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Register a new userOn the existence of non-trivial steady-state size-distributions for a class of flocculation equations
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Flocculation is the process whereby particles (i.e., flocs) in suspension reversibly combine and separate. The process is widespread in soft matter and aerosol physics as well as environmental science and engineering. We consider a general size-structured flocculation model, which describes the evolution of floc size distribution in an aqueous environment. Our work provides a unified treatment for many size-structured models in the environmental, industrial, medical, and marine engineering literature. In particular, the mathematical model considered in this work accounts for basic biological phenomena in a population of microorganisms including growth, death, sedimentation, predation, surface erosion, renewal, fragmentation and aggregation. The central objective of this work is to prove existence of positive steady states of this generalized flocculation model. Using results from fixed point theory we derive conditions for the existence of continuous, non-trivial stationary solutions. We further develop a numerical scheme based on spectral collocation method to approximate these positive stationary solutions. We explore the stationary solutions of the model for various biologically relevant parameters and give valuable insights for the efficient removal of suspended particles.
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Flocculation is the process whereby particles (i.e., flocs) in suspension reversibly combine and separate. The process is widespread in soft matter and aerosol physics as well as environmental science and engineering. We consider a general size-structured flocculation model, which describes the evolution of flocs in an aqueous environment. Our work provides a unified treatment for many size-structured models in the environmental, industrial, medical, and marine engineering literature. In particular, our model accounts for basic biological phenomena in a population of microorganisms including growth, death, sedimentation, predation, renewal, fragmentation and aggregation. Our central goal in this paper is to rigorously investigate the long-term behavior of this generalized flocculation model. Using results from fixed point theory we derive conditions for the existence of continuous, non-trivial stationary solutions. We further apply the principle of linearized stability and semigroup compactness arguments to provide sufficient conditions for local exponential stability of stationary solutions as well as sufficient conditions for instability. Abstract. The end results of this analytical development are relatively simple inequality-criteria which thus allows for the rapid evaluation of the existence and stability of a non-trivial stationary solution. To our knowledge, this work is the first to derive precise existence and stability criteria for such a generalized model. Lastly, we also provide an illustrating application of this criteria to several flocculation models.
In this paper, we study the existence, stability and bifurcation of random complete and periodic solutions for stochastic parabolic equations with multiplicative noise. We first prove the existence and uniqueness of tempered random attractors for the stochastic equations and characterize the structures of the attractors by random complete solutions. We then examine the existence and stability of random complete quasi-solutions and establish the relations of these solutions and the structures of tempered attractors. When the stochastic equations are incorporated with periodic forcing, we obtain the existence and stability of random periodic solutions. For the stochastic Chafee-Infante equation, we further establish the multiplicity and stochastic bifurcation of complete and periodic solutions.
The purpose of this paper is extend recent results of Bonder-Groisman and Foondun-Nualart to the stochastic wave equation. In particular, a suitable integrability condition for non-existence of global solutions is derived.
We find the complete equivalence group of a class of (1+1)-dimensional second-order evolution equations, which is infinite-dimensional. The equivariant moving frame methodology is invoked to construct, in the regular case of the normalization procedure, a moving frame for a group related to the equivalence group in the context of equivalence transformations among equations of the class under consideration. Using the moving frame constructed, we describe the algebra of differential invariants of the former group by obtaining a minimum generating set of differential invariants and a complete set of independent operators of invariant differentiation.
We devise an explicit method to integrate $alpha$-stable stochastic differential equations (SDEs) with non-Lipschitz coefficients. To mitigate against numerical instabilities caused by unbounded increments of the Levy noise, we use a deterministic map which has the desired SDE as its homogenised limit. Moreover, our method naturally overcomes difficulties in expressing the Marcus integral explicitly. We present an example of an SDE with a natural boundary showing that our method respects the boundary whereas Euler-Maruyama discretisation fails to do so. As a by-product we devise an entirely deterministic method to construct $alpha$-stable laws.
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