Analysis

Coclite, G.M. et al. (2024) ‘Oleinik-type estimates for nonlocal conservation laws and applications to the nonlocal-to-local limit’, Journal of Hyperbolic Differential Equations, 21(3), pp. 681–705. Available at: https://doi.org/10.1142/S021989162440006X.

Brué, Elia et al. (2024) ‘Onsager critical solutions of the forced Navier-Stokes equations’, Communications on Pure and Applied Analysis, 23(10), pp. 1350–1366. Available at: https://doi.org/10.3934/cpaa.2023071.

Ciampa, Gennaro, Crippa, Gianluca and Spirito, Stefano (2024) ‘Propagation of logarithmic regularity and inviscid limit for the 2D Euler equations’, Mathematics in Engineering, 6(4), pp. 494–509. Available at: https://doi.org/10.3934/mine.2024020.

Colombo, Maria, Crippa, Gianluca and Sorella, Massimo (2023) ‘Anomalous Dissipation and Lack of Selection in the Obukhov–Corrsin Theory of Scalar Turbulence’, Annals of PDE, 9(2). Available at: https://doi.org/10.1007/s40818-023-00162-9.

Colombo, Maria et al. (2023) ‘Nonlocal Traffic Models with General Kernels: Singular Limit, Entropy Admissibility, and Convergence Rate’, Archive for Rational Mechanics and Analysis, 247(2). Available at: https://doi.org/10.1007/s00205-023-01845-0.

Crippa, Gianluca and Schulze, Christian (2023) ‘Sub-exponential mixing of generalized cellular flows with bounded palenstrophy†’, Mathematics In Engineering, 5(1). Available at: https://doi.org/10.3934/mine.2023006.

Bonicatto, Paolo, Ciampa, Gennaro and Crippa, Gianluca (2022) ‘On the advection-diffusion equation with rough coefficients: weak solutions and vanishing viscosity’, Journal de Mathématiques Pures et Appliquées, 167, pp. 204–224. Available at: https://doi.org/10.1016/j.matpur.2022.09.005.

Crippa, Gianluca et al. (2022) ‘Growth of Sobolev norms and loss of regularity in transport equations’, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 380(2225), p. 20210024. Available at: https://doi.org/10.1098/rsta.2021.0024.

De Rosa, Luigi and Haffter, Silja (2022) ‘Dimension of the singular set of wild Hölder solutions of the incompressible Euler equations’, Nonlinearity, 35(10), pp. 5150–5192.

Caravenna, Laura and Crippa, Gianluca (2021) ‘A directional Lipschitz extension lemma, with applications to uniqueness and Lagrangianity for the continuity equation’, Communications Partial Differential Equations, 46(8), pp. 1488–1520. Available at: https://doi.org/10.1080/03605302.2021.1883650.

Ciampa, Gennaro, Crippa, Gianluca and Spirito, Stefano (2021) ‘Strong convergence of the vorticity for the 2D Euler equations in the inviscid limit’, Archive for Rational Mechanics and Analysis, 240(1), pp. 295–326. Available at: https://doi.org/10.1007/s00205-021-01612-z.

Colombo, Maria et al. (2021) ‘On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws’, ESAIM: Mathematical Modelling and Numerical Analysis, 55(6), pp. 2705–2723. Available at: https://doi.org/10.1051/m2an/2021073.

Colombo, Maria et al. (2021) ‘Local limit of nonlocal traffic models: Convergence results and total variation blow-up’, Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 38(5), pp. 1653–1666. Available at: https://doi.org/10.1016/j.anihpc.2020.12.002.

Schulze, Christian (2021) Inviscid mixing by fluid flows: optimal stirring with cellular and radial velocity fields. Dissertation.

Ciampa, Gennaro, Crippa, Gianluca and Spirito, Stefano (2020) ‘Smooth approximation is not a selection principle for the transport equations with rough vector field’, Calculus of Variations and Partial Differential Equations, 59, p. 13. Available at: https://doi.org/10.1007/s00526-019-1659-0.

Ciampa, Gennaro, Crippa, Gianluca and Spirito, Stefano (2020) ‘Weak Solutions Obtained by the Vortex Method for the 2D Euler Equations are Lagrangian and Conserve the Energy’, Journal of Nonlinear Science, 30(6), pp. 2787–2820. Available at: https://doi.org/10.1007/s00332-020-09635-8.

Crippa, Gianluca and Ligabue, Silvia (2020) ‘A Note on the Lagrangian Flow Associated to a Partially Regular Vector Field’, Differential Equations and Dynamical Systems, pp. 1–20. Available at: https://doi.org/10.1007/s12591-020-00530-y.

Alberti, Giovanni, Crippa, Gianluca and Mazzucato, Anna L. (2019) ‘Exponential self-similar mixing by incompressible flows’, Journal of the American Mathematical Society, 32(2), pp. 445–490. Available at: https://doi.org/10.1090/jams/913.

Alberti, Giovanni, Crippa, Gianluca and Mazzucato, Anna L. (2019) ‘Loss of regularity for the continuity equation with non-Lipschitz velocity field’, Annals of PDE, 5(1), p. 9. Available at: https://doi.org/10.1007/s40818-019-0066-3.

Colombo, Maria, Crippa, Gianluca and Spinolo, Laura Valentina (2019) ‘On the singular local limit for conservation laws with nonlocal fluxes’, Archive for Rational Mechanics and Analysis, 233(3), pp. 1131–1167. Available at: https://doi.org/10.1007/s00205-019-01375-8.

Crippa, Gianluca, Lucà, Renato and Schulze, Christian (2019) ‘Polynomial mixing under a certain stationary Euler flow’, Physica D: Nonlinear Phenomena, 394, pp. 44–55. Available at: https://doi.org/10.1016/j.physd.2019.01.009.

Ligabue, Silvia (2019) Anisotorpic vector fields: quantitative estimates and applications to the Vlasov-Poisson equation. Dissertation.

Crippa, Gianluca, Ligabue, Silvia and Saffirio, Chiara (2018) ‘Lagrangian solutions to the Vlasov-Poissosystem with a point charge’, Kinetic and related models, 11(6), pp. 1277–1299. Available at: https://doi.org/10.3934/krm.2018050.

Lamy, Xavier and Ghiraldin, Francesco (2018) ‘Optimal besov differentiability for entropy solutions of the eikonal equation’, Communications in Pure and Applied Mathematics [Preprint]. Available at: https://archiv.math.unibas.ch/uploads/tx_x4epublication/2018-02-eikonal-veryfinal.pdf.

Bianchini, Stefano et al. (2017) ‘Optimality of integrability estimates for advection-diffusion equations’, Nonlinear Differential Equations and Applications, 24(4), p. 19. Available at: https://doi.org/10.1007/s00030-017-0455-9.

Choudhury, Anupam, Crippa, Gianluca and Spinolo, Laura Valentina (2017) ‘Initial-boundary value problems for nearly incompressible vector fields, and applications to the Keyfitz and Kranzer system’, Zeitschrift für Angewandte Mathematik und Physik, 68(6), p. 19. Available at: https://doi.org/10.1007/s00033-017-0883-8.

Crippa, Gianluca et al. (2017) ‘Failure of the chain rule for the divergence of bounded vector fields’, Annali della Scuola Normale di Pisa - Classe di Scienze, 17(1), pp. 1–18. Available at: https://doi.org/10.2422/2036-2145.201506_006.

Crippa, Gianluca and Mazzucato, Anna (2017) Transport, Fluids, and Mixing, Partial differential equations and measure theory. Berlin: De Gruyter Open (Partial differential equations and measure theory). Available at: https://www.degruyter.com/view/product/497138.

Crippa, Gianluca and Mazzucato, Anna (2017) ‘Introduction’, Transport, Fluids, and Mixing: Open Access Partial Differential Equations and Measure Theory, pp. 1–7. Available at: https://doi.org/10.1515/9783110571240-001.

Crippa, Gianluca et al. (2017) Transport, Fluids, and Mixing: Open Access Partial Differential Equations and Measure Theory. Available at: https://doi.org/10.1515/9783110571240.

Crippa, Gianluca et al. (2017) ‘Eulerian and Lagrangian Solutions to the Continuity and Euler Equations with L 1 Vorticity’, SIAM Journal on Mathematical Analysis, 49(5), pp. 3973–3998. Available at: https://doi.org/10.1137/17m1130988.

Crippa, Gianluca and Schulze, Christian (2017) ‘Cellular mixing with bounded palenstrophy’, Mathematical Models and Methods in Applied Sciences, 27(12), pp. 2297–2320. Available at: https://doi.org/10.1142/s0218202517500452.

Bohun, Anna, Bouchut, Francois and Crippa, Gianluca (2016) ‘Lagrangian flows for vector fields with anisotropic regularity’, Annales de l’Institut Henri Poincaré (C) Analyse non linéaire, 33(6), pp. 1409–1429. Available at: https://doi.org/10.1016/j.anihpc.2015.05.005.

Bohun, Anna, Bouchut, François and Crippa, Gianluca (2016) ‘Lagrangian solutions to the Vlasov-Poisson system with L-1 density’, Journal of differential equations, 260(4), pp. 3576–3597. Available at: https://doi.org/10.1016/j.jde.2015.10.041.

Bohun, Anna, Bouchut, François and Crippa, Gianluca (2016) ‘Lagrangian solutions to the 2D Euler system with L1 vorticity and infinite energy’, Nonlinear Analysis: Theory, Methods & Applications, 132, pp. 160–172. Available at: https://doi.org/10.1016/j.na.2015.11.004.

Caravenna, Laura and Crippa, Gianluca (2016) ‘Uniqueness and Lagrangianity for solutions with lack of integrability of the continuity equation’, Comptes rendus mathematique, 354(12), pp. 1168–1173. Available at: https://doi.org/10.1016/j.crma.2016.10.009.

Colombo, Maria, Crippa, Gianluca and Stefano, Spirito (2016) ‘Logarithmic estimates for continuity equations’, Networks and Heterogeneous Media, 11(2), pp. 301–311. Available at: https://doi.org/10.3934/nhm.2016.11.301.

Crippa, Gianluca et al. (2016) ‘Flows of vector fields with point singularities and the vortex-wave system’, Discrete and continuous dynamical systems, 36(5), pp. 2405–2417. Available at: https://doi.org/10.3934/dcds.2016.36.2405.

Bohun, Anna (2015) Flows of singular vector fields and applications to fluid and kinetic equations. Dissertation.

Colombo, Maria, Crippa, Gianluca and Spirito, Stefano (2015) ‘Renormalized solutions to the continuity equation with an integrable damping term’, Calculus of variations and partial differential equations, 54(2), pp. 1831–1845. Available at: https://doi.org/10.1007/s00526-015-0845-y.

Crippa, Gianluca et al. (2015) ‘Non-uniqueness and prescribed energy for the continuity equation’, Communications in mathematical sciences, 13(7), pp. 1937–1947. Available at: https://doi.org/10.4310/cms.2015.v13.n7.a12.

Crippa, Gianluca, Semenova, Elizaveta and Spirito, Stefano (2015) ‘Strong continuity for the 2D Euler equations’, Kinetic and related models, 8(4), pp. 685–689. Available at: https://doi.org/10.3934/krm.2015.8.685.

Crippa, Gianluca and Spirito, Stefano (2015) ‘Renormalized solutions of the 2D Euler equations’, Communications in mathematical physics, 339(1), pp. 191–198. Available at: https://doi.org/10.1007/s00220-015-2411-z.

Alberti, Giovanni, Bianchini, Stefano and Crippa, Gianluca (2014) ‘A uniqueness result for the continuity equation in two dimensions’, Journal of the European Mathematical Society, 16(2), pp. 201–234. Available at: https://doi.org/10.4171/jems/431.

Alberti, Giovanni, Bianchini, Stefano and Crippa, Gianluca (2014) ‘On the Lp-differentiability of certain classes of functions’, Revista matemática Iberoamericana, 30(1), pp. 349–367. Available at: https://doi.org/10.4171/rmi/782.

Alberti, Giovanni, Crippa, Gianluca and Mazzucato, Anna L. (2014) ‘Exponential self-similar mixing and loss of regularity for continuity equations’, Comptes rendus mathematique, 352(11), pp. 901–906. Available at: https://doi.org/10.1016/j.crma.2014.08.021.

Ambrosio, Luigi and Crippa, Gianluca (2014) ‘Continuity equations and ODE flows with non-smooth velocity’, Proceedings of the Royal Society of Edinburgh. Section A, Mathematics, 144(6), pp. 1191–1244. Available at: https://doi.org/10.1017/s0308210513000085.

Crippa, Gianluca (2014) ‘Ordinary differential equations and singular integrals’, in AIMS on Applied Mathematics. American Institute of Mathematical Sciences: American Institute of Mathematical Sciences (AIMS on Applied Mathematics). Available at: https://aimsciences.org/books/am/AMVol8.html.

Crippa, Gianluca, Donadello, Carlotta and Spinolo, Laura V. (2014) ‘Initial-boundary value problems for continuity equations with BV coefficients’, Journal de mathématiques pures et appliquées, 102(1), pp. 79–98. Available at: https://doi.org/10.1016/j.matpur.2013.11.002.

Crippa, Gianluca, Donadello, Carlotta and Spinolo, Laura Valentina (2014) ‘A note on the initial-boundary value problem for continuity equations with rough coefficients’, in AIMS on Applied Mathematics. American Institute of Mathematical Sciences: American Institute of Mathematical Sciences (AIMS on Applied Mathematics). Available at: https://aimsciences.org/books/am/AMVol8.html.

Alberti, Giovanni, Bianchini, Stefano and Crippa, Gianluca (2013) ‘Structure of level sets and Sard-type properties of Lipschitz maps’, Annali della Scuola Normale di Pisa - Classe di Scienze, 12(4), pp. 863–902. Available at: https://doi.org/10.2422/2036-2145.201107_006.

Bouchut, Francois and Crippa, Gianluca (2013) ‘Lagrangian flows for vector fields with gradient given by a singular integral’, Journal of hyperbolic differential equations, 10(2), pp. 235–282. Available at: https://doi.org/10.1142/s0219891613500100.

Crippa, Gianluca and Lecureux-Mercier, Magali (2013) ‘Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow’, NoDEA: nonlinear differential equations and applications, 20(3), pp. 523–537. Available at: https://doi.org/10.1007/s00030-012-0164-3.

Acerbi, Emilio, Crippa, Gianluca and Mucci, Domenico (2012) ‘A variational problem for multifunctions with interaction between leaves’, ESAIM: Control, Optimisation and Calculus of Variations, 18(4), pp. 1178–1206. Available at: https://doi.org/10.1051/cocv/2011195.

Crippa, Gianluca (2011) ‘Lagrangian flows and the one-dimensional Peano phenomenon for ODEs’, Journal of Differential Equations, 250(7), pp. 3135–3149. Available at: https://doi.org/10.1016/j.jde.2010.12.007.

Ambrosio, Luigi et al. (2009) ‘Some new well-posedness results for continuity and transport equations, and applications to the chromatography system’, SIAM Journal on Mathematical Analysis, 41(5), pp. 1890–1920. Available at: https://doi.org/10.1137/090754686.

Crippa, Gianluca (2009) The flow associated to weakly differentiable vector fields, Theses / Scuola Normale Superiore di Pisa. Pisa: Edizioni della Normale (Theses / Scuola Normale Superiore di Pisa).

Crippa, Gianluca, Jimenez, Chloé and Pratelli, Aldo (2009) ‘Optimum and equilibrium in a transport problem with queue penalization effect’, Advances in Calculus of Variations, 2(3), pp. 207–246. Available at: https://doi.org/10.1515/acv.2009.009.

Ambrosio, Luigi, Crippa, Gianluca and LeFloch, Philippe G. (2008) ‘Leaf superposition property for integer rectifiable currents’, Networks and Heterogeneous Media, 3(1), pp. 85–95. Available at: https://doi.org/10.3934/nhm.2008.3.85.

Crippa, Gianluca and De Lellis, Camillo (2008) ‘Estimates and regularity results for the DiPerna-Lions flow’, Journal für die reine und angewandte Mathematik, 616(616), pp. 15–46. Available at: https://doi.org/10.1515/crelle.2008.016.

Ambrosio, Luigi and Crippa, Gianluca (2008) ‘Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields’, in Ancona, Fabio; Bianchini, Stefano; Colombo, Rinaldo M.; De Lellis, Camillo; Marson, Andrea; Montanari, Annamaria (ed.) Transport equations and multi-D hyperbolic conservation laws. Berlin: Springer (Transport equations and multi-D hyperbolic conservation laws), pp. 3–57. Available at: https://doi.org/10.1007/978-3-540-76781-7_1.

Crippa, Gianluca, Otto, Felix and Westdickenberg, Michael (2008) ‘Regularizing effect of nonlinearity in multidimensional scalar conservation laws’, in Ancona, Fabio; Bianchini, Stefano; Colombo, Rinaldo M.; De Lellis, Camillo; Marson, Andrea; Montanari, Annamaria (ed.) Transport equations and multi-D hyperbolic conservation laws. Berlin: Springer (Transport equations and multi-D hyperbolic conservation laws), pp. 77–128. Available at: https://doi.org/10.1007/978-3-540-76781-7_3.

Bouchut, François and Crippa, Gianluca (2006) ‘Uniqueness, renormalization, and smooth approximations for linear transport equations’, SIAM Journal on Mathematical Analysis, 38(4), pp. 1316–1328. Available at: https://doi.org/10.1137/06065249x.

Colombini, Ferruccio, Crippa, Gianluca and Rauch, Jeffrey (2006) ‘A note on two-dimensional transport with bounded divergence’, Communications Partial Differential Equations, 31(7), pp. 1109–1115. Available at: https://doi.org/10.1080/03605300500455933.

Crippa, Gianluca and De Lellis, Camillo (2006) ‘Oscillatory solutions to transport equations’, Indiana University Mathematics Journal, 55(1), pp. 1–13. Available at: https://doi.org/10.1512/iumj.2006.55.2793.

Ambrosio, Luigi, Crippa, Gianluca and Maniglia, Stefania (2005) ‘Traces and fine properties of a BD class of vector fields and applications’, Annales de la Faculté des Sciences de Toulouse. Mathématiques, 14(4), pp. 527–561. Available at: https://doi.org/10.5802/afst.1102.