Section 2.1 General facts about PDEs
A general smooth Partial Differential Equation of order \(k\) on \(\bR^n\text{,}\) with coordinates \(x^1,\dots,x^n\text{,}\) has the form
\begin{equation*}
\Phi(x^\alpha,y^a,\partial_\beta y^a,\dots,\partial^k_{\beta_1\dots\beta_k} y^a)=0,
\end{equation*}
where the index \(a\) ranges from 1 to \(m\text{,}\) for some smooth function \(\Phi\text{.}\) A solution of this PDE is a map
\begin{equation*}
f:\bR^n\to\bR^m
\end{equation*}
such that
\begin{equation*}
\Phi(x^\alpha,f^a(x^\alpha),\partial_\beta f^a(x^\alpha),\dots,\partial^k_{\beta_1\dots\beta_k} f^a(x^\alpha))=0
\end{equation*}
for each point \(x = (x^1,\dots,x^n)\text{.}\)
That said, all PDEs we will meet in this course will involve scalar fields (i.e. functions) in either two or three variables. Below are a few examples of important elementary PDEs in two and three variables: Poisson Equation.
\begin{equation*}
u_{xx}+u_{yy}=\rho
\end{equation*}
Often this equation is written in its more compact form \(\Delta u=\rho\text{.}\) In Newtonian gravitation (resp. electrostatic) theory, this is the equation satisfied by the potential of a given distribution of masses (resp. electric charges) of density \(\rho\text{.}\) In Fluid dynamics, an equation of this type is satisfied by the pressure field of an incompressible fluid.
Advection Equation.
\begin{equation*}
u_{t}+c u_{x}=0
\end{equation*}
This equation models the transport of a substance by bulk motion of a fluid and plays an important role in meterology and oceanography. In the version above, \(u\) is the density of the substance and \(c\) is the 1-dimensional speed of the fluid, assumed incompressible.
Wave Equation.
\begin{equation*}
u_{tt}=u_{xx}+u_{yy}
\end{equation*}
This PDE models both mechanical and electromagnetic scalar waves. Here \(u\) represents the intensity of the wave.
Heat Equation.
\begin{equation*}
u_{t}=u_{xx}+u_{yy}
\end{equation*}
This PDE models diffusive processes such as heat distribution or the evolution of the density in a solution. In these cases, \(u\) represents the temperature or the density of the solution.
Schrodinger Equation.
\begin{equation*}
iu_{t}=u_{xx}+u_{yy}
\end{equation*}
This PDE models the motion of a quantum free particle or underwater acustic waves. Unlike the Heat PDE, this is not diffusive but rather dispersive, namely the solution tends to break up into oscillatory wave packets.
Sine-Gordon Equation.
\begin{equation*}
u_{tt}=\alpha^2 u_{xx}-\mu^2\sin u
\end{equation*}
This PDE appears in many unrelated context among which Riemannian geometry, DNA mechanical models and Josephsons Junctions.
KdV Equation.
\begin{equation*}
u_{t}+u_{xxx}-6u u_x=0
\end{equation*}
This PDE models non-linear waves and it was the first where the existence of solitons was detected analytically.
Navier-Stokes Equations.
\begin{equation*}
\mathbf u_t+(\mathbf{u\cdot\nabla})\mathbf{u}+\mathbf{\nabla} p=\nu\Delta \mathbf{u}
\end{equation*}
This PDE models the motion of viscous incompressible substances. Here \(\mathbf u=(u^1,u^2,u^3)\) is the flow velocity, \(p\) the pressure and \(\nu\) the viscosity of the fluid. Solving numerically this PDE system is one of the greatest challenges for contemporary numerical analysis. 
Subsection 2.1.1 Existence and uniqueness of solutions
. Just as it happens in case of ODEs, any PDE of order \(k\) can be re-written as a system of \(k\) PDEs of order 1. The analogue of the Picard theorem for order-1 PDEs, written in case of scalar fields of three variables, is the following:Theorem 2.1.2 (Cauchy-Kovaleskaya).
If \(a(t,x,y)\text{,}\) \(b(t,x,y)\text{,}\) \(c(t,x,y)\) and \(d(x,y)\) are analytical in all arguments, then the PDE
\begin{equation*}
u_t = a u_x + b u_y +c\,,\;u(0,x,y) = d(x,y)
\end{equation*}
has a unique analytical solution for small enough \(t\text{,}\) \(x\) and \(y\text{.}\)Subsection 2.1.2 Hyperbolic, Parabolic and Elliptic PDEs
Consider first the case of 1-st order PDEs with constant coefficients. It is enough to consider the case of two variables:
\begin{equation*}
au_x+bu_y=0.
\end{equation*}
There is an ODE strictly related to this PDE:
\begin{equation*}
\left\{
\begin{aligned}
\dot x&=a\\
\dot y&=b\\
\end{aligned}
\right..
\end{equation*}
The relation between the ODE and the PDE is the following. Let
\begin{equation*}
\gamma(t)=(x(t),y(t))
\end{equation*}
be any solution of the ODE and
\begin{equation*}
u(x,y)
\end{equation*}
any solution of the PDE. Then the derivative of \(u\) along \(\gamma\) is given by
\begin{equation*}
\frac{d\phantom{t}}{dt}u(x(t),y(t)) = \dot x(t) u_x(x(t),y(t)) + \dot y(t) u_y(x(t),y(t)) = a u_x+bu_y = 0.
\end{equation*}
This means that the derivative of \(u\) along \(\gamma\) is completely determined by the PDE. The curves with this property are called characteristic curves.
Important consequences. When the PDE problem is well-posed, namely there is a unique solution defined at every point, \(\gamma\) will intersect the set on which the initial/boundary condition is set and so the values of the solution at all points of \(\gamma\) is determined only by the values of the initial/boundary condition that lie on \(\gamma\) itself. In other words, when there are characteristic curves the dependance of the solution from the initial/boundary condition is localized. On the contrary, when there are no characteristics, the value of the solution at each point is determined by all values of the function at the boundary. Definition 2.1.3.
We say that a PDE of order \(k\) is hyperbolic when it has as \(k\) families of characteristic curves. We say that is elliptic when it has no characteristics. Finally, we say that it is parabolic when it is neither hyperbolic neither elliptic.
\begin{equation*}
A(x,y) u_{xx} + 2B(x,y) u_{xy} + C(x,y) u_{yy} = D(x,y,u_x,u_y).
\end{equation*}
Consider any curve \(\gamma(t)=(x(t),y(t))\) and the corresponding directional derivative of the gradient of \(u\) (notice that we in the formula below we drop the arguments \((x(t),y(t))\) for readability sake).
\begin{equation*}
\begin{aligned}
\frac{d\phantom{t}}{dt}u_x &= \dot x(t) u_{xx} + \dot y(t) u_{xy}\\
\frac{d\phantom{t}}{dt}u_y &= \dot x(t) u_{xy} + \dot y(t) u_{yy}\\
\end{aligned}
\end{equation*}
We have therefore the following relations among the second partial derivative of \(u\text{:}\)
\begin{equation*}
\begin{pmatrix}
A&B&C\\
\dot x&\dot y&0\\
0&\dot x&\dot y\\
\end{pmatrix}
\begin{pmatrix}
u_{xx}\\
u_{xy}\\
u_{yy}\\
\end{pmatrix}
=
\begin{pmatrix}
D\\
du_x/dt\\
du_y/dt\\
\end{pmatrix}
\end{equation*}
Hence, a curve \(\gamma\) is characteristic when the determinant of the matrix above vanishes, since that means that there is a linear relation between its rows and, equivalently, that \(du_x/dt\) and \(du_y/dt\) are not independent. This condition reads explicitly as
\begin{equation*}
0=
\det
\begin{pmatrix}
A&B&C\\
\dot x&\dot y&0\\
0&\dot x&\dot y\\
\end{pmatrix}
=
A\dot y^2-B\dot x\dot y+C\dot x^2.
\end{equation*}
In any neighborhood of \((x(t),y(t))\) where the curve can be expressed as \(y=f(x)\text{,}\) namely nearby any point where \(\dot x\neq0\text{,}\) we can re-write the condition above as
\begin{equation*}
A \left(\frac{df}{dx}\right)^2 - B \frac{df}{dx} + C = 0,
\end{equation*}
whose (formal) solution is
\begin{equation*}
\frac{df}{dx} = \frac{B\pm\sqrt{B^2-4AC}}{2A}.
\end{equation*}
Hence: - when \(B^2-4AC>0\text{,}\) then the PDE has two characteristic curves and so it is hyperbolic;
- when \(B^2-4AC=0\text{,}\) there is a single characteristic curve and so the PDE is parabolic;
- when \(B^2-4AC<0\) there is no characteristic curve and so the PDE is elliptic.
| Hyperbolic | \(u_{tt}=u_{xx}\) |
| Parabolic | \(u_{t}=u_{xx}\) |
| Elliptic | \(u_{xx}+u_{yy}=0\) |
- Elliptic PDEs describe processes that have already reached a steady state, and so are time-independent.
- Hyperbolic PDEs describe time-dependent conservative processes, such as convection, that are not evolving towards a steady state.
- Parabolic PDEs describe time-dependent dissipative processes, such as diffusion, that are evolving towards a steady state.