Section 2.10 Partial differential equations
A partial differential equation (PDE) is an equation involving an unknown function of two or more variables and some of its partial derivatives. More precisely:
Definition 2.52.
A \(k\)-th order PDE is an equation of the form
\begin{equation}
F(D^ku(x), D^{k-1}u(x),\ldots,Du(x),u(x),x)=0 \tag{2.8}
\end{equation}
for \(x \in \Omega\text{,}\) where here \(F\) is a given real-valued function and \(u\colon \Omega \to \R\) is the unknown.
\begin{gather*}
F(D^ku, D^{k-1}u,\ldots,Du,u,x)=0.
\end{gather*}
As with algebraic equations and ordinary differential equations (ODEs), we can talk about the PDE (2.8) being linear or nonlinear:
Definition 2.53.
The PDE (2.8) is called linear if it has the form
\begin{equation}
\sum_{|\alpha| \le k}a_{\alpha}(x)\partial^{\alpha}u(x)=f(x) \tag{2.9}
\end{equation}
for given functions \(a_{\alpha}\text{,}\) \(f\text{,}\) and where the summation ranges over all multiindices \(\alpha\) with order \(\abs \alpha \le k\text{.}\) The linear PDE (2.9) is homogeneous if \(f \equiv 0\text{.}\) The PDE (2.8) is semilinear if it has the form:
\begin{gather*}
\sum_{|\alpha| = k}a_{\alpha}(x)\partial^{\alpha}u(x) +
a_0(D^{k-1}u(x),\dots,Du(x),u(x),x)=0.
\end{gather*}
Roughly speaking, (2.8) is semilinear if looks like a linear PDE involving \(D^k u\) plus nonlinear terms which do not involve \(D^k
u\text{.}\)
Example 2.54.
- The wave equation is the second-order (homogeneous) linear PDE\begin{equation*} u_{tt} - \Delta u = 0\text{.} \end{equation*}
- The sine-Gordon equation is the second-order semilinear PDE\begin{equation*} \Delta u - \sin u = 0\text{.} \end{equation*}
- The (inviscid) Burgers’ equation is the first-order nonlinear PDE\begin{equation*} u_t + uu_x = 0\text{.} \end{equation*}It is not semilinear.
- The mean curvature equation is the nonlinear second-order PDE\begin{equation*} (1+u_y^2) u_{xx} - 2 u_x u_y u_{xy} +(1+u_x^2) u_{yy} = 0\text{.} \end{equation*}It is not semilinear.
Notation 2.55. Differential operator.
It is often useful to think of the left hand side of (2.9) as a \(k\)-th order linear differential operator
\begin{equation*}
L = \sum_{|\alpha| \le k}a_{\alpha}(x)\partial^{\alpha}
\end{equation*}
being applied to the function \(u\text{.}\) If the coefficients \(a_\alpha\) are continuous on \(\Omega\text{,}\) then indeed \(L\) defines a linear mapping \(C^k(\Omega) \to C^0(\Omega)\text{.}\) Similarly if the coefficients are continuous on \(\overline \Omega\text{,}\) then \(L\) defines a mapping \(C^k(\overline \Omega) \to
C^0(\overline \Omega)\text{.}\)
1
When \(\Omega\) is bounded and \(C^0(\overline\Omega)\) and \(C^k(\overline\Omega)\) are equipped with the norm from Theorem 2.20, this mapping is not just well-defined but (Lipschitz) continuous.
As in Notation 2.14, our general convention is that the operator \(L\) is applied to a named function \(u\) before that function is evaluated on its arguments. So, for instance, \(Lu(2x)\) denotes the function \(Lu\) obtained by applying \(L\) to \(u\text{,}\) evaluated at the point \(2x\text{.}\)
Definition 2.56.
A \(k\)-th order system of PDEs is an equation
\begin{equation*}
F(D^ku(x), D^{k-1}u(x),\dots,Du(x),u(x),x)=0
\end{equation*}
for \(x \in \Omega\) where now \(F=(F_1,\ldots,F_m)\) is a given function taking values in \(\R^m\) and \(u\maps \Omega \to \R^m\text{,}\) \(u = (u_1, \ldots, u_m)\) is the unknown.
Example 2.57.
- The Cauchy–Riemann equations from complex analysis can be written as\begin{align*} \partial_1 u_1 - \partial_2 u_2 \amp = 0,\\ \partial_1 u_2 + \partial_2 u_1 \amp = 0\text{.} \end{align*}This is a system of two first-order (homogeneous) linear partial differential equations.
- The Navier–Stokes equations are a nonlinear system of PDEs for a velocity field \(u \maps \R^3 \to \R^3\) and a pressure field \(p \maps \R^3 \to \R\text{.}\) They can be written\begin{align*} \partial_t u_i + u_j \partial_j u_i \amp = -\partial_i p + \nu \Delta u_i, \qquad i=1,2,3\\ \nabla \cdot u \amp = 0, \end{align*}where \(\nu \gt 0\) is a constant.
To ‘solve’ a PDE is to find all \(u\) satisfying (2.8), possibly among those functions satisfying certain boundary conditions on \(\partial \Omega\text{.}\) In this course we do not primarily aim at finding (more or less) explicit solutions, but instead at deriving properties (existence, uniqueness, regularity, estimates, symmetry etc.) of these solutions using rigorous mathematical analysis. There is no general theory for all types of PDE, and most work is concentrated directed towards equations with some physical or other external motivation. In this course we will consider two specific classes of second-order equations for which maximum principle techniques are especially useful:
Elliptic equations.
These generalise the Poisson and Laplace equation \(\Delta u = f\) for \(u\maps \Omega \to \R\text{,}\) complemented by boundary conditions for \(u\) on \(\partial\Omega\text{.}\) They are ubiquitous in electrostatics and continuum mechanics, as well as other areas of mathematics such as differential geometry. Elliptic equations also govern the steady (time independent) solutions of parabolic equations.
Parabolic equations.
These generalise the heat and diffusion equation \(\partial_tu - \Delta u = f\) for \(u=u(x,t)\text{,}\) complemented by boundary conditions on \(\partial \Omega\) and initial conditions for \(t=0\text{.}\) Such equations model temporal changes in chemical concentrations or heat. Semilinear versions with right hand side \(f=f(u(x,t),x,t)\) model reactions, and systems of such equations are of interest in chemistry and biology. Like elliptic equations, parabolic equations appear in other areas of mathematics with some frequency. For instance, they are connected to stochastic ODEs via the Feynmann–Kac formula.
Exercises Exercises
1. Compositions and commutators.
Consider first-order differential operators \(L = a_1 \partial_1 + a_2 \partial_2\) and \(S = b_1 \partial_1 + b_2 \partial_2\) acting on smooth functions \(u \in C^\infty(\R^2)\) and with smooth coefficients \(a_1,a_2,b_1,b_2 \in C^\infty(\R^2)\text{.}\)
(a)
Show that the composition \(LS\) is also a differential operator, in general a second order operator.
(b)
Show that the commutator \(LS-SL\) is a first order differential operator.
2. Linearisation.
The linearisation of a nonlinear PDE
\begin{equation*}
F(D^ku, D^{k-1}u,\ldots,Du,u,x)=0
\end{equation*}
about a solution \(u_0\) is the linear PDE
\begin{equation*}
\frac d{d\varepsilon} F(D^k(u_0+\varepsilon u), D^{k-1}(u_0+\varepsilon u),\ldots,D(u_0+\varepsilon u),u_0+\varepsilon u,x) \bigg|_{\varepsilon=0}
=0 \text{.}
\end{equation*}
Calculate the linearisations of the PDEs in Example 2.54 about \(u_0 \equiv 0\text{.}\)
