29 research outputs found
Regularity estimates for scalar conservation laws in one space dimension
In this paper we deal with the regularizing effect that, in a scalar
conservation laws in one space dimension, the nonlinearity of the flux function
has on the entropy solution. More precisely, if the set
is dense, the regularity of the solution can be expressed in terms of
spaces, where depends on the nonlinearity of . If
moreover the set is finite, under the additional polynomial
degeneracy condition at the inflection points, we prove that for every and that this can be
improved to regularity except an at
most countable set of singular times. Finally we present some examples that
shows the sharpness of these results and counterexamples to related questions,
namely regularity in the kinetic formulation and a property of the fractional
BV spaces
Differentiability properties of the flow of 2d autonomous vector fields
We investigate under which assumptions the flow associated to autonomous
planar vector fields inherits the Sobolev or BV regularity of the vector field.
We consider nearly incompressible and divergence-free vector fields, taking
advantage in both cases of the underlying Hamiltonian structure. Finally we
provide an example of an autonomous planar Sobolev divergence-free vector
field, such that the corresponding regular Lagrangian flow has no bounded
variation
On the concentration of entropy for scalar conservation laws
We prove that the entropy for an -solution to a scalar conservation laws with continuous initial data is concentrated on a countably -rectifiable set. To prove this result we introduce the notion of Lagrangian representation of the solution and give regularity estimates on the solution
Stability of the vortex in micromagnetics and related models
We consider line-energy models of Ginzburg-Landau type in a two-dimensional
simply-connected bounded domain. Configurations of vanishing energy have been
characterized by Jabin, Otto and Perthame: the domain must be a disk, and the
configuration a vortex. We prove a quantitative version of this statement in
the class of domains, improving on previous results by Lorent. In
particular, the deviation of the domain from a disk is controlled by a power of
the energy, and that power is optimal. The main tool is a Lagrangian
representation introduced by the second author, which allows to decompose the
energy along characteristic curves
New regularity results for scalar conservation laws and applications to a source–destination model for traffic flows on networks
We focus on entropy admissible solutions of scalar conservation laws in one
space dimension and establish new regularity results with respect to time.
First, we assume that the flux function is strictly convex and show that,
for every , the total variation of the composite function is controlled by the total variation of the initial datum.
Next, we assume that is monotone and, under no convexity assumption, we
show that, for every , the total variation of the left and right trace
is controlled by the total variation of the initial datum. We
also exhibit a counter-example showing that in the first result the total
variation bound does not extend to the function , or equivalently that in
the second result we cannot drop the monotonicity assumption. We then discuss
applications to a source-destination model for traffic flows on road networks.
We introduce a new approach, based on the analysis of transport equations with
irregular coefficients, and, under the assumption that the network only
contains so-called T-junctions, we establish existence and uniqueness results
for merely bounded data in the class of solutions where the traffic is not
congested. Our assumptions on the network and the traffic congestion are
basically necessary to obtain well-posedness in view of a counter-example due
to Bressan and Yu. We also establish stability and propagation of
regularity, and this is again interesting in view of recent counter-examples.Comment: 28 pages, 2 figure
Stability of quasi-entropy solutions of non-local scalar conservation laws
We prove the stability of entropy solutions of nonlinear conservation laws
with respect to perturbations of the initial datum, the space-time dependent
flux and the entropy inequalities.
Such a general stability theorem is motivated by the study of problems in
which the flux depends possibly non-locally on the solution
itself. For these problems we show the conditional existence and uniqueness of
entropy solutions.
Moreover, the relaxation of the entropy inequality allows to treat
approximate solutions arising from various numerical schemes. This can be used
to derive the rate of convergence of the recent particle method introduced in
[Radici-Stra 2021] to solve a one-dimensional model of traffic with congestion,
as well as recover already known rates for some other approximation methods