Self-similar solutions for the LSW model with encounters

The LSW model with encounters has been suggested by Lifshitz and Slyozov as a regularization of their classical mean-field model for domain coarsening to obtain universal self-similar long-time behavior. We rigorously establish that an exponentially decaying self-similar solution to this model exist, and show that this solutions is isolated in a certain function space. Our proof relies on setting up a suitable fixed-point problem in an appropriate function space and careful asymptotic estimates of the solution to a corresponding homogeneous problem.Comment: 22 page

On a thermodynamically consistent modification of the Becker-Doering equations

Recently, Dreyer and Duderstadt have proposed a modification of the Becker--Doering cluster equations which now have a nonconvex Lyapunov function. We start with existence and uniqueness results for the modified equations. Next we derive an explicit criterion for the existence of equilibrium states and solve the minimization problem for the Lyapunov function. Finally, we discuss the long time behavior in the case that equilibrium solutions do exist

Self-similar solutions with fat tails for Smoluchowski's coagulation equation with locally bounded kernels

The existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation has so far only been established for the solvable and the diagonal kernel. In this paper we prove the existence of such self-similar solutions for continuous kernels $K$ that are homogeneous of degree $\gamma \in [0,1)$ and satisfy $K(x,y) \leq C (x^{\gamma} + y^{\gamma})$. More precisely, for any $\rho \in (\gamma,1)$ we establish the existence of a continuous weak self-similar profile with decay $x^{-(1{+}\rho)}$ as $x \to \infty$

On thermodynamically consistent Stefan problems with variable surface energy

A thermodynamically consistent two-phase Stefan problem with temperature-dependent surface tension and with or without kinetic undercooling is studied. It is shown that these problems generate local semiflows in well-defined state manifolds. If a solution does not exhibit singularities, it is proved that it exists globally in time and converges towards an equilibrium of the problem. In addition, stability and instability of equilibria is studied. In particular, it is shown that multiple spheres of the same radius are unstable if surface heat capacity is small; however, if kinetic undercooling is absent, they are stable if surface heat capacity is sufficiently large.Comment: To appear in Arch. Ration. Mech. Anal. The final publication is available at Springer via http://dx.doi.org/10.1007/s00205-015-0938-y. arXiv admin note: substantial text overlap with arXiv:1101.376

Alteration in elemental and functional composition of heated peat humic acids

The article analyzes the effect of thermal modification of different-type peat on the alteration of elemental and functional composition of peat humic acids. Based on the data of IR-spectra and readings of electron paramagnetic resonance, structural alterations are identified. It is shown that the impact of peat characteristics on humic acids is preserved after thermal modification. It is revealed that the strongest alteration of humic acid composition and properties caused by peat heating are typical to humic acid samples extracted from the peat with low decomposition degree

Asymptotics of self-similar solutions to coagulation equations with product kernel

We consider mass-conserving self-similar solutions for Smoluchowski's coagulation equation with kernel $K(\xi,\eta)= (\xi \eta)^{\lambda}$ with $\lambda \in (0,1/2)$. It is known that such self-similar solutions $g(x)$ satisfy that $x^{-1+2\lambda} g(x)$ is bounded above and below as $x \to 0$. In this paper we describe in detail via formal asymptotics the qualitative behavior of a suitably rescaled function $h(x)=h_{\lambda} x^{-1+2\lambda} g(x)$ in the limit $\lambda \to 0$. It turns out that $h \sim 1+ C x^{\lambda/2} \cos(\sqrt{\lambda} \log x)$ as $x \to 0$. As $x$ becomes larger $h$ develops peaks of height $1/\lambda$ that are separated by large regions where $h$ is small. Finally, $h$ converges to zero exponentially fast as $x \to \infty$. Our analysis is based on different approximations of a nonlocal operator, that reduces the original equation in certain regimes to a system of ODE

Thalamocortical Connectivity Correlates with Phenotypic Variability in Dystonia

Dystonia is a brain disorder characterized by abnormal involuntary movements without defining neuropathological changes. The disease is often inherited as an autosomal-dominant trait with incomplete penetrance. Individuals with dystonia, whether inherited or sporadic, exhibit striking phenotypic variability, with marked differences in the somatic distribution and severity of clinical manifestations. In the current study, we used magnetic resonance diffusion tensor imaging to identify microstructural changes associated with specific limb manifestations. Functional MRI was used to localize specific limb regions within the somatosensory cortex. Microstructural integrity was preserved when assessed in subrolandic white matter regions somatotopically related to the clinically involved limbs, but was reduced in regions linked to clinically uninvolved (asymptomatic) body areas. Clinical manifestations were greatest in subjects with relatively intact microstructure in somatotopically relevant white matter regions. Tractography revealed significant phenotype-related differences in the visualized thalamocortical tracts while corticostriatal and corticospinal pathways did not differ between groups. Cerebellothalamic microstructural abnormalities were also seen in the dystonia subjects, but these changes were associated with genotype, rather than with phenotypic variation. The findings suggest that the thalamocortical motor system is a major determinant of dystonia phenotype. This pathway may represent a novel therapeutic target for individuals with refractory limb dystonia

Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction

We study a singular-limit problem arising in the modelling of chemical reactions. At finite {\epsilon} > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/{\epsilon}, and in the limit {\epsilon} -> 0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savar\'e, and Veneroni, SIAM Journal on Mathematical Analysis, 42(4):1805-1825, 2010, using the linear structure of the equation. In this paper we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the {\epsilon}-problem converge to a solution of the limiting problem.Comment: Added two sections, corrected minor typos, updated reference