This paper deals with a boundary-value problem in two-dimensional smoothly bounded domains for the coupled Keller-Segel-Stokes systemHere, one of the novelties is that the chemotactic sensitivity S is not a scalar function but rather attains values in R 2×2 , and satisfies |S(x, n, c)| ≤ C S (1 + n) −α with some C S > 0 and α > 0. We shall establish the existence of global bounded classical solutions for arbitrarily large initial data. In contrast to the corresponding case of scalar-valued sensitivities, this system does not possess any gradient-like structure due to the appearance of such matrix-valued S. To overcome this difficulty, we will derive a series of a priori estimates involving a new interpolation inequality.To the best of our knowledge, this is the first result on global existence and boundedness in a KellerSegel-Stokes system with tensor-valued sensitivity, in which production of the chemical signal is involved.
This paper deals with the Keller–Segel–Navier–Stokes system [Formula: see text] in a bounded domain [Formula: see text] with smooth boundary, where [Formula: see text] and [Formula: see text] are given functions. We shall develop a weak solution concept which requires solutions to satisfy very mild regularity hypotheses only, especially for the component [Formula: see text]. Under the assumption that there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] it is finally shown that for all suitably regular initial data an associated initial-boundary value problem possesses a globally defined weak solution. In comparison to the result for the corresponding fluid-free system, it is easy to see that the restriction on [Formula: see text] here is optimal. This result extends previous studies on global solvability for this system in the two-dimensional domain and for the associated chemotaxis-Stokes system obtained on neglecting the nonlinear convective term in the fluid equation.
This paper deals with convergence of solutions to a class of parabolic Keller-Segel systems, possibly coupled to the (Navier-)Stokes equations in the framework of the full modelto solutions of the parabolic-elliptic counterpart formally obtained on taking ε ց 0. In smoothly bounded physical domains Ω ⊂ R N with N ≥ 1, and under appropriate assumptions on the model ingredients, we shall first derive a general result which asserts certain strong and pointwise convergence properties whenever asserting that supposedly present bounds on ∇c ε and u ε are bounded in L λ ((0, T ); L q (Ω)) and in L ∞ ((0, T ); L r (Ω)), respectively, for some λ ∈ (2, ∞], q > N and r > max{2, N } such that 1 λ + N 2q < 1 2 . To our best knowledge, this seems to be the first rigorous mathematical result on a fast signal diffusion limit in a chemotaxis-fluid system. This general result will thereafter be concretized in the context of two examples: Firstly, for an unforced Keller-Segel-Navier-Stokes system we shall establish a statement on global classical solutions under suitable smallness conditions on the initial data, and show that these solutions approach a global classical solution to the respective parabolic-elliptic simplification.We shall secondly derive a corresponding convergence property for arbitrary solutions to fluid-free Keller-Segel systems with logistic source terms, which in spatially one-dimensional settings turn out to allow for a priori estimates compatible with our general theory. Building on the latter in conjunction with a known result on emergence of large densities in the associated parabolic-elliptic limit system, we will finally discover some quasi-blowup phenomenon for the fully parabolic Keller-Segel system with logistic source and suitably small parameter ε > 0.
Hyperchaotic system, as an important topic, has become an active research subject in nonlinear science. Over the past two decades, hyperchaotic system between nonlinear systems has been extensively studied. Although many kinds of numerical methods of the system have been announced, simple and efficient methods have always been the direction that scholars strive to pursue. Based on this problem, this paper introduces another novel numerical method to solve a class of hyperchaotic system. Barycentric Lagrange interpolation collocation method is given and illustrated with hyperchaotic system (x˙=ax+dz-yz,y˙=xz-by, 0≤t≤T,z˙=cx-z+xy,w˙=cy-w+xz,) as examples. Numerical simulations are used to verify the effectiveness of the present method.
Partial differential equations (PDEs) are widely used in mechanics, control processes, ecological and economic systems, chemical cycling systems, and epidemiology. Although there are some numerical methods for solving PDEs, simple and efficient methods have always been the direction that scholars strive to pursue. Based on this problem, we give the meshless barycentric interpolation collocation method (MBICM) for solving a class of PDEs. Four numerical experiments are carried out and compared with other methods; the accuracy of the numerical solution obtained by the present method is obviously improved.
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