Publications of Alexis VASSEUR

Submitted preprints:

[109] Moon-Jin Kang, Alexis F. Vasseur, Yi Wang, Time-asymptotic stability of composite waves of viscous shock and rarefaction for barotropic Navier-Stokes equations. [Arxiv]
Preprint.

[108] François Golse, Cyril Imbert, Alexis F. Vasseur, Partial regularity for the space homogeneous Landau equation with very soft potentials. [Arxiv]
Preprint.

[107] Alexis F. Vasseur and Jincheng Yang, Boundary Vorticity Estimates for Navier-Stokes and Application to the Inviscid Limit. [Arxiv]
Preprint.

[106] W. Golding, S. Krupa, A. Vasseur, Sharp a-contraction estimates for small extremal shocks. [Arxiv]
submitted.

[105] Moon-Jin Kang and Alexis Vasseur, Well-posedness of the Riemann problem with two shocks for the isentropic Euler System in a class of vanishing physical viscosity limits. [pdf]
J. Differential Equations 338 (2022), 128--226.

[104] Didier Bresch, Marguerite Gisclon, Ingrid Lacroix-Violet, and Alexis Vasseur, On the Exponential decay for Compressible Navier-Stokes-Korteweg equations with a Drag Term, [pdf]
J. Math. Fluid Mech. 24 (2022), no. 1, Paper No. 11, 16 pp.

[103] Geng Chen, Sam G. Krupa, and Alexis F. Vasseur, Uniqueness and weak-BV stability for 2×2 conservation laws, [pdf]
to appear in ARMA.

[102] R. M. Chen, A.Vasseur, and Ch. Yu, Global ill-posedness for a dense set of initial data to the Isentropic system of gas dynamics. [Arxiv]
Adv. Math. 393 (2021), Paper No. 108057, 46 pp.

[101] A. Vasseur, and J. Yang, Second derivatives estimate of suitable solutions to the 3D Navier-Stokes equations, [Arxiv]
Arch. Ration. Mech. Anal. 241 (2021), no. 2, 683--727.

[100] L. Lafleche, A. Vasseur, and Misha Vishik, Instability for Axisymmetric Blow-up Solutions to Incompressible Euler Equations, [Arxiv]
J. Math. Pures Appl. (9) 155 (2021), 140--154.

[99] M.-J. Kang, A. Vasseur, Y. Wang, Uniqueness of a planar contact discontinuity for 3D compressible Euler system in a class of zero dissipation limits from Navier-Stokes-Fourier system [Arxiv]
Comm. Math. Phys. 384 (2021), no. 3, 1751--1782.

[98] F. Golse, M. P. Gualdani, C. Imbert, A. Vasseur, Partial Regularity in Time for the Space Homogeneous Landau Equation with Coulomb Potential, [pdf]
To appear in Annales scientifiques de l'Ecole Normale Superieure.

[97] D. Bresch, A. Vasseur, Ch. Yu, Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities, [pdf]
J. Eur. Math. Soc. (JEMS) 24 (2022), no. 5, 1791--1837.

[96] M.-J. Kang, A. Vasseur, Uniqueness and stability of entropy shocks to the isentropic Euler system in a class of inviscid limits from a large family of Navier-Stokes systems, [pdf]
Invent. Math. 224 (2021), no. 1, 55--146.

[95] K. Choi, M.-J. Kang, A. Vasseur, Global well-posedness of large perturbations of traveling waves in a hyperbolic-parabolic system arising from a chemotaxis model [pdf]
J. Math. Pures Appl. (9) 142 (2020), 266--297.

[94] A. Vasseur, M. Vishik, Blow-up solutions to 3D Euler are hydrodynamically unstable, [pdf]
Comm. Math. Phys. 378 (2020), no. 1, 557--568.

[93] S. Akopian, M.-J. Kang, A. Vasseur, Inviscid limit to the shock waves for the fractal Burgers equation, [ pdf]
Commun. Math. Sci. 18 (2020), no. 6, 1477--1491.

[92] M. Novack, A. Vasseur, Classical Solutions for the 3D Quasi-Geostrophic System on a Bounded Domain, [pdf]
Phys. D 404 (2020), 132362.

[91] K.Choi, M.-J. Kang, Y.-S. Kwon, A. Vasseur, Contraction for large perturbations of travelling waves in a hyperbolic-parabolic system arising from a Chemotaxis model, [pdf]
Math. Models Methods Appl. Sci. 30 (2020), no. 2, 387--437.

[90] L. Stokols, A. Vasseur, Holder Regularity up to the Boundary for Critical SQG on Bounded Domains, [pdf]
Arch. Ration. Mech. Anal. 236 (2020), no. 3, 1543--1591.

[89] M.-J. Kang, A. Vasseur, Global smooth solutions for 1D barotropic Navier-Stokes equations with a large class of degenerate viscosities, [pdf]
J. Nonlinear Sci. 30 (2020), no. 4, 1703--1721.

[88] S. Krupa, A. Vasseur, Stability and uniqueness for piecewise smooth solutions to Burgers-Hilbert among a large class of solutions, [pdf]
SIAM J. Math. Anal. 52 (2020), no. 3, 2491--2530.

[87] M. Novack, A. Vasseur, The inviscid 3D quasi-geostrophic system on bounded domains, [pdf]
Arch. Ration. Mech. Anal. 235 (2020), no. 2, 973--1010.

[86] M.-J. Kang, A. Vasseur, Contraction property for large perturbations of shocks of the barotropic Navier-Stokes systems, [pdf]
J. Eur. Math. Soc. (JEMS) 23 (2021), no. 2, 585--638.

[85] S. Krupa, A. Vasseur, Single Entropy Condition for Burgers Equation via the Relative Entropy Method, [pdf]
J. Hyperbolic Differ. Equ. 16 (2019), no. 1, 157--191.

[84] M.-J. Kang, A. Vasseur, and Y. Wang, L^2 contraction of large planar shock waves for multi-dimensional scalar viscous conservation laws, [pdf]
J. Differential Equations 267 (2019), no. 5, 2737--2791.

[83] C. Caputo, Th. Goudon, A. Vasseur, Solutions of the 4-species quadratic reaction-diffusion system are bounded and $C^\infty$-smooth, in any space dimension, [pdf]
Anal. PDE 12 (2019), no. 7, 1773--1804.

[82] Th. Goudon, A. Vasseur, Statistical stability for transport in random media. [pdf]
Multiscale Model. Simul. 17 (2019), no. 1, 507--530.

[81] F. Golse, C. Imbert, C. Mouhot, and A. Vasseur, Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation, [pdf]
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), no. 1, 253--295.

[80] A. Vasseur, H. Wen, and Ch. Yu, Global weak solution to the viscous two-phase model with finite energy, [pdf]
J. Math. Pures Appl. (9) 125 (2019), 247--282.

[79] M. Novack and A. Vasseur, Global in time classical solutions to the 3D quasigeostrophic system for large initial data, [pdf]
Comm. Math. Phys. 358 (2018), no. 1, 237--267.

[78] C. Chainais-Hillairet, B. Merlet, and A. Vasseur, Positive lower bound for the numerical solution of a convection-diffusion equation. [pdf]
Finite volumes for complex applications VIII -- methods and theoretical aspects, 331--339, Springer Proc. Math. Stat., 199, Springer, Cham, 2017.

[77] L. Stokols and A. Vasseur, De Giorgi techniques applied to Hamilton-Jacobi equations with unbounded right-hand side, [pdf]
Commun. Math. Sci. 16 (2018), no. 6, 1465--1487.

[76] I. Lacroix-Violet and A. Vasseur, Global weak solutions to the compressible quantum Navier-Stokes equation and its semi-classical limit, [pdf]
J. Math. Pures Appl. (9) 114 (2018), 191--210.
An error was present in the statement of Lemma 3.1 in the published version. The version available on this webpage fixes this oversight, and the induced ripples through the paper.

[75] Ch.H. Chan and A. Vasseur, De Giorgi techniques applied to the Holder regularity of solutions to Hamilton-Jacobi equations, [pdf]
From particle systems to partial differential equations, 117--137, Springer Proc. Math. Stat., 209, Springer, Cham, 2017.

[74] M. Allen, L. Caffarelli and A. Vasseur, Porous medium flow with both fractional potential pressure and fractional time derivative, [pdf]
Chin. Ann. Math. Ser. B 38 (2017), no. 1, 45--82.

[73] A. Vasseur and L. Yao, Nonlinear stability of viscous shock wave to one-dimensional compressible isentropic Navier-Stokes equations with density dependent viscous coefficients, [pdf]
Commun. Math. Sci. 14 (2016), no. 8, 2215--2228.

[72] M.J. Kang and A. Vasseur, Criteria on contractions for entropic discontinuities of systems of conservation laws, [pdf]
Arch. Ration. Mech. Anal. 222 (2016), no. 1, 343--391.

[71] A. Vasseur and Ch. Yu, Existence of global weak solutions for 3D degenerate compressible Navier--Stokes equations. [pdf]
Invent. Math. 206 (2016), no. 3, 935--974.

[70] A. Vasseur and Ch. Yu, Global weak solutions to compressible quantum Navier-Stokes equations with damping. [pdf]
SIAM J. Math. Anal. 48 (2016), no. 2, 1489--1511.

[69] M. Allen, L. Caffarelli and A. Vasseur, A parabolic problem with a fractional time derivative, [pdf]
Arch. Ration. Mech. Anal. 221 (2016), no. 2, 603--630.

[68] D. Serre and A. Vasseur, The relative entropy method for the stability of intermediate shock waves ; the rich case, [pdf]
Discrete Contin. Dyn. Syst. 36 (2016), no. 8, 4569--4577.

[67] A. Vasseur and Y. Wang, The inviscid limit to a contact discontinuity for the compressible Navier-Stokes-Fourier system using the relative entropy method. [pdf]
SIAM J. Math. Anal. 47 (2015), no. 6, 4350--4359.

[66] M.J. Kang and A. Vasseur, L^2-contraction for shock waves of scalar viscous conservation laws, [pdf]
Ann. Inst. H. Poincare Anal. Non Lineaire 34 (2017), no. 1, 139--156.

[65] D. Serre and A. Vasseur, About the relative entropy method for hyperbolic systems of conservation laws, [pdf]
Contemporary Mathematics 658 (2016), 237--248.

[64] A. Vasseur, Relative entropy and contraction for extremal shocks of Conservation Laws up to a shift [pdf]
Recent advances in partial differential equations and applications, 385--404, Contemp. Math., 666, Amer. Math. Soc., Providence, RI, 2016.

[63] A. Vasseur, The De Giorgi method for elliptic and parabolic equations and some applications, [pdf]
Part 4, 195--222, Morningside Lect. Math., 4, Int. Press, Somerville, MA, 2016.

[62] Th. Goudon, A. Vasseur, On a Model for Mixture Flows: Derivation, Dissipation and Stability Properties, [pdf]
Arch. Ration. Mech. Anal. 220 (2016), no. 1, 1--35.

[61] M. Puel and A. Vasseur, Global weak solutions to the inviscid 3D quasi-geostrophic equation, [pdf]
Comm. Math. Phys. 339 (2015), no. 3, 1063--1082.

[60] E.Feireisl, O.Kreml and A. Vasseur, Stability of the isentropic Riemann solutions of the full multidimensional Euler system, [pdf]
SIAM J. Math. Anal. 47 (2015), no. 3, 2416--2425.

[59] M. J.Kang, and A. Vasseur, Asymptotic analysis of Vlasov-type equations under strong local alignment regime, [pdf]
Math. Models Methods Appl. Sci. 25 (2015), no. 11, 2153--2173.

[58] K.Choi, A. Vasseur, Short-time stability of scalar viscous shocks in the inviscid limit by the relative entropy method. [pdf]
SIAM J. Math. Anal. 47 (2015), no. 2, 1405--1418.

[57] Y.-S. Kwon, A. Vasseur, Asymptotic limit to a shock for BGK models using the relative entropy method [pdf]
Nonlinearity 28 (2015) 531--543.

[56] D. Serre, A. Vasseur, L2-type contraction for systems of conservation laws, [pdf]
Journal de l'École polytechnique -- Mathématiques, 1 (2014), p. 1-28.

[55] K. Choi, A. Vasseur, Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations [pdf]
Annales de l'institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 899 - 945.

[54] E. Baer, A. Vasseur, A Bound from Below on the Temperature for the Navier-Stokes-Fourier System. [pdf]
SIAM J. Math. Anal. 45 (2013), no. 4, 2046-2063.

[53] A. Vasseur, A rigorous derivation of the coupling of a kinetic equation and Burgers' equation. [pdf]
Arch. Ration. Mech. Anal. 206 (2012), no. 1, 1-30.

[52] B. Perthame, A. Vasseur, Regularization in Keller-Segel type systems and the De Giorgi method. [pdf]
Commun. Math. Sci. 10 (2012), no. 2, 463-476.

[51] L. Caffarelli, Ch.-H. Chan, A. Vasseur, Regularity theory for nonlinear integral operators. [pdf]
J. Amer. Math. Soc. 24 (2011), no. 3, 849--869.

[51] L. Caffarelli, F. Golse, Y. Guo, C. Kenig, A. Vasseur, Nonlinear partial differential equations. Selected lecture notes from the School "Topics in PDE's and Applications 2008."
Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser/Springer Basel AG, Basel, 2012. viii+149 pp.

[49] L. Caffarelli, A. Vasseur, The De Giorgi method for nonlocal fluid dynamics. Nonlinear partial differential equations,
1--38, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer Basel AG, Basel, 2012.

[48] Nicholas Leger, A. Vasseur, Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non BV perturbations. [pdf]
Arch. Ration. Mech. Anal. 201 (2011), no. 1, 271--302.

[47] C. Bjorland, A. Vasseur, Weak in Space, Log in Time Improvement of the Ladyzenskaja-Prodi-Serrin Criteria. [pdf],
J. Math. Fluid Mech. 13 (2011), no. 2, 259-269.

[46] B. Ducomet, S. Necasova, A. Vasseur, "On spherically symmetric motions of a viscous compressible barotropic and self-graviting gas" [pdf],
J. Math. Fluid Mech. 13 (2011), no. 2, 191?211.

[45] E. Feireisl, A. Vasseur, New perspectives in fluid dynamics: Mathematical analysis of a model proposed by Howard Brenner. [pdf],
New directions in mathematical fluid mechanics, 153 --179, Adv. Math. Fluid Mech., Birkhauser Verlag, Basel, 2010.

[44] B. Ducomet, and S. Necasova, A. Vasseur, On global motions of a compressible barotropic and selfgravitating gas with density-dependent viscosities. [ pdf]
Z. Angew. Math. Phys. 61 (2010), no. 3, 479--491.

[43] L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. [pdf],
Annals of Math., Vol. 171 (2010), No. 3, 1903-1930.

[42] A. Vasseur, Higher derivatives estimate for the 3D Navier-Stokes equation. [pdf],
Annales de l'institut Henri Poincare (C) Non Linear Analysis, 27(5) (2010), 1189-1204.

[41] L. Caffarelli, A. Vasseur, The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics.
Discrete and Continuous Dynamical Systems - Series S (DCDS-S), 3 (3) (2010), 409 - 427.

[40] Th. Goudon, A.Vasseur, Regularity analysis for systems of reaction-diffusion equations. [pdf],
Ann. Sci. Ec. Norm. Super. (4) 43 (2010), no. 1, 117--14.

[39] M. Bostan, M. Gamba, Th. Goudon, A. Vasseur, Boundary value problems for the stationary Vlasov-Boltzmann-Poisson equation. [pdf],
Indiana Univ. Math. J. 59 (2010), no. 5, 1629?1660.

[38] J. Evans, C. Michoski, P. Schmitz, A. Vasseur, "A discontinuous Galerkin method for viscous compressible multifluids."
J. Comput. Phys. 229 (2010), no. 6, 2249--2266.

[37] C.Caputo, A. Vasseur, Global regularity of solutions to systems of reaction-diffusion wity sub-quadratic growth in any dimension. [pdf],
Comm. Partial Differential Equations 34 (2009), no. 10-12, 1228--1250.

[36] J. Evans, C. Michoski, P. Schmitz, A. Vasseur, "Quantum Hydrodynamics with Trajectories: The Nonlinear Conservation Form Mixed/Discontinuous Galerkin Method with Applications in Chemistry" [pdf],
J. Comput. Phys. 228 (2009), no. 23, 8589--8608.

[35] I. Gamba, A. Jungel, A. Vasseur, Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations. [pdf]
J. Differential Equations 247 (2009), no. 11, 3117--3135.

[34] F.Berthelin, A.E.Tzavaras, A. Vasseur, From discrete velocity Boltzmann equations to gas dynamics before shocks. [pdf],
J. Stat. Phys. 135 (2009), no. 1, 153--173.

[33] A. Mellet, A. Vasseur, L^p estimates for quantities advected by a compressible flow. [pdf],
J. Math. Anal. Appl. 355 (2009), no. 2, 548--563.

[32] N.Leger, A. Vasseur, Study of a generalized fragmentation model for sprays. [pdf],
J. Hyperbolic Differ. Equ. 6 (2009), no. 1, 185--206.

[31] C.Michoski, A. Vasseur, Existence and Uniqueness of strong solutions for a compressible multiphase Navier-Stokes Miscible Fluid-Flow Problem in dimension 1. [pdf]
Math. Models Methods Appl. Sci. 19 (2009), no. 3, 443--476.

[30] A. Vasseur, Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity. [pdf]
Appl. Math. 54 (2009), no. 1, 47--52.

[29] A. Mellet, A. Vasseur, A bound from below for the temperature in compressible Navier-Stokes equations. [pdf]
Monatsh. Math. 157 (2009), no. 2, 143--161.

[28] A. Mellet, A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/Compressible Navier-Stokes system of equations. [pdf]
Comm. Math. Phys. 281 (2008), no. 3, 573--596.

[27] A. Vasseur, Recent results on hydrodynamic limits.
Handbook of differential equations: evolutionary equations. Vol. IV, 323--376, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008.

[26] A. Mellet and A. Vasseur, Existence and Uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations. [pdf]
SIAM J. Math. Anal. 39 (2007/08), no. 4, 1344--1365.

[25] Ch.-H. Chan, A. Vasseur, Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations. [pdf]
Methods Appl. Anal. 14 (2007), no. 2, 197--212.

[24] A. Mellet and A. Vasseur, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations. [pdf]
Math. Models Methods Appl. Sci. 17 (2007), no. 7, 1039--1063.

[23] A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations. [pdf]
Comm. Partial Differential Equations 32 (2007), no. 1-3, 431--452.

[22] Y.-S. Kwon and A. Vasseur, Strong traces for solutions to scalar conservation laws with general flux. [pdf]
Arch. Ration. Mech. Anal. 185 (2007), no. 3, 495--513.

[21] A. Vasseur, A new proof of partial regularity of solutions to Navier-Stokes equations. [pdf]
NoDEA Nonlinear Differential Equations Appl. 14 (2007), no. 5-6, 753--785.

[20] A. Mellet and A. Vasseur, Homogenization of a nonlinear transport equation, [pdf]
Asymptot. Anal. 51 (2007), no. 2, 157--166.

[19] F. Berthelin and A. Vasseur, From kinetic equations to multidimensional isentropic dynamics before shocks. [pdf]
SIMA Vol. 36 Number 6, pp. 180-183. 2005.

[18] Th. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. Parts I: Light particles regime.
Indiana Univ. Math. J. 53 No. 6 (2004), 1495--1516.

[17] Th. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. Parts II: fine particles regime.
Indiana Univ. Math. J. 53 No. 6 (2004), 1517--1536.

[16] G. Loeper, and A.Vasseur, Electric turbulence in a plasma subject to a strong magnetic field. [pdf]
Asymptotic Analysis, Vol. 40, Number 1/2004 pages 51-65.

[15] F.Poupaud and A.Vasseur, Classical and quantum transport in random media. [pdf]
J. Math. pures Appl.(9) 82 (2003), no. 6, 711--748.

[14] R.Botchorishvili, B.Perthame and A.Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources. [pdf]
Math. of Comp. 72 (2003), no.241, 131--157.

[13] T. Horsin, S. Mischler and A. Vasseur, On the convergence of numerical schemes for the Boltzmann equation. [pdf]
Ann. Inst. H. Poincare, Anal. Non Lin\'eaire 20 (2003), no.5, 731--758.

[12] A.Vasseur, Well-posedness of scalar conservation laws with singular sources. [pdf]
Methods Appl. Anal. 9 (2002), no.2, 291--312.

[11] J.-F. Collet, T. Goudon, F. Poupaud and A. Vasseur, The Becker-Doring system and its Lifshitz-Slyozov limit. [pdf]
SIAM Journal of Applied Math., 2002, vol 62, 5, p. 1488--1500.

[10] J.-F. Collet, T. Goudon and A. Vasseur, Some remarks on large-time asymptotic of the Lifshitz-Slyozov equations.
J. Stat. Phys., 2002, vol. 108, no 1-2, 341--359.

[9] A. Vasseur, Strong traces for solutions to multidimensional scalar conservation laws. [pdf]
Archive for Rational Mechanics and Analysis, 160 (2001), no. 3, 181--193.

[8] A. Vasseur, Existence and properties of semi-discrete shock profiles for the isentropic gas dynamic system with $\gamma=3$. [pdf]
SIAM J. Numer. Anal. 38 (2001), no. 6, 1886--1901 (electronic).

[7] A.Vasseur, Convergence of a semi-discrete kinetic scheme for the isentropic gas dynamic system with $\gamma=3$.
Indiana University Mathematics Journal 48 (1999), no. 1, 347--364.

[6] A. Vasseur, Time regularity for the system of isentropic gas dynamics with $\gamma=3$.
Communications in Partial Differential Equations (1999) 11-12, 1987--1997.

[5] A. Vasseur, Kinetic semidiscretization of scalar conservation laws and convergence by using averaging lemmas. [pdf]
SIAM J. Numer. Anal. 36 (1999), no. 2, 465--474 (electronic).

Published articles in non refereed Journal:

[4] J.-F. Collet, T. Goudon, S. Hariz, F. Poupaud, A. Vasseur. Some recent results on the kinetic theory of phase transitions.
Transport in transition regimes (Minneapolis,MN,2000), 103--120, IMA Vol. Math. Appl., 135, Springer, New York, 2004.

[3] A.Vasseur, Interface cinetique/fluide: un modele siplifie. (French) [kinetic/fluid interface: a simplified model]
Seminaire: Equations aux Derivees Partielles 2002--2003, Exp. No. III, 15pp., Ecole Poytech., Palaiseau, 2003.

[2] Th. Goudon, P.-E. Jabin, A. Vasseur, Limites hydrodynamiques pour les equations de Vlasov-Stokes. (French)
[Hydrodynamic limits for the Vlasov-Stokes equations]
Journees Equations aux Derivees partielles (Forges-les-eaux, 2002) Exp.No.VII, 15pp.,Univ.Nantes, 2002.

[1] D. Besnard, F. Ducros, Ph. Loreaux, S. Mimouni and A. Vasseur, Turbulent mixing modeling and simulation.
Proceedings of the Fifth International Workshop on Compressible Turbulent Mixing (Stony Brook, NY, 1995), 294--302.