1.2. EXAMPLES 3

NOTATION. We write “PDE” as an abbreviation for both the singular

“partial diﬀerential equation” and the plural “partial diﬀerential equations”.

1.2. EXAMPLES

There is no general theory known concerning the solvability of all partial

diﬀerential equations. Such a theory is extremely unlikely to exist, given

the rich variety of physical, geometric, and probabilistic phenomena which

can be modeled by PDE. Instead, research focuses on various particular

partial diﬀerential equations that are important for applications within and

outside of mathematics, with the hope that insight from the origins of these

PDE can give clues as to their solutions.

Following is a list of many speciﬁc partial diﬀerential equations of in-

terest in current research. This listing is intended merely to familiarize the

reader with the names and forms of various famous PDE. To display most

clearly the mathematical structure of these equations, we have mostly set

relevant physical constants to unity. We will later discuss the origin and

interpretation of many of these PDE.

Throughout x ∈ U, where U is an open subset of

Rn,

and t ≥ 0. Also

Du = Dxu = (ux1 , . . . , uxn ) denotes the gradient of u with respect to the

spatial variable x = (x1, . . . , xn). The variable t always denotes time.

1.2.1. Single partial diﬀerential equations.

a. Linear equations.

1. Laplace’s equation

Δu =

n

i=1

uxixi = 0.

2. Helmholtz’s (or eigenvalue) equation

−Δu = λu.

3. Linear transport equation

ut +

n

i=1

biuxi

= 0.

4. Liouville’s equation

ut −

n

i=1

(biu)xi

= 0.