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Barbara Niethammer

Publications and source records attributed to Barbara Niethammer.

At least 37 records · Page 2Linked to original sources
arXiv2018

Self-similar gelling solutions for the coagulation equation with diagonal kernel

We consider Smoluchowski's coagulation equation in the case of the diagonal kernel with homogeneity $γ>1$. In this case the phenomenon of gelation occurs and solutions lose mass at some finite time. The problem of the existence of self-similar solutions involves a free parameter $b$, and one expects that a physically relevant solution (i.e. nonnegative and with sufficiently fast decay at infinity) exists for a single value of $b$, depending on the homogeneity $γ$. We prove this picture rigorously for large values of $γ$. In the general case, we discuss in detail the behaviour of solutions to the self-similar equation as the parameter $b$ changes.

math.AP↗
arXiv2018

Self-similar asymptotic behavior for the solutions of a linear coagulation equation

In this paper we consider the long time asymptotics of a linear version of the Smoluchowski equation which describes the evolution of a tagged particle moving at constant speed in a random distribution of fixed particles. The volumes $v$ of the particles are independently distributed according to a probability distribution which decays asymptotically as a power law $v^{-σ}$. The validity of the equation has been rigorously proved in \cite{NoV} for values of the exponent $σ>3$. The solutions of this equation display a rich structure of different asymptotic behaviours according to the different values of the exponent $σ$. Here we show that for $\frac{5}{3}<σ<2$ the linear Smoluchowski equation is well posed and that there exists a unique self-similar profile which is asymptotically stable.

math.AP↗
arXiv2018

Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity smaller than one

We prove the existence of a one-parameter family of self-similar solutions with time-dependent tails for Smoluchowski's coagulation equation, for a class of rate kernels $K(x,y)$ which are homogeneous of degree $γ\in(-\infty,1)$ and satisfy $K(x,1)\sim x^{-a}$ as $x\to 0$, for $a=1-γ$. In particular, for small values of a parameter $ρ>0$ we establish the existence of a positive self-similar solution with finite mass and asymptotics $A(t)x^{-(2+ρ)}$ as $x\to\infty$, with $A(t)\simρt^\fracρ{1-γ}$.

math.AP↗
arXiv2018

Mass transport in Fokker-Planck equations with tilted periodic potential

We consider Fokker-Planck equations with tilted periodic potential in the subcritical regime and characterize the spatio-temporal dynamics of the partial masses in the limit of vanishing diffusion. Our convergence proof relies on suitably defined substitute masses and bounds the approximation error using the energy-dissipation relation of the underlying Wasserstein gradient structure. In the appendix we also discuss the case of an asymmetric double-well potential and derive the corresponding limit dynamics in an elementary way.

math.AP↗
arXiv2018

Stability of concentrated suspensions under Couette and Poiseuille flow

The stability of two-dimensional Poiseuille flow and plane Couette flow for concentrated suspensions is investigated. Linear stability analysis of the two-phase flow model for both flow geometries shows the existence of a convectively driven instability with increasing growth rates of the unstable modes as the particle volume fraction of the suspension increases. In addition it is shown that there exists a bound for the particle phase viscosity below which the two-phase flow model may become ill-posed as the particle phase approaches its maximum packing fraction. The case of two-dimensional Poiseuille flow gives rise to base state solutions that exhibit a jammed and unyielded region, due to shear-induced migration, as the maximum packing fraction is approached. The stability characteristics of the resulting Bingham-type flow is investigated and connections to the stability problem for the related classical Bingham-flow problem are discussed.

physics.flu-dyn↗
arXiv2017

A revised proof of uniqueness of self-similar profiles to Smoluchowski's coagulation equation for kernels close to constant

In this article we correct the proof of a uniqueness result for 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 \begin{equation*} -\varepsilon \leq K(x,y)-2 \leq \varepsilon \left( \Big(\frac{x}{y}\Big)^α + \Big(\frac{y}{x}\Big)^α\right) \end{equation*} for $α\in [0,\frac 1 2)$. Assuming in addition that $K$ has an analytic extension to $\mathbb{C}\setminus(-\infty,0]$ and prescribing the precise asymptotic behaviour of $K$ at the origin, we prove that self-similar solutions with given mass are unique if $\varepsilon$ is sufficiently small.

math.AP↗
arXiv2017

Oscillatory travelling wave solutions for coagulation equations

We consider Smoluchowski's coagulation equation with kernels of homogeneity one of the form $K_{\varepsilon }(ξ,η) =\big( ξ^{1-\varepsilon }+η^{1-\varepsilon }\big)\big ( ξη\big) ^{\frac{\varepsilon }{2}}$. Heuristically, in suitable exponential variables, one can argue that in this case the long-time behaviour of solutions is similar to the inviscid Burgers equation and that for Riemann data solutions converge to a traveling wave for large times. Numerical simulations in \cite{HNV16} indeed support this conjecture, but also reveal that the traveling waves are oscillatory and the oscillations become stronger with smaller $\varepsilon$. The goal of this paper is to construct such oscillatory traveling wave solutions and provide details of their shape via formal matched asymptotic expansions.

math.AP↗
arXiv2016

Gradient flow formulation and longtime behaviour of a constrained Fokker-Planck equation

We consider a Fokker-Planck equation which is coupled to an externally given time-dependent constraint on its first moment. This constraint introduces a Lagrange-multiplier which renders the equation nonlocal and nonlinear. In this paper we exploit an interpretation of this equation as a Wasserstein gradient flow of a free energy ${\mathcal{F}}$ on a time-constrained manifold. First, we prove existence of solutions by passing to the limit in an explicit Euler scheme obtained by minimizing $h {\mathcal{F}}(\varrho)+W_2^2(\varrho^0,\varrho)$ among all $\varrho$ satisfying the constraint for some $\varrho^0$ and time-step $h>0$. Second, we provide quantitative estimates for the rate of convergence to equilibrium when the constraint converges to a constant. The proof is based on the investigation of a suitable relative entropy with respect to minimizers of the free energy chosen according to the constraint. The rate of convergence can be explicitly expressed in terms of constants in suitable logarithmic Sobolev inequalities.

math.AP↗
arXiv2016

Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel

We characterize the long-time behaviour of solutions to Smoluchowski's coagulation equation with a diagonal kernel of homogeneity $γ< 1$. Due to the property of the diagonal kernel, the value of a solution depends only on a discrete set of points. As a consequence, the long-time behaviour of solutions is in general periodic, oscillating between different rescaled versions of a self-similar solution. Immediate consequences of our result are a characterization of the set of data for which the solution converges to self-similar form and a uniqueness result for self-similar profiles.

math.AP↗
arXiv2016

Instabilities and oscillations in coagulation equations with kernels of homogeneity one

We discuss the long-time behaviour of solutions to Smoluchowski's coagulation equation with kernels of homogeneity one, combining formal asymptotics, heuristic arguments based on linearization, and numerical simulations. The case of what we call diagonally dominant kernels is particularly interesting. Here one expects that the long-time behaviour is, after a suitable change of variables, the same as for the Burgers equation. However, for kernels that are close to the diagonal one we obtain instability of both, constant solutions and traveling waves and in general no convergence to N-waves for integrable data. On the other hand, for kernels not close to the diagonal one these structures are stable, but the traveling waves have strong oscillations. This has implications on the approach towards an N-wave for integrable data, which is also characterized by strong oscillations near the shock front.

math.AP↗
arXiv2016

Mathematical analysis of a coarsening model with local interactions

We consider particles on a one-dimensional lattice whose evolution is governed by nearest-neighbor interactions where particles that have reached size zero are removed from the system. Concentrating on configurations with infinitely many particles, we prove existence of solutions under a reasonable density assumption on the initial data and show that the vanishing of particles and the localized interactions can lead to non-uniqueness. Moreover, we provide a rigorous upper coarsening estimate and discuss the generic statistical properties as well as some non-generic behavior of the evolution by means of heuristic arguments and numerical observations.

math.AP↗
arXiv2014

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

We show the existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation for homogeneous kernels satisfying $C_1 \left(x^{-a}y^{b}+x^{b}y^{-a}\right)\leq K\left(x,y\right)\leq C_2\left(x^{-a}y^{b}+x^{b}y^{-a}\right)$ with $a>0$ and $b<1$. This covers especially the case of Smoluchowski's classical kernel $K(x,y)=(x^{1/3} + y^{1/3})(x^{-1/3} + y^{-1/3})$. For the proof of existence we first consider some regularized kernel $K_ε$ for which we construct a sequence of solutions $h_ε$. In a second step we pass to the limit $ε\to 0$ to obtain a solution for the original kernel $K$. The main difficulty is to establish a uniform lower bound on $h_ε$. The basic idea for this is to consider the time-dependent problem and choosing a special test function that solves the dual problem.

math.AP↗
arXiv2014

Rate-independent dynamics and Kramers-type phase transitions in nonlocal Fokker-Planck equations with dynamical control

The hysteretic behavior of many-particle systems with non-convex free energy can be modeled by nonlocal Fokker-Planck equations that involve two small parameters and are driven by a time- dependent constraint. In this paper we consider the fast reaction regime related to Kramers-type phase transitions and show that the dynamics in the small-parameter limit can be described by a rate-independent evolution equation with hysteresis. For the proof we first derive mass-dissipation estimates by means of Muckenhoupt constants, formulate conditional stability estimates, and char- acterize the mass flux between the different phases in terms of moment estimates that encode large deviation results. Afterwards we combine all these partial results and establish the dynamical sta- bility of localized peaks as well as sufficiently strong compactness results for the basic macroscopic quantities.

math.AP↗
arXiv2013

Exponential tail behaviour of self-similar solutions to Smoluchowski's coagulation equation

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 $\lim_{x \to \infty} \frac{1}{x} \log \Big(\frac{1}{f(x)}\Big)$ exists. In addition we prove properties of the prefactor if the kernel is uniformly bounded below.

math.AP↗
arXiv2012

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 $γ\in [0,1)$ and satisfy $K(x,y) \leq C (x^γ + y^γ)$. More precisely, for any $ρ\in (γ,1)$ we establish the existence of a continuous weak self-similar profile with decay $x^{-(1{+}ρ)}$ as $x \to \infty$.

math.AP↗
arXiv2011

On Global Stability for Lifschitz-Slyozov-Wagner like equations

This paper is concerned with the stability and asymptotic stability at large time of solutions to a system of equations, which includes the Lifschitz-Slyozov-Wagner (LSW) system in the case when the initial data has compact support. The main result of the paper is a proof of weak global asymptotic stability for LSW like systems. Previously strong local asymptotic stability results were obtained by Niethammer and Velázquez for the LSW system with initial data of compact support. Comparison to a quadratic model plays an important part in the proof of the main theorem when the initial data is critical. The quadratic model extends the linear model of Carr and Penrose, and has a time invariant solution which decays exponentially at the edge of its support in the same way as the infinitely differentiable self-similar solution of the LSW model.

math.AP↗
arXiv2011

Self-similar Solutions to a Kinetic Model for Grain Growth

We prove the existence of self-similar solutions to the Fradkov model for two-dimensional grain growth, which consists of an infinite number of nonlocally coupled transport equations for the number densities of grains with given area and number of neighbours (topological class). For the proof we introduce a finite maximal topological class and study an appropriate upwind-discretization of the time dependent problem in self-similar variables. We first show that the resulting finite dimensional differential system has nontrivial steady states. Afterwards we let the discretization parameter tend to zero and prove that the steady states converge to a compactly supported self-similar solution for a Fradkov model with finitely many equations. In a third step we let the maximal topology class tend to infinity and obtain self-similar solutions to the original system that decay exponentially. Finally, we use the upwind discretization to compute self-similar solutions numerically.

math.AP↗