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.
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.
2014
Alexei Borodin, Ivan Corwin, Patrik L. Ferrari and Balint Vető Height fluctuations for the stationary KPZ equation Math. Phys. Anal. Geom., 18(1, Art. 20): 1-95 2014 http://arxiv.org/abs/1407.6977
Abstract: We compute the one-point probability distribution for the stationary KPZ equation (i.e. initial data $\mathcal{H}(0,X)=B(X)$, for $B(X)$ a two-sided standard Brownian motion) and show that as time $T$ goes to infinity, the fluctuations of the height function $\mathcal{H}(T,X)$ grow like $T^{1/3}$ and converge to those previously encountered in the study of the stationary totally asymmetric simple exclusion process, polynuclear growth model and last passage percolation. The starting point for this work is our derivation of a Fredholm determinant formula for Macdonald processes which degenerates to a corresponding formula for Whittaker processes. We relate this to a polymer model which mixes the semi-discrete and log-gamma random polymers. A special case of this model has a limit to the KPZ equation with initial data given by a two-sided Brownian motion with drift $β$ to the left of the origin and $b$ to the right of the origin. The Fredholm determinant has a limit for $β>b$, and the case where $β=b$ (corresponding to the stationary initial data) follows from an analytic continuation argument.
Miguel 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.
Miguel Escobedo and Juan J. L. Velázquez On the Blow Up and Condensation of Supercritical Solutions of the Nordheim Equation for Bosons Communications in Mathematical Physics, 330(1): 331-365 2014 http://dx.doi.org/10.1007/s00220-014-2034-9
Abstract: In this paper we prove that the solutions of the isotropic, spatially homogeneous Nordheim equation for bosons with bounded initial data blow up in finite time in the L ∞ norm if the values of the energy and particle density are in the range of values where the corresponding equilibria contain a Dirac mass. We also prove that, in the weak solutions, whose initial data are measures with values of particle and energy densities satisfying the previous condition, a Dirac measure at the origin forms in finite time.
Barbara Niethammer and Juan J. L. Velázquez Exponential tail behaviour of self-similar solutions to Smoluchowski's coagulation equation Communications in Partial Differential Equations, 39(12): 2314-2350 2014 http://dx.doi.org/10.1080/03605302.2014.918143
Abstract: We consider self-similar solutions with finite mass to Smoluchowski's coagulation equation for rate kernels that have homogeneity zero but are possibly singular such as Smoluchowski's original kernel. We prove pointwise exponential decay of these solutions under rather mild assumptions on the kernel. If the support of the kernel is sufficiently large around the diagonal we also proof that \( \lim_{x\rightarrow\infty}\frac{1}{x}\log\left(\frac{1}{f(x)}\right)\) exists. In addition we prove properties of the prefactor if the kernel is uniformly bounded below.
Barbara Niethammer and Juan J. L. Velázquez Uniqueness of self-similar solutions to Smoluchowski's coagulation equations for kernels that are close to constant J. Stat. Phys., 157(1): 158-181 2014 http://dx.doi.org/10.1007/s10955-014-1070-3
Abstract: We consider self-similar solutions to Smoluchowski's coagulation equation for kernels K=K(x,y) that are homogeneous of degree zero and close to constant in the sense that \[ -\varepsilon \leq K(x,y)-2 \leq \varepsilon \Big(\Big(\frac{x}{y}\Big)^{\alpha} + \Big(\frac{y}{x}\Big)^{\alpha}\Big) \] for \(\alpha \in [0,1)\). We prove that self-similar solutions with given mass are unique if \(\varepsilon\) is sufficiently small which is the first such uniqueness result for kernels that are not solvable. Our proof relies on a contraction argument in a norm that measures the distance of solutions with respect to the weak topology of measures.
Yoshio Sugiyama, Yohei Tsutsui and Juan J. L. Velázquez Global solutions to a chemotaxis system with non-diffusive memory J. Math. Anal. Appl., 410(2): 908-917 2014 http://dx.doi.org/10.1016/j.jmaa.2013.08.065
Abstract: In this article, an existence theorem of global solutions with small initial data belonging to L1∩Lp, (n