1,141 research outputs found
Lagrangian flows driven by fields in Wiener spaces
We establish the renormalization property for essentially bounded solutions
of the continuity equation associated to fields in Wiener spaces, with
values in the associated Cameron-Martin space; thus obtaining, by standard
arguments, new uniqueness and stability results for correspondent Lagrangian
-flows. An example related to Neumann elliptic problems is also
discussed
Some new well-posedness results for continuity and transport equations, and applications to the chromatography system
We obtain various new well-posedness results for continuity and transport
equations, among them an existence and uniqueness theorem (in the class of
strongly continuous solutions) in the case of nearly incompressible vector
fields, possibly having a blow-up of the BV norm at the initial time. We apply
these results (valid in any space dimension) to the k x k chromatography system
of conservation laws and to the k x k Keyfitz and Kranzer system, both in one
space dimension.Comment: 33 pages, minor change
Renormalized solutions for stochastic transport equations and the regularization by bilinear multiplicative noise
A linear stochastic transport equation with non-regular coefficients is
considered. Under the same assumption of the deterministic theory, all weak
-solutions are renormalized. But then, if the noise is nondegenerate,
uniqueness of weak -solutions does not require essential new
assumptions, opposite to the deterministic case where for instance the
divergence of the drift is asked to be bounded. The proof gives a new
explanation why bilinear multiplicative noise may have a regularizing effect
Lecture notes on the DiPerna-Lions theory in abstract measure spaces
These notes closely correspond to a series of lectures given by the first
author in Toulouse, on the recent extension of the theory of ODE well-posedness
to abstract spaces, jointly obtained by the two authors. In the last part, we
describe some further developments with respect to the theory of (possibly
degenerate) diffusion processes, in a similar setting, contained in the second
author's PhD thesis
Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient
The Cauchy problem for a multidimensional linear transport equation with
discontinuous coefficient is investigated. Provided the coefficient satisfies a
one-sided Lipschitz condition, existence, uniqueness and weak stability of
solutions are obtained for either the conservative backward problem or the
advective forward problem by duality. Specific uniqueness criteria are
introduced for the backward conservation equation since weak solutions are not
unique. A main point is the introduction of a generalized flow in the sense of
partial differential equations, which is proved to have unique jacobian
determinant, even though it is itself nonunique.Comment: 19-03-200
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