79 research outputs found

    Moment bounds for the Smoluchowski equation and their consequences

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    We prove uniform bounds on moments X_a = \sum_{m}{m^a f_m(x,t)} of the Smoluchowski coagulation equations with diffusion, valid in any dimension. If the collision propensities \alpha(n,m) of mass n and mass m particles grow more slowly than (n+m)(d(n) + d(m)), and the diffusion rate d(\cdot) is non-increasing and satisfies m^{-b_1} \leq d(m) \leq m^{-b_2} for some b_1 and b_2 satisfying 0 \leq b_2 < b_1 < \infty, then any weak solution satisfies X_a \in L^{\infty}(\mathbb{R}^d \times [0,T]) \cap L^1(\mathbb{R}^d \times [0,T]) for every a \in \mathbb{N} and T \in (0,\infty), (provided that certain moments of the initial data are finite). As a consequence, we infer that these conditions are sufficient to ensure uniqueness of a weak solution and its conservation of mass.Comment: 30 page

    Eternal solutions to a singular diffusion equation with critical gradient absorption

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    The existence of nonnegative radially symmetric eternal solutions of exponential self-similar type u(t,x)=e−pÎČt/(2−p)fÎČ(∣x∣e−ÎČt;ÎČ)u(t,x)=e^{-p\beta t/(2-p)} f_\beta(|x|e^{-\beta t};\beta) is investigated for the singular diffusion equation with critical gradient absorption \begin{equation*} \partial_{t} u-\Delta_{p} u+|\nabla u|^{p/2}=0 \quad \;\;\hbox{in}\;\; (0,\infty)\times\real^N \end{equation*} where 2N/(N+1)<p<22N/(N+1) < p < 2. Such solutions are shown to exist only if the parameter ÎČ\beta ranges in a bounded interval (0,ÎČ∗](0,\beta_*] which is in sharp contrast with well-known singular diffusion equations such as ∂tϕ−Δpϕ=0\partial_{t}\phi-\Delta_{p} \phi=0 when p=2N/(N+1)p=2N/(N+1) or the porous medium equation ∂tϕ−Δϕm=0\partial_{t}\phi-\Delta\phi^m=0 when m=(N−2)/Nm=(N-2)/N. Moreover, the profile f(r;ÎČ)f(r;\beta) decays to zero as r→∞r\to\infty in a faster way for ÎČ=ÎČ∗\beta=\beta_* than for ÎČ∈(0,ÎČ∗)\beta\in (0,\beta_*) but the algebraic leading order is the same in both cases. In fact, for large rr, f(r;ÎČ∗)f(r;\beta_*) decays as r−p/(2−p)r^{-p/(2-p)} while f(r;ÎČ)f(r;\beta) behaves as (log⁥r)2/(2−p)r−p/(2−p)(\log r)^{2/(2-p)} r^{-p/(2-p)} when ÎČ∈(0,ÎČ∗)\beta\in (0,\beta_*)

    Convergence to equilibrium for the discrete coagulation-fragmentation equations with detailed balance

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    Under the condition of detailed balance and some additional restrictions on the size of the coefficients, we identify the equilibrium distribution to which solutions of the discrete coagulation-fragmentation system of equations converge for large times, thus showing that there is a critical mass which marks a change in the behavior of the solutions. This was previously known only for particular cases as the generalized Becker-D\"oring equations. Our proof is based on an inequality between the entropy and the entropy production which also gives some information on the rate of convergence to equilibrium for solutions under the critical mass.Comment: 28 page

    Refined asymptotics for the infinite heat equation with homogeneous Dirichlet boundary conditions

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    The nonnegative viscosity solutions to the infinite heat equation with homogeneous Dirichlet boundary conditions are shown to converge as time increases to infinity to a uniquely determined limit after a suitable time rescaling. The proof relies on the half-relaxed limits technique as well as interior positivity estimates and boundary estimates. The expansion of the support is also studied

    The discrete fragmentation equations : semigroups, compactness and asynchronous exponential growth

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    In this paper we present a class of fragmentation semigroups which are compact in a scale of spaces defined in terms of finite higher moments. We use this compactness result to analyse the long time behaviour of such semigroups and, in particular, to prove that they have the asynchronous growth property. We note that, despite compactness, this growth property is not automatic as the fragmentation semigroups are not irreducible

    A parabolic free boundary problem modeling electrostatic MEMS

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    The evolution problem for a membrane based model of an electrostatically actuated microelectromechanical system (MEMS) is studied. The model describes the dynamics of the membrane displacement and the electric potential. The latter is a harmonic function in an angular domain, the deformable membrane being a part of the boundary. The former solves a heat equation with a right hand side that depends on the square of the trace of the gradient of the electric potential on the membrane. The resulting free boundary problem is shown to be well-posed locally in time. Furthermore, solutions corresponding to small voltage values exist globally in time while global existence is shown not to hold for high voltage values. It is also proven that, for small voltage values, there is an asymptotically stable steady-state solution. Finally, the small aspect ratio limit is rigorously justified

    Rigorous derivation of a nonlinear diffusion equation as fast-reaction limit of a continuous coagulation-fragmentation model with diffusion

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    Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann-Bak model of coagulation-breakup within a bounded domain with homogeneous Neumann boundary conditions are shown to converge, in the fast reaction limit, towards local equilibria determined by their mass. Moreover, this mass is the solution of a nonlinear diffusion equation whose nonlinearity depends on the (size-dependent) diffusion coefficient. Initial data are assumed to have integrable zero order moment and square integrable first order moment in size, and finite entropy. In contrast to our previous result [CDF2], we are able to show the convergence without assuming uniform bounds from above and below on the number density of clusters

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

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    We consider mass-conserving self-similar solutions for Smoluchowski's coagulation equation with kernel K(Ο,η)=(Οη)λK(\xi,\eta)= (\xi \eta)^{\lambda} with λ∈(0,1/2)\lambda \in (0,1/2). It is known that such self-similar solutions g(x)g(x) satisfy that x−1+2λg(x)x^{-1+2\lambda} g(x) is bounded above and below as x→0x \to 0. In this paper we describe in detail via formal asymptotics the qualitative behavior of a suitably rescaled function h(x)=hλx−1+2λg(x)h(x)=h_{\lambda} x^{-1+2\lambda} g(x) in the limit λ→0\lambda \to 0. It turns out that h∌1+Cxλ/2cos⁥(λlog⁥x)h \sim 1+ C x^{\lambda/2} \cos(\sqrt{\lambda} \log x) as x→0x \to 0. As xx becomes larger hh develops peaks of height 1/λ1/\lambda that are separated by large regions where hh is small. Finally, hh converges to zero exponentially fast as x→∞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

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

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    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 KK that are homogeneous of degree γ∈[0,1)\gamma \in [0,1) and satisfy K(x,y)≀C(xÎł+yÎł)K(x,y) \leq C (x^{\gamma} + y^{\gamma}). More precisely, for any ρ∈(Îł,1)\rho \in (\gamma,1) we establish the existence of a continuous weak self-similar profile with decay x−(1+ρ)x^{-(1{+}\rho)} as x→∞x \to \infty
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