Proceedings of Symposia in Applied Mathematics
Two–dimensional transport equation
with Hamiltonian vector fields
Giovanni Alberti, Stefano Bianchini, and Gianluca
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
∂tu(t, x) + b(t, x) · ∇u(t, x) = 0
u(0,x) = ¯(x) u ,
where b : [0,T ] ×

is a vector field, ¯ u
is the initial data, and
the unknown u belongs to
] ×
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
˙ 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)
Proceedings of Symposia in Applied Mathematics
Volume 67.2, 2009
2009 American Mathematical Society
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