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