| 2017Marco Bonacini, B. Niethammer and J.J. L. Velázquez
Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity smaller than one
2017
https://arxiv.org/abs/1704.08905
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| Marco Bonacini, B. Niethammer and J.J. L. Velázquez
Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity one
2017
https://arxiv.org/abs/1612.06610
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| Carlota M. Cuesta, Hans Knüpfer and J.J. L. Velázquez
Self-similar lifting and persistent touch-down points in the thin film equation
2017
https://arxiv.org/abs/1708.00243
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| Alessia Nota, Sergio Simonella and Juan J.L. Velázquez
On the theory of the Lorentz gases with long range interactions
2017
https://arxiv.org/abs/1707.04193
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| J.J. L. Velázquez and Raphael Winter
From a non-Markovian system to the Landau equation
2017
https://arxiv.org/abs/1707.07544
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| 2016Michael Herrmann, Barbara Niethammer and Juan J. L. Velázquez
Instabilities and oscillations in coagulation equations with kernels of homogeneity one
2016
http://arxiv.org/abs/1606.09405
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| Richard Höfer and Juan JL Velázquez
The Method of Reflections, Homogenization and Screening for Poisson and Stokes Equations in Perforated Domains
2016
http://arxiv.org/abs/1603.06750
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| P. Laurençot, B. Niethammer and J. J. L. Velázquez
Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel
2016
http://arxiv.org/abs/1603.02929
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| A. Nota and J.J.L. Velázquez
On the growth of a particle coalescing in a Poisson distribution of obstacles
2016
http://arxiv.org/abs/1608.08118
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| Alan D. Rendall and Juan J. L. Velázquez
Veiled singularities for the spherically symmetric massless Einstein-Vlasov system
2016
http://arxiv.org/abs/1604.06576
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| 2015Carlota M. Cuesta, Maria Calle and Juan J. L. Velázquez
Interfaces determined by capillarity and gravity in a two-dimensional porous medium
2015
http://arxiv.org/abs/1505.03676
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| Juhi Jang, Juan J. L. Velázquez and Hyung Ju Hwang
On the structure of the singular set for the kinetic Fokker-Planck equations in domains with boundaries
2015
http://arxiv.org/abs/1509.03366
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| Arthur H. M. Kierkels and Juan J. L. Velázquez
On the transfer of energy towards infinity in the theory of weak turbulence for the nonlinear Schrödinger equation
J. Stat. Phys., 159(3): 668-712 2015
http://dx.doi.org/10.1007/s10955-015-1194-0
Abstract: We study the mathematical properties of a kinetic equation which describes the long time behaviour of solutions to the weak turbulence equation associated to the cubic nonlinear Schrödinger equation.In particular, we give a precise definition of weak solutions and prove global existence of solutions for all initial data with finite mass. We also prove that any nontrivial initial datum yields the instantaneous onset of a condensate, i.e.~a Dirac mass at the origin for any positive time. Furthermore we show that the only stationary solutions with finite total measure are Dirac masses at the origin. We finally construct solutions with finite energy, which is transferred to infinity in a self-similar manner. |
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| Barbara Niethammer, Sebastian Throm and Juan J. L. Velázquez
A revised proof of uniqueness of self-similar profiles to Smoluchowski's coagulation equation for kernels close to constant
2015
http://arxiv.org/abs/1510.03361
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| Barbara Niethammer, Juan J. L. Velázquez and Michael Helmers
Mathematical analysis of a coarsening model with local interactions
2015
http://arxiv.org/abs/1509.04917
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| Barbara Niethammer, Sebastian Throm and Juan J. L. Velázquez
Self-similar solutions with fat tails for Smoluchowski's coagulation equation with singular kernels
Ann. I. H. Poincaré - AN 2015
http://dx.doi.org/10.1016/j.anihpc.2015.04.002
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| Juan J. L. Velázquez and Arthur H. M. Kierkels
On self-similar solutions to a kinetic equation arising in weak turbulence theory for the nonlinear Schrödinger equation
2015
http://arxiv.org/abs/1511.01292
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| Juan J. L. Velázquez and Miguel Escobedo
On the theory of Weak Turbulence for the Nonlinear Schrödinger Equation
Memoirs of the AMS, 238 2015
http://dx.doi.org/10.1090/memo/1124
Abstract: We study the Cauchy problem for a kinetic equation arising in the weak turbulence theory for the cubic nonlinear Schrödinger equation. We define suitable concepts of weak and mild solutions and prove local and global well posedness results. Several qualitative properties of the solutions, including long time asymptotics, blow up results and condensation in finite time are obtained. We also prove the existence of a family of solutions that exhibit pulsating behavior. |
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| Juan J. L. Velázquez and Jogia Bandyopadhyay
Blow-up rate estimates for the solutions of the bosonic Boltzmann-Nordheim equation
J. Math. Phys., 56(Art. 063302): 1-29 2015
http://arxiv.org/abs/1411.5460
Abstract: In this paper, we study the behavior of a class of mild solutions of the homogeneous and isotropic bosonic Boltzmann-Nordheim equation near the blow-up. We obtain some estimates on the blow-up rate of the solutions and prove that, as long as a solution is bounded above by the critical singularity 1x1x (the equilibrium solutions behave like this power law near the origin), it remains bounded in the uniform norm. In Sec. III of the paper, we prove a local existence result for a class of measure-valued mild solutions, which is of independent interest and which allows us to solve the Boltzmann-Nordheim equation for some classes of unbounded densities. |
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| 2014Miguel Escobedo and Juan J. L. Velázquez
Finite time blow-up and condensation for the bosonic Nordheim equation
Inventiones mathematicae, 200(3): 761-847 2014
http://dx.doi.org/10.1007/s00222-014-0539-7
Abstract: The homogeneous bosonic Nordheim equation is a kinetic equation describing the dynamics of the distribution of particles in the space of moments for a homogeneous, weakly interacting, quantum gas of bosons. We show the existence of classical solutions of the homogeneous bosonic Nordheim equation that blow up in finite time. We also prove finite time condensation for a class of weak solutions of the kinetic equation. |
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