1Basics of PDEs
III Analysis of Partial Differential Equations
1 Basics of PDEs
It might be wise to define what a partial differential equation is.
Definition
(Partial differential equation)
.
Suppose
U ⊆ R
n
is open. A partial
differential equation (PDE ) of order k is a relation of the form
F (x, u(x), Du(x), . . . , D
k
u(x)) = 0, (∗)
where
F
:
U ×R ×R
n
×R
n
2
×···×R
n
k
→ R
is a given function, and
u
:
U → R
is the “unknown”.
Definition
(Classical solution)
.
We say
u ∈ C
k
(
U
) is a classical solution of a
PDE if in fact the PDE is identically satisfied on
U
when
u,
D
u, . . . ,
D
k
u
are
substituted in.
More generally, we can allow
u
and
F
to take values in a vector space. In
this case, we say it is a system of PDEs.
We can now entertain ourselves by writing out a large list of PDEs that are
naturally found in physics and mathematics.
Example
(Transport equation)
.
Suppose
v
:
R
4
× R → R
3
and
f
:
R
4
→ R
are
given. The transport equation is
∂u
∂t
(x, t) + v(x, t, u(x, t)) · D
x
u(x, t) = f (x, t)
where we think of
x ∈ R
3
and
t ∈ R
. This describes the evolution of the density
u of some chemical being advected by a flow v and produced at a rate f.
We see that this is a PDE of order 1, and a relatively straightforward solution
method exists, namely the method of characteristics.
Example
(Laplace’s and Poissson’s equations)
.
Taking
u
:
R
n
→ R
, Laplace’s
equation is
∆u(x) =
n
X
i=1
∂
2
u
∂x
i
∂x
i
(x) = 0.
This describes, for example, the electrostatic potential in vacuum and the static
distribution of heat inside a uniform solid body. It also has applications to
steady flows in 2d fluids.
There is an inhomogeneous version of this:
∆u(x) = f(x),
where
f
:
R
n
→ R
is a fixed function. This is known as Poisson’s equation, and
describes, for example, the electrostatic field due to a charge distribution, and
the gravitational field in Newtonian gravity.
Example (Heat/diffusion equation). This is given by
∂u
∂t
= ∆u,
where
u
:
R
n
× R → R
is now a function of space and time. This describes the
evolution of temperature inside a uniform body, or equivalently the diffusion of
some chemical (where u is the density).
Example (Wave equation). The wave equation is given by
∂
2
u
∂t
2
= ∆u,
where
u
:
R
n
× R → R
is again a function of space and time. This describes
oscillations of
– strings (n = 1)
– membrane/drum (n = 2)
– air density in a sound wave (n = 3)
Example
(Schr¨odinger’s equation)
.
Let
u
:
R
n
× R → C
∼
=
R
2
. Up to choices
of units and convention, the Schr¨odinger’s equation is
i
∂u
∂t
+ ∆u − V u = 0.
Here u is the wavefunction of a particle moving in a potential V : R
n
→ R.
Example
(Maxwell’s equations)
.
The unknowns here are
E, B
:
R
3
× R → R
3
.
They satisfy Maxwell’s equations
∇ · E = ρ ∇ · B = 0
∇ × E +
∂B
∂t
= 0 ∇ × B −
∂E
∂t
= J,
where
ρ
is the electric charge density,
J
is the electric current,
E
is the electric
field and B is the magnetic field.
This is a system of 6 equations and 6 unknowns.
Example (Einstein’s equations). The Einstein’s equation in vacuum are
R
µν
[g] = 0,
where
g
is a Lorentzian metric (encoding the gravitational field), and
R
µν
[
g
] is
the Ricci curvature of g.
Since we haven’t said what
g
and
R
µν
are, it is not clear that this is a partial
differential equation, but it is.
Example (Minimal surface equation). The minimal surface equation is
Div
Du
p
1 + |Du|
2
!
= 0,
where
u
:
R
n
→ R
is some function. This is the condition that the graph of
u
,
{(x, u(x))} ⊆ R
n
× R, is locally an extremizer of area.
Example
(Ricci flow)
.
Let
g
be a Riemannian metric on some manifold. The
Ricci flow is a PDE that evolves this metric:
∂g
ij
∂t
= R
ij
[g],
where R
ij
is again the Ricci curvature.
The most famous application is in proving the Poincar´e conjecture, which is
a topological conjecture about 3-manifolds.
These PDEs exhibit a wide variety of behaviours. For example, waves behave
very differently from the evolution of temperature. This means it is unlikely
that we can say anything about PDEs as a whole, since everything we say must
be true for both the heat equation and the wave equation. We must restrict
to some particular classes of PDEs to say something useful. Thus, we seek to
classify our PDEs into different types. We first introduce some notation.
In this course, the natural numbers start at 0.
Notation
(Multi-index/Schwartz notation)
.
We say an element
α ∈ N
n
is a
multi-index. Writing α = (α
1
, . . . , α
n
). We write
|α| = α
1
+ α
2
+ ··· + α
n
.
Also, we have
D
α
f =
∂
|α|
f
∂x
α
1
1
∂x
α
2
2
···∂x
α
n
n
.
If x = (x
1
, . . . , x
n
) ∈ R
n
, then
x
α
= x
α
1
1
x
α
2
2
···x
α
n
n
.
We also write
α! = α
1
!α
2
! ···α
n
!.
We now try to crudely classify the PDEs we have written down. Recall that
our PDEs take the general form
F (x, u(x), Du(x), . . . , D
k
u(x)) = 0.
Definition
(Linear PDE)
.
We say a PDE is linear if
F
is a linear function of
u
and its derivatives. In this case, we can re-write it as
X
|α|≤k
a
α
(x)D
α
u = 0.
Definition
(Semi-linear PDE)
.
We say a PDE is semi-linear if it is of the form
X
|α|=k
a
α
(x)D
α
u(x) + a
0
[x, u, Du, . . . , D
k−1
u] = 0.
In other words, the terms involving the highest order derivatives are linear.
Generalizing further, we have
Definition
(Quasi-linear PDE)
.
We say a PDE is quasi-linear if it is of the
form
X
|α|=k
a
α
[x, u, Du, . . . , D
k−1
u]D
α
u(x) + a
0
[x, u, . . . , D
k−1
u] = 0.
So the highest order derivative still appears linearly, but the coefficients can
depend on lower-order derivatives of u.
Finally, we have
Definition
(Fully non-linear PDE)
.
A PDE is fully non-linear if it is not
quasi-linear.
Example. Laplace’s equation ∆u = 0 is linear.
Example. The equation u
xx
+ u
yy
= u
2
x
is semi-linear.
Example. The equation uu
xx
+ u
yy
= u
2
x
is quasi-linear.
Example. The equation u
xx
u
yy
− u
2
xy
= 0 is fully non-linear.