10 research outputs found
Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient
Full text linkThe 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
Lagrangian solutions to the 2D euler system with L^1 vorticity and infinite energy
Get PDFWe consider solutions to the two-dimensional incompressible Euler system with
only integrable vorticity, thus with possibly locally infinite energy. With
such regularity, we use the recently developed theory of Lagrangian flows
associated to vector fields with gradient given by a singular integral in order
to define Lagrangian solutions, for which the vorticity is transported by the
flow. We prove strong stability of these solutions via strong convergence of
the flow, under only the assumption of L^1 weak convergence of the initial
vorticity. The existence of Lagrangian solutions to the Euler system follows
for arbitrary L^1 vorticity. Relations with previously known notions of
solutions are established
An introduction to finite volume methods for hyperbolic conservation laws
No full textHydrodynamic transport problems often take the form of
systems of hyperbolic conservation laws.
This minicourse intends to introduce the main notions and tools
for the numerical approximation of such systems by finite volumes.
The notions of consistency, stability,
and approximate Riemann solvers are explained in particular.
The main ingredients to go to
second-order and multidimension are given
