@article {ipa|Initial-boundary value problems for nearly incompressible vector fields, and applications to the Keyfitz and Kranzer system|2017, title = {Initial-boundary value problems for nearly incompressible vector fields, and applications to the Keyfitz and Kranzer system}, year = {2017}, month = {January}, pages = {19 p.}, institution = {CNR-IMATI}, type = {Working paper}, address = {Pavia}, abstract = {

We establish existence and uniqueness results for initial boundary value problems with nearly incompressible vector fields. We then apply our results to establish well-posedness of the initial-boundary value problem for the Keyfitz and Kranzer system of conservation laws in several space dimensions.

}, keywords = {Systems of conservation laws}, url = {http://irs.imati.cnr.it/reports/irs17-02}, author = {A.P. Choudhury and G. Crippa and L.V. Spinolo} } @article {irs17-04, title = {Optimality of integrability estimates for advection-diffusion equations}, number = {17-04}, year = {2017}, month = {February}, pages = {19}, publisher = {CNR-IMATI}, type = {Working paper}, address = {Pavia}, abstract = {

We discuss integrability estimates for the solution of an advection-diffusion equation. We first summarize some classical results proving such estimates for certain ranges of the space and time summability of the velocity field. Afterwards we prove the optimality of such ranges by means of new original examples.

}, keywords = {Advection-diffusion equation, Duhamel formula, integrability estimates and their optimality, parabolic equations, self-similar solutions}, url = {http://irs.imati.cnr.it/reports/irs17-04}, author = {S. Bianchini and M. Colombo and G. Crippa and L.V. Spinolo} } @article {ipa|Quantitative estimates on localised finite differences for the fractional Poisson problem, and applications to regularity and spectral stability|2017, title = {Quantitative estimates on localised finite differences for the fractional Poisson problem, and applications to regularity and spectral stability}, year = {2017}, month = {January}, pages = {38 p.}, institution = {CNR-IMATI}, type = {Working paper}, address = {Pavia}, abstract = {

We establish new quantitative estimates for localized finite differences of solutions to the Poisson problem for the fractional Laplace operator with homogeneous Dirichlet conditions of solid type settled in bounded domains satisfying the Lipschitz cone regularity condition. We then apply these estimates to obtain (i) regularity results for solutions of fractional Poisson problems in Besov spaces; (ii) quantitative stability estimates for solutions of fractional Poisson problems with respect to domain perturbations; (iii) quantitative stability estimates for eigenvalues and eigenfunctions of fractional Laplace operators with respect to domain perturbations.

}, keywords = {Spectral stability}, url = {http://irs.imati.cnr.it/reports/irs17-03}, author = {G. Akagi and G. Schimperna and A. Segatti and L.V. Spinolo} } @article {irs16-01, title = {Optimal strategies for a time-dependent harvesting problem}, number = {16-01}, year = {2016}, month = {February}, pages = {44 p.}, publisher = {IMATI-CNR}, type = {Working paper}, address = {Pavia}, abstract = {

We focus on an optimal control problem, introduced by Bressan and Shen in [5] as a model for fish harvesting. We consider the time-dependent case and we establish existence and uniqueness of an optimal strategy, and sufficient conditions for optimality. We also consider a related differential game that models the situation where there are several competin fish companies and we prove existence of Nash equilibria. From the technical viewpoint, the most relevant point is establishing the uniqueness result. This amounts to prove precise a-priori estimates for solutions of suitable parabolic equations with measure-valued coefficients. All the analysis is developed in the case when the fishing domain is one-dimensional.

}, keywords = {Optimal control}, url = {http://irs.imati.cnr.it/reports/irs16-01}, author = {G.M. Coclite and M. Garavello and L.V. Spinolo} } @article {irs15-02, title = {Schaeffer{\textquoteright}s regularity theorem for scalar conservation laws does not extend to systems}, number = {15-02}, year = {2015}, month = {4th May 2015}, pages = {47}, publisher = {CNR-IMATI}, type = {Working paper}, edition = {Imati Report Series}, address = {Pavia}, abstract = {

Schaeffer{\textquoteright}s regularity theorem for scalar conservation laws can be loosely speaking formulated as follows. Assume that the flux is uniformly convex, then for a generic smooth initial datum the admissible solution is smooth outside a locally finite number of curves in the (t, x) plane. Here the term "generic{\textquotedblright} is to be interpreted in a suitable sense, related to the Baire Category Theorem. Whereas other regularity results valid for scalar conservation laws with convex fluxes have been extended to systems of conservation laws with genuinely nonlinear characteristic fields, in this work we exhibit an explicit counterexample which rules out the possibility of extending Schaeffer{\textquoteright}s Theorem. The analysis relies on careful interaction estimates and uses fine properties of the wave front-tracking approximation.

}, keywords = {Conservation laws, Regularity, Schaeffer theorem, Shock formation}, url = {http://irs.imati.cnr.it/reports/irs15-02}, author = {L.V. Spinolo and L. Caravenna} }