Proceedings of Symposia in Applied Mathematics

Two–dimensional transport equation

with Hamiltonian vector fields

Giovanni Alberti, Stefano Bianchini, and Gianluca

Crippa∗

Abstract. We illustrate the main steps in the proof of a sharp result of well-

posedness for the two-dimensional transport equation whose vector field is

bounded, autonomous, divergence free and satisfies the so-called “weak Sard

property”. We also remark on the fact that the weak Sard property we identify

is indeed equivalent to the well-posedness.

1. Introduction and main notation

The study of the well-posedness of the transport equation

(1.1)

∂tu(t, x) + b(t, x) · ∇u(t, x) = 0

u(0,x) = ¯(x) u ,

where b : [0,T ] ×

Rd

→

Rd

is a vector field, ¯ u ∈

L∞(Rd)

is the initial data, and

the unknown u belongs to

L∞([0,T

] ×

Rd),

is of great importance in the theory

of nonlinear evolutionary partial diﬀerential equations, due to the appearance of

equations of this form in many physical phenomena; we refer for instance to [16]

and [26] for a general overview on the theory of conservation laws. The theory

is classical and well-understood in the case when b is suﬃciently smooth (at least

Lipschitz with respect to the spatial variable, uniformly with respect to the time),

and is strongly based on the so-called theory of characteristics, i.e. on the connection

between (1.1) and the ordinary diﬀerential equation

(1.2)

˙ γ (t) = b(t, γ(t))

γ(0) = x .

However, in many applications motived by physical models, non-smooth vector

fields show up as velocity fields of transport equations. Thus a great interest has

arisen in the study of (1.1) when b is only in some classes of weak diﬀerentiabil-

ity. We mention in this context the two seminal papers by DiPerna and Lions

[20] and by Ambrosio [3], in which the Sobolev and the BV cases respectively are

considered, in both cases under boundedness assumptions on the spatial divergence

1991 Mathematics Subject Classification. Primary 35F10; Secondary 35A05, 28A80.

Key words and phrases. Transport equation, two-dimensional transport equation, Sard

lemma, Hamiltonian function, coarea formula.

c 0000 (copyright holder)

1

Proceedings of Symposia in Applied Mathematics

Volume 67.2, 2009

c

2009 American Mathematical Society

337

http://dx.doi.org/10.1090/psapm/067.2/2605229