Page 599 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 599
A JOURNEY FROM PURE TO APPLIED MATHEMATICS (MS-53)
means space-dependence and anisotropy yields dependence of M on ∇u not necessarily via its
length |∇u|.
The datum is too poorly regular for weak solutions to exist, thus a more delicate notion of
very weak solutions is necessary. Besides existence and uniqueness they share some regularity
properties of the weak ones. The generality we admit covers classical linear and polynomial
growth operators possibly with measurable coefficients, generalizations of p-Laplacian involv-
ing log-Hölder continuous variable exponent, as well as problems posed in fully anisotropic
Orlicz spaces (under no growth conditions), and double-phase spaces within the range of param-
eters sharp for density of smooth functions. The conditions for the density and their importance
will be stressed.
The talk will be based on series of joint papers with: Youssuf Ahmida, Angela Alberico,
Andrea Cianchi, Piotr Gwiazda, Ahmed Youssfi, and Anna Zatorska-Goldstein.
Flows of nonsmooth vector fields: new results on non uniqueness and
commutativity
Maria Colombo, maria.colombo@epfl.ch
EPFL Lausanne, Switzerland
Given a vector field in Rd, the classical Cauchy-Lipschitz theorem shows existence and unique-
ness of its flow (namely, the solution X(t) of the ODE X (t) = b(t, X(t)) from any initial datum
x ∈ Rd) provided the vector field is sufficiently smooth. The theorem looses its validity as soon
as v is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized
notion of flow, consisting of a suitable selection among the trajectories of the associated ODE,
and they showed existence, uniqueness and stability for this notion of flow for much less regular
vector fields.
The talk presents an overview and new results in the context of the celebrated DiPerna-
Lions and Ambrosio’s theory on flows of Sobolev vector fields, including a negative answer to
the following long-standing open question: are the trajectories of the ODE unique for a.e. initial
datum in Rd
for vector fields as in Di Perna and Lions theorem? We will exploit the connection between
the notion of flow and an associated PDE, the transport equation, and combine ingredients from
probability theory, harmonic analysis, and the “convex integration” method for the construction
of nonunique solutions to certain PDEs.
597
means space-dependence and anisotropy yields dependence of M on ∇u not necessarily via its
length |∇u|.
The datum is too poorly regular for weak solutions to exist, thus a more delicate notion of
very weak solutions is necessary. Besides existence and uniqueness they share some regularity
properties of the weak ones. The generality we admit covers classical linear and polynomial
growth operators possibly with measurable coefficients, generalizations of p-Laplacian involv-
ing log-Hölder continuous variable exponent, as well as problems posed in fully anisotropic
Orlicz spaces (under no growth conditions), and double-phase spaces within the range of param-
eters sharp for density of smooth functions. The conditions for the density and their importance
will be stressed.
The talk will be based on series of joint papers with: Youssuf Ahmida, Angela Alberico,
Andrea Cianchi, Piotr Gwiazda, Ahmed Youssfi, and Anna Zatorska-Goldstein.
Flows of nonsmooth vector fields: new results on non uniqueness and
commutativity
Maria Colombo, maria.colombo@epfl.ch
EPFL Lausanne, Switzerland
Given a vector field in Rd, the classical Cauchy-Lipschitz theorem shows existence and unique-
ness of its flow (namely, the solution X(t) of the ODE X (t) = b(t, X(t)) from any initial datum
x ∈ Rd) provided the vector field is sufficiently smooth. The theorem looses its validity as soon
as v is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized
notion of flow, consisting of a suitable selection among the trajectories of the associated ODE,
and they showed existence, uniqueness and stability for this notion of flow for much less regular
vector fields.
The talk presents an overview and new results in the context of the celebrated DiPerna-
Lions and Ambrosio’s theory on flows of Sobolev vector fields, including a negative answer to
the following long-standing open question: are the trajectories of the ODE unique for a.e. initial
datum in Rd
for vector fields as in Di Perna and Lions theorem? We will exploit the connection between
the notion of flow and an associated PDE, the transport equation, and combine ingredients from
probability theory, harmonic analysis, and the “convex integration” method for the construction
of nonunique solutions to certain PDEs.
597
