02 Mar 2021-Archive for Rational Mechanics and Analysis (Springer Berlin Heidelberg)-Vol. 240, Iss: 2, pp 809-875

Abstract: We study coagulation equations under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. We consider both discrete and continuous coagulation equations, and allow for a large class of coagulation rate kernels, with the main restriction being boundedness from above and below by certain weight functions. The weight functions depend on two power law parameters, and the assumptions cover, in particular, the commonly used free molecular and diffusion limited aggregation coagulation kernels. Our main result shows that the two weight function parameters already determine whether there exists a stationary solution under the presence of a source term. In particular, we find that the diffusive kernel allows for the existence of stationary solutions while there cannot be any such solutions for the free molecular kernel. The argument to prove the non-existence of solutions relies on a novel power law lower bound, valid in the appropriate parameter regime, for the decay of stationary solutions with a constant flux. We obtain optimal lower and upper estimates of the solutions for large cluster sizes, and prove that the solutions of the discrete model behave asymptotically as solutions of the continuous model.

... read more

Topics: Continuous modelling (52%), Weight function (51%), Upper and lower bounds (51%)

More

8 results found

••

Abstract: We modify the usual Erdos-Renyi random graph evolution by letting connected clusters 'burn down' (i.e. fall apart to disconnected single sites) due to a Poisson flow of lightnings. In a range of the intensity of rate of lightnings the system sticks to a permanent critical state.

... read more

Topics: Random graph (56%)

5 Citations

••

Abstract: We study multicomponent coagulation via the Smoluchowski coagulation equation under non-equilibrium stationary conditions induced by a source of small clusters. The coagulation kernel can be very general, merely satisfying certain power law asymptotic bounds in terms of the total number of monomers in a cluster. The bounds are characterized by two parameters and we extend previous results for one-component systems to classify the parameter values for which the above stationary solutions do or do not exist. Moreover, we also obtain criteria for the existence or non-existence of solutions which yield a constant flux of mass towards large clusters.

... read more

Topics: Smoluchowski coagulation equation (60%)

2 Citations

••

Abstract: In this paper we prove the existence of a family of self-similar solutions for a class of coagulation equations with a constant flux of particles from the origin. These solutions are expected to describe the longtime asymptotics of Smoluchowski's coagulation equations with a time independent source of clusters concentrated in small sizes. The self-similar profiles are shown to be smooth, provided the coagulation kernel is also smooth. Moreover, the self-similar profiles are estimated from above and from below by $x^{-(\gamma+3)/2}$ as $x \to 0$ and are proven to decay at least exponentially as $x \to \infty$.

... read more

1 Citations

••

Abstract: Existence and non-existence of integrable stationary solutions to Smoluchowski's coagulation equation with source are investigated when the source term is integrable with an arbitrary support in (0, $\infty$) Besides algebraic upper and lower bounds, a monotonicity condition is required for the coagulation kernel Connections between integrability properties of the source and the corresponding stationary solutions are also studied

... read more

Topics: Kernel (statistics) (63%), Upper and lower bounds (50%)

1 Citations

More

41 results found

••

01 Jan 1968-

Abstract: Weak Convergence in Metric Spaces. The Space C. The Space D. Dependent Variables. Other Modes of Convergence. Appendix. Some Notes on the Problems. Bibliographical Notes. Bibliography. Index.

... read more

Topics: Modes of convergence (72%), Weak convergence (69%), Convergence of measures (62%) ... read more

12,859 Citations

••

01 Jan 1955-

Abstract: The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable. It has been developed from courses given by the authors and probably contains more material than will ordinarily be covered in a one-year course. It is hoped that the book will be a useful text in the application of differential equations as well as for the pure mathematician.

... read more

Topics: Ordinary differential equation (62%), Differential equation (59%)

7,066 Citations

••

07 Sep 2011-

Abstract: Preface.- 1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions.- 2. The Uniform Boundedness Principle and the Closed Graph Theorem. Unbounded Operators. Adjoint. Characterization of Surjective Operators.- 3. Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity.- 4. L^p Spaces.- 5. Hilbert Spaces.- 6. Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators.- 7. The Hille-Yosida Theorem.- 8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension.- 9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions.- 10. Evolution Problems: The Heat Equation and the Wave Equation.- 11. Some Complements.- Problems.- Solutions of Some Exercises and Problems.- Bibliography.- Index.

... read more

Topics: Interpolation space (67%), Compact operator on Hilbert space (67%), Sobolev inequality (66%) ... read more

3,753 Citations

••

Mikael Ehn^{1}, Joel A. Thornton^{2}, Einhard Kleist^{3}, Mikko Sipilä^{1} +29 more•Institutions (5)

Abstract: Forests emit large quantities of volatile organic compounds (VOCs) to the atmosphere. Their condensable oxidation products can form secondary organic aerosol, a significant and ubiquitous component of atmospheric aerosol, which is known to affect the Earth's radiation balance by scattering solar radiation and by acting as cloud condensation nuclei. The quantitative assessment of such climate effects remains hampered by a number of factors, including an incomplete understanding of how biogenic VOCs contribute to the formation of atmospheric secondary organic aerosol. The growth of newly formed particles from sizes of less than three nanometres up to the sizes of cloud condensation nuclei (about one hundred nanometres) in many continental ecosystems requires abundant, essentially non-volatile organic vapours, but the sources and compositions of such vapours remain unknown. Here we investigate the oxidation of VOCs, in particular the terpene α-pinene, under atmospherically relevant conditions in chamber experiments. We find that a direct pathway leads from several biogenic VOCs, such as monoterpenes, to the formation of large amounts of extremely low-volatility vapours. These vapours form at significant mass yield in the gas phase and condense irreversibly onto aerosol surfaces to produce secondary organic aerosol, helping to explain the discrepancy between the observed atmospheric burden of secondary organic aerosol and that reported by many model studies. We further demonstrate how these low-volatility vapours can enhance, or even dominate, the formation and growth of aerosol particles over forested regions, providing a missing link between biogenic VOCs and their conversion to aerosol particles. Our findings could help to improve assessments of biosphere-aerosol-climate feedback mechanisms, and the air quality and climate effects of biogenic emissions generally.

... read more

Topics: Aerosol (55%), Cloud condensation nuclei (53%)

1,083 Citations

•

01 Jan 2000-

784 Citations