. This paper investigates an initial-boundary value problem for the compressible full Navier-Stokes equations in a three-dimensional bounded domain subject to vorticity-slip boundary conditions. The incompressible limit at low Mach number is established for all-time strong solutions with ill-prepared initial data. The proofs rely on subtle weighted uniform estimates concerning both the Mach number & varepsilon; is an element of (0, 1] and all time t is an element of [0, +infinity), as well as on the derivation of a nonlinear integral inequality with decay. In contrast to the case of well-prepared initial data, the temporal derivatives of velocity become unbounded due to the presence of ill-prepared initial data, which leads to the loss of strong convergence for the velocity. The key novelties of this paper lie in the careful selection of distinct weights in various norms to cancel the large operators, as well as in the combination of estimates for fast and slow variables, particularly for the high-order spatial derivatives of the fast components. By employing the Helmholtz decomposition and utilizing the strong convergence of the incompressible part of velocity, we demonstrate the convergence to the global strong solution of the incompressible Navier-Stokes equations.
This work studies the global behavior of a fully parabolic chemotaxis system that combines attraction-repulsion mechanisms with nonlinear density-dependent diffusion: {u(t)( )= d triangle u + u(a(1)-a(2)u-a(3)v), x is an element of ohm, t > 0, v(t) = V center dot (D(v)Vv + chi vVw-xi vVu) + rho v (1-v) + ea(3)uv, x is an element of ohm, t > 0, w(t) = eta triangle w + ru-gamma w, x is an element of ohm, t > 0, del u center dot nu= Vv center dot nu= Vw center dot nu = (0), x is an element of partial derivative ohm, t > 0, (u, v, w)(x, 0) = (u(0), v(0), w(0))(x), x is an element of ohm, posed in a smooth bounded domain ohm subset of R-n, n > 3, subject to homogeneous Neumann boundary conditions. The diffusion coefficient of v satisfies D(v) > D(0)v theta for some D0 > 0. Under the condition theta > 1-4/n+2 , we establish the existence of a globally bounded classical solution. Furthermore, for sufficiently small chemotactic coefficients, the asymptotic behavior of these bounded solutions is completely described: when a(1) > a(3), exponential convergence to the coexistence equilibrium (u(& lowast;),v(& lowast;),w(& lowast;)) takes place; when a(1) < a(3), exponential convergence to the semi-trivial state (0, 1, 0) occurs; and when a(1) = a(3), convergence to the same state is algebraic. In comparison with previous works, our analysis provides global boundedness for the full attraction-repulsion system with nonlinear diffusion in higher dimensions and gives sharper L-infinity-decay estimates due to the repulsive chemotaxis effect.
In this article, we are concerned with the inflow problems of a viscous ions model, which is governed by the compressible Navier-Stokes-Poisson equations, in the half space. When the far field states and the boundary values of the density and the electron potential satisfy the quasi-neutral condition, and far field states and the boundary values of the density and the velocity can be connected by the rarefaction wave of the initial problem of the corresponding hyperbolic system, it is shown that the rarefaction wave is asymptotically stable in the case that 0 < u- < delta(0) and the initial data is a suitable small perturbation of the rarefaction ware. The proof is the L-2-energy method, which takes into account both the effect of the self-consistent electrostatic potential and the time decay of the rarefaction wave.
. In this paper, we aim to establish the decay rate for the semi-group it a|del|+b|del|3-c partial derivative 2 Tabc(t)f := e x1 f associated with linearized electrohydro dynamic waves in two dimensions, where a, b, c is an element of {0, 1}. Compared to the decay estimate for the semi-group of linearized water waves, the primary obstruction arises because the operator a|del| + b|del|3 - c partial derivative 2x1 is anisotropic, preventing the direct application of the van der Corput lemma. By applying Stein's theorem on oscillatory integrals, we establish the decay estimate for Tabc(t). These results provide crucial groundwork for establishing the global well-posedness of the electrohydro dynamic waves model.
In this paper, by means of the Poincare compactification of R3, we describe the global dynamics of the Sprott dynamical system (Case A) with an additional linear antidamping term, x(center dot) = y, y(center dot)= -x-yz, z(center dot)=y2-a + bz, where (x, y, z) is an element of R3 are the state variables and (a, b) is an element of R2 are real parameters. For suitable values of the parameters, this system exhibits invariant algebraic surfaces that play a key role in its global dynamics. We characterize the dynamics of this differential system in the finite region and at infinity, which is represented by the sphere S2 in the Poincare ball, and highlighting the influence of the antidamping term bz on this dynamics.
. We investigate a tick metapopulation model in a fragmented mpatch ecosystem, incorporating time-varying coefficients and delays. Employing differential inequality techniques, the fluctuation lemma, and discrete dynamical system theory, we perform a comprehensive analysis of the global dynamics, establishing two main results: (i) the global (exponential) attractivity of the zero equilibrium for all nonnegative initial conditions; and (ii) the mutual attraction among all persistent solutions. We then apply this theoretical framework to two ecologically relevant tick metapopulation models, featuring Ricker-type and Mackey-Glass-type reproduction functions, respectively, and for the first time rigorously establish the global attractivity of positive periodic solutions in these models. Our findings not only significantly extend the existing results in the literature but also offer novel insights into the long-term dynamics of non-autonomous delay-structured tick metapopulation systems. Numerical simulations are provided and are in excellent agreement with the theoretical results.
