Ngô Quốc Anh

March 14, 2010

Classification of a system of n first-order PDEs

Filed under: Nghiên Cứu Khoa Học, PDEs — Ngô Quốc Anh @ 11:18

The classification of a system of n first-order PDEs is based on whether there are n directions along which the PDEs reduce to n ODEs. To be more precise, assume that we are given a system of n equations in n unknowns u_1, u_2,...,u_n which we write in matrix form as

\displaystyle \mathbf{u}_t + A(x,t,\mathbf{u})\mathbf{u}_x = \mathbf{b}(x,t,\mathbf{u}),

where \mathbf{u}=(u_1,...,u_n)^t, \mathbf{b}=(b_1,...,b_n)^t, and A=(a_{ij}(x,t,\mathbf{u})) is an n \times n matrix.

Now we ask whether there is a family of curves along which the PDEs reduce to a system of ODEs, that is, in which the directional derivative of each u_i occurs in the same direction. We consider a row vector \gamma = (\gamma_1,...,\gamma_n)^t to be determined later. Then

\displaystyle \mathbf{\gamma}^t\mathbf{u}_t + \mathbf{\gamma}^tA(x,t,\mathbf{u})\mathbf{u}_x =  \mathbf{\gamma}^t\mathbf{b}(x,t,\mathbf{u}).

We want the above system to have the form of a linear combination of total derivatives of the u_i in the same direction \lambda, that is, we want our system to have the form

\displaystyle \mathbf{m}^t \left( {{{\mathbf{u}}_t} + \lambda {{\mathbf{u}}_x}} \right) = \mathbf{\gamma}^t{\mathbf{b}}

for some \mathbf{m}. Consequently, we require

\displaystyle \mathbf{m}=\gamma, \quad \mathbf{m}^t\lambda=\gamma^tA

or

\displaystyle \gamma^t A=\lambda \gamma^t.

This means that \lambda is an eigenvalue of A and \gamma^t is a corresponding left eigenvector. Note that \lambda as well as \gamma can depend on x, t, and \mathbf{u}. So, if (\lambda, \gamma^t) is an eigenpair, then

\displaystyle \gamma^t \frac{d\mathbf{u}}{dt}=\gamma^t\mathbf{b}

along

\displaystyle \frac{dx}{dt}=\lambda(x,t,\mathbf{u})

and the system of PDEs is reduced to a single ODE along the family of curves, called characteristics, defined by \frac{dx}{dt}=\lambda. The eigenvalue \lambda is called the characteristics direction. Clearly, because there are n unknowns, it would appear that n ODEs are required; but if A has n distinct real eigenvalues, there are n ODEs, each holding along a characteristics direction defined by an eigenvalue. In this case we say that the system is hyperbolic.

Definition. The quasilinear system

\displaystyle \mathbf{u}_t + A(x,t,\mathbf{u})\mathbf{u}_x =  \mathbf{b}(x,t,\mathbf{u})

is hyperbolic if A has n real eigenvalues and n linearly independent left eigenvectors. Once these eigenvectors are distinct, the system is called stricly hyperbolic.

The system is called elliptic if A has no real eigenvalues, and it is parabolic if A has n real eigenvalues but fewer then n independent left eigenvectors.

No exhaustive classification is made in the case that A has both real and complex eigenvalues. Note that once matrix A has n distinct, real eigenvalues it has n independent left eigenvectors, because distinct eigenvalues have independent eigenvectors.

More general systems of the form

\displaystyle B(x,t,\mathbf{u})\mathbf{u}_t + A(x,t,\mathbf{u})\mathbf{u}_x =   \mathbf{b}(x,t,\mathbf{u})

can be considered as well. We refer the reader to a book entitled “An introduction to nonlinear partial differential equations” due to J.D. Logan.

We are now in a position to see why a single first-order quasilinear PDE is hyperbolic. The coefficient matrix for the equation

\displaystyle u_t + c(x,t,u)u_x=b(x,t,u)

is just the real scalar function c(x,t,u) which has the single eigenvalue c(x,t,u) and its corresponding eigenvector 1, a constant function. In this direction, once \frac{dx}{dt}=c(x,t,u) the PDE reduces to the ODE \frac{du}{dt}=b(x,t,u). We refer the reader to the following topic, called characteristic curves, where we consider when the equation has constant coefficients and variable coefficients.

We place here three more examples.

Example 1 (The shallow-water equations). The following system

\displaystyle\begin{gathered} {h_t} + u{h_x} + h{u_x} = 0, \hfill \\ {u_t} + u{u_x} + g{h_x} = 0, \hfill \\ \end{gathered}

is trictly hyperbolic.

Example 2. The following system

\displaystyle\begin{gathered} {u_t} - {v_x} = 0, \hfill \\ {v_t} - c{u_x} = 0, \hfill \\ \end{gathered}

is elliptic if c<0 and is hyperbolic if c>0.

Example 3 (The diffusion equations). The following equation

\displaystyle u_t=u_{xx}

may be written as the first-order system

\displaystyle\begin{gathered}u_t-v_x=0, \hfill \\u_x-v = 0, \hfill \\ \end{gathered}

and thus is parabolic.

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