2D TRANSPORT EQUATION WITH HAMILTONIAN VECTOR FIELDS 9
Lemma 6.3. Every ψ Lipc([0,T [×[α,
β]∗)
can be approximated uniformly
with a sequence of functions { ˜n} ϕ of the form above and such that Lip( ˜n) ϕ is equi-
bounded.
This means that we can write (6.6) with ˜ ϕ = ˜n ϕ for every n and passing to the
limit we get
(6.7)
T
0
β
α
˜ u
(
∂tψ + ∂sψ
)
dsdt = 0
for any ψ Lipc([0,T [×, [α,
β]∗).
This is now a distributional equation on [0,T ] ×
[α,
β]∗.
Step 3. Uniqueness on C. Now it suffices to notice that (6.7) is the distri-
butional form of the Cauchy problem
(6.8)
∂t ˜ u + ∂s ˜ u = 0
˜(0, u ·) = 0 .
By the smooth theory for the transport equation (see [7]) we know that the only
solution to this problem is ˜ u 0. From the definition of ˜ u we see that this precisely
implies (6.2), thus we have shown the desired thesis.
We close this note by presenting two particular cases in which the weak Sard
property (4.1) is satisfied by the function H Lipc(R2) associated to b as in (1.3)
and thus the uniqueness result of Theorem 6.2 holds. See [2] for the proof.
Corollary 6.4. Let b
L∞(R2; R2)
with compact support and assume that
div b = 0 and that b is approximately differentiable
L2-a.e.
in
R2.
Then, for eve-
ry initial data ¯ u
L∞(R2),
the Cauchy problem (1.1) has a unique solution u
L∞([0,T
] ×
R2).
We observe that the approximate differentiability assumption on b in Corollary
6.4 is of “qualitative” type, in contrast with the usual “quantitative” weak regularity
assumptions, for instance Sobolev or BV . In the second corollary we deal with the
case in which we assume a control on the distributional curl of the vector field.
Corollary 6.5. Let b
L∞(R2; R2)
with compact support and assume that
div b = 0 and that the distributional curl of b is a measure. Then, for every initial
data ¯ u
L∞(R2),
the Cauchy problem (1.1) has a unique solution u
L∞([0,T
] ×
R2).
References
[1] M. Aizenman: On vector fields as generators of flows: a counterexample to Nelson’s conjec-
ture. Ann. Math., 107 (1978), 287–296.
[2] G. Alberti, S. Bianchini & G. Crippa: Work in preparation.
[3] L. Ambrosio: Transport equation and Cauchy problem for BV vector fields. Invent. Math.,
158 (2004), 227–260.
[4] L. Ambrosio: Transport equation and Cauchy problem for non-smooth vector fields and
applications. Lecture Notes in Mathematics “Calculus of Variations and Non-Linear Partial
Differential Equation” (CIME Series, Cetraro, 2005) 1927, B. Dacorogna and P. Marcellini
eds., 2–41, 2008.
[5] L. Ambrosio, F. Bouchut & C. De Lellis: Well-posedness for a class of hyperbolic systems
of conservation laws in several space dimensions. Comm. Partial Differential Equations, 29
(2004), 1635–1651.
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