System of linear first order homogeneous PDEs

Definition

A system of first order linear homogeneous PDEs can be written in the general matrix form as

$$ A_0 \mathbf u_t + A_1 \mathbf u_x + A_2 \mathbf u_y + A_3 \mathbf u_z = 0, $$

where $x, y, z, t$ are independent variables, $\mathbf u = (u_1, \ldots, u_N)^T$ is an $N$-dimensional vector of unknown functions, and $A_i$ are $M \times N$ matrices whose entries depend on the independent variables $x,y,z,t$. Both $M$ and $N$ are positive integers satisfying $M \geq N$.

$\blacksquare$

It is a system of DEs of the form

$$ \sum_{i=1}^{m} \sum_{j=1}^{n} a_{i j k}\left(x_{1}, x_{2}, \ldots, x_{n}\right) \frac{\partial V_{i}}{\partial x_{j}}=0, \quad k=1,2, \ldots, r $$

where $x_j$ are the independent variables, $V_i$ the dependent ones and $a_{ijk}$ smooth functions. Suppose the $r$ equations are linearly independent. We say the system is overdetermined if $r>m$, determined if $r=m$ and underdetermined if $r

The simpler case when $m=1$ is important. When also $r=1$ we are in the situation of characteristics of first-order partial differential equation, that is, a homogeneous linear PDE.

We are going to show that when $m=1$ we can always reduce to the case $r=1$ (Jonasson_2007 page 20).

If $m=1$ we can consider $r$ vector fields $X_k$ such that the system is giving by

$$ \begin{aligned} X_{1}(V) &:=a_{11} \frac{\partial V}{\partial x_{1}}+a_{12} \frac{\partial V}{\partial x_{2}}+\cdots+a_{1 n} \frac{\partial V}{\partial x_{n}}=0 \\ X_{2}(V):=& a_{21} \frac{\partial V}{\partial x_{1}}+a_{22} \frac{\partial V}{\partial x_{2}}+\cdots+a_{2 n} \frac{\partial V}{\partial x_{n}}=0 \\ & \vdots \\ X_{r}(V):=& a_{r 1} \frac{\partial V}{\partial x_{1}}+a_{r 2} \frac{\partial V}{\partial x_{2}}+\cdots+a_{r n} \frac{\partial V}{\partial x_{n}}=0 \end{aligned} $$

blablabla... to be continued.... Jonasson_2007 page 20.

Dependence on parameters

I think that from here it can be shown that if the coefficients of the system depends smoothly on parameters, the solution must depends smoothly on these parameters. Because we can convert the system to a system of ODEs, and there we have continuous and differentiable dependency on parameters.

Existence of solution

Since this is a particular case of system of first order inhomogeneous linear PDEs, the existence is concluded from the result existence of solutions for a system of first order linear inhomogeneous PDEs.

Also

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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