y)v, \\ v_y &= c_{21}(x,y)u + c_{22}(x,y)v. \end{align}

As you may can see that this system can written as:

\begin{align} \nabla \begin{pmatrix} u\\v\end{pmatrix} = \begin{pmatrix} c_{11}+c_{21}\\c_{12}+c_{22}\end{pmatrix} \begin{pmatrix} u\\v\end{pmatrix} \end{align}

The well-psedness conditions of this system can be determined by Lax-Milgram theorem.