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In this paper, we study the Cauchy problem of a two-phase flow system consisting of the compressible isothermal Euler equations and the incompressible Navier-Stokes equations coupled through the drag force, which can be formally derived from the Vlasov-Fokker-Planck/incompressible Navier-Stokes equations. When the initial data is a small perturbation around an equilibrium state, we prove the global well-posedness of the classical solutions to this system and show the solutions tends to the equilibrium state as time goes to infinity. In order to resolve the main difficulty arising from the pressure term of the incompressible Navier-Stokes equations, we properly use the Hodge decomposition, spectral analysis, and energy method to obtain the L 2 time decay rates of the solution when the initial perturbation belongs to L 1 space. Furthermore, we show that the above time decay rates are optimal. [ABSTRACT FROM AUTHOR]

Asymptotic expansion in far-field for the incompressible Navier–Stokes flow are established. It is well known that a velocity decays slowly in far-field. This property prevents classical procedure giving asymptotic expansions of solutions with high-order. In this paper, under natural settings and moment conditions on the initial vorticity, technique of renormalization together with Biot–Savart law derives an asymptotic expansion for velocity with high-order. Especially scalings and large-time behaviors of the expansions are clarified. By employing them, time evolution of velocity in far-field is drawn. As an appendix, asymptotic behavior of solutions as time variable tends to infinity is given. In this assertion, large-time behavior of velocity is discovered clearly. [ABSTRACT FROM AUTHOR]

In this paper, we consider 3D compressible magneto-micropolar fluids without resistivity and spin viscosity in a strip domain. The prominent character of the governing model is the presence of the microstructure, a linear coupling structure involving derivatives of the velocity fields, which along with the lack of spin viscosity brings several challenges to the analysis. By exploiting the two-tier energy method developed in Guo and Tice (Arch Ration Mech Anal, 2013), we prove the global existence of classical solutions to the governing model around a uniform magnetic field that is non-parallel to the horizontal boundary. Moreover, we show that the solution converges to the steady state at an almost exponential rate as time goes to infinity. One of the main ingredients in our analysis, compared with previous works on micropolar fluids, is that we deal with the microstructure by establishing some delicate estimates based on the analysis of the div-curl decomposition, and the coupling between the fluid velocity and the vorticity of angular velocity. [ABSTRACT FROM AUTHOR]

In the present paper, the primitive equations, which can be used to simulate the large-scale motion of the ocean and atmosphere, are considered in the three-dimensional domain bounded below by a fixed solid boundary and above by a free-moving boundary. The global existence and uniqueness of strong solutions are established, and the long-time convergence to the equilibrium state is shown to be either at an exponential rate for the horizontal periodic domain or at an algebraic rate for the horizontal whole space. [ABSTRACT FROM AUTHOR]

The decay behavior of a class of equation w t = - ∇ ⋅ ( w n ⁢ ∇ ⁡ Δ ⁢ w + α ⁢ w n - 1 ⁢ Δ ⁢ w ⁢ ∇ ⁡ w + β ⁢ w n - 2 ⁢ | ∇ ⁡ w | 2 ⁢ ∇ ⁡ w ) is considered under the Neumann boundary condition. The equation can be viewed as a generalization of the thin film equation w t + (w n ⁢ w x ⁢ x ⁢ x ) x = 0 , which can be used to describe the movement of the skinny viscous layer of compressible fluid along the slope. We obtain that the solution decays exponentially in L 1 -norm in the multi-dimensional case, and decays algebraically in L ∞ -norm in the one-dimensional case. The critical step solving the problem is to construct appropriate dissipative entropies. [ABSTRACT FROM AUTHOR]

We consider the barotropic Euler equations with pairwise attractive Riesz interactions and linear velocity damping in the periodic domain. We establish the global-in-time well-posedness theory for the system near an equilibrium state if the coefficient of the Riesz interaction term is small. We also analyze the large-time behavior of solutions showing the exponential rate of convergence toward the equilibrium state as time goes to infinity. [ABSTRACT FROM AUTHOR]

In this study, the initial-boundary value problem for cylindrically symmetric Navier-Stokes equations was considered with temperature-dependent viscosity and heat conductivity. Firstly, we established the existence and uniqueness of a strong solution when the viscosity and heat conductivity were both power functions of temperature. Moreover, the large-time behavior of the strong solution was obtained with large initial data, since all of the estimates in this paper were independent of time. It is worth noting that we identified the relationship between the initial data and the power of the temperature in the viscosity for the first time.

Recently, Niche [J. Differential Equations, 260 (2016), 4440–4453] established upper bounds on the decay rates of solutions to the 3D incompressible Navier–Stokes–Voigt equations in terms of the decay character r ∗ of the initial data in H 1 (R 3). Motivated by this work, we focus on characterizing the large-time behavior of all space-time derivatives of the solutions for the 2D case and establish upper bounds and lower bounds on their decay rates by making use of the decay character and Fourier splitting methods. In particular, for the case − n 2 < r ∗ ⩽ 1 , we verify the optimality of the upper bounds, which is new to the best of our knowledge. Similar improved decay results are also true for the 3D case. [ABSTRACT FROM AUTHOR]

In this paper, we shall investigate the large-time behavior of the solution to an outflow problem of the one-dimensional bipolar quantum Navier-Stokes-Poisson system in the half space. Under some suitable assumptions on the boundary data and the space-asymptotic states, we successfully construct a nonlinear wave which is the superposition of the stationary solution and the 2-rarefaction wave. Then, by means of the $ L^2 $-energy method, we prove that this nonlinear wave is asymptotically stable provided that the initial perturbation and the strength of the stationary solution are small enough, while the strength of the 2-rarefaction wave can be arbitrarily large. [ABSTRACT FROM AUTHOR]

In this paper, we shall investigate the large-time behavior of solutions to the Cauchy problem for the three-dimensional bipolar isentropic Euler–Poisson equations with time-dependent damping effects $ -\frac {\mu }{(1+t)^{\lambda }}n_iu_i(i=1, 2) $ − μ (1 + t) λ n i u i (i = 1 , 2) for $ -1 − 1 < λ < 1 , μ > 0. Here, we consider a more general case that the two pressure functions are different and the doping profile is non-zero. Under the assumption that the initial data are close to the constant equilibrium states, we show that the smooth solutions to the Cauchy problem exist uniquely and globally. The algebraic time-decay rates of the solutions toward the constant equilibrium states are also obtained. The main ingredient of the proof is the time-weighted energy method with artfully chosen time-weighted functions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the global well-posedness and large-time behavior of the compressible Navier-Stokes equations with hyperbolic heat conduction, where the heat flux is assumed to satisfy Cattaneo's law. When the initial perturbation is small and regular, we establish the global existence of the classical solution without any additional assumption on the relaxation time τ. Furthermore, assuming the initial perturbation belongs to a negative Sobolev space, we obtain the optimal time-decay rates of the high-order spatial derivatives of solutions by using a pure energy method instead of complicated spectral analysis. It is also observed that the heat flux, due to its damping structure, decays to the motionless state at a faster time-decay rate compared to density, velocity, and temperature. [ABSTRACT FROM AUTHOR]

The global existence of mass‐conserving weak solutions to the Safronov–Dubovskiǐ coagulation equation is shown for the coagulation kernels satisfying at most linear growth for large sizes. In contrast to previous works, the proof mainly relies on the de la Vallée–Poussin theorem, which only requires the finiteness of the first moment of the initial condition. By showing the necessary regularity of solutions, it is shown that the weak solutions constructed herein are indeed classical solutions. Under additional restrictions on the initial data, the uniqueness of solutions is also shown. Finally, the continuous dependence on the initial data and the large‐time behavior of solutions are also addressed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show the global wellposedness of the Navier-Stokes equations, including surface tension and gravity, with a free surface in an unbounded domain such as bottomless ocean. In addition, it is proved that the solution decays polynomially as time t tends to infinity. To show these results, we first use the Hanzawa transformation in order to reduce the problem in a time-dependent domain Ω t ⊂ R 3 , t > 0 , to a problem in the lower half-space R − 3. We then establish some time-weighted estimate of solutions, in an L p -in-time and L q -in-space setting, for the linearized problem around the trivial steady state with the help of L r - L s time decay estimates of semigroup. Next, the time-weighted estimate, combined with the contraction mapping principle, shows that the transformed problem in R − 3 admits a global-in-time solution in the L p - L q setting and that the solution decays polynomially as time t tends to infinity under the assumption that p , q satisfy the conditions: 2 < p < ∞ , 3 < q < 16 / 5 , and (2 / p) + (3 / q) < 1. Finally, we apply the inverse transformation of Hanzawa's one to the solution in R − 3 to prove our main results mentioned above for the original problem in Ω t. Here we want to emphasize that it is not allowed to take p = q in the above assumption about p , q , which means that the different exponents p , q of L p - L q setting play an essential role in our approach. [ABSTRACT FROM AUTHOR]

It is well-known that solutions to deterministic nonlocal aggregation-diffusion models may blow up in two or higher dimensions. Various mechanisms hence have been proposed to "regularize" the deterministic aggregation-diffusion equations in a manner that allows pattern formation without blow-up. However, stochastic effect has not been ever considered among other things. In this work, we consider a nonlocal aggregation-diffusion model with multiplicative noise and establish the local existence and uniqueness of strong solutions on ℝ d (d ≥ 2). If the noise is non-autonomous and linear, we establish the global existence and large-time behavior of strong solutions with decay properties by combining the Moser-Alikakos iteration technique and some decay estimates of Girsanov type processes. If the noise is nonlinear and strong enough, we show that blow-up can be prevented. As such, our results assert that certain multiplicative noise can also regularize the aggregation-diffusion model. [ABSTRACT FROM AUTHOR]

The stability and large-time behavior problem on the magneto-micropolar equations has evoked a considerable interest in recent years. In this paper, we study the stability and exponential decay near magnetic hydrostatic equilibrium to the two-dimensional magneto-micropolar equations with partial dissipation in the domain $ \Omega =\mathbb {T}\times \mathbb {R} $ Ω = T × R . In particular, we takes advantage of the geometry of the domain $ \mathbb {T}\times \mathbb {R} $ T × R to divide u into zeroth mode and the nonzero modes, and obey a strong version of the Poincaré's inequality, which plays a crucial role in controlling the nonlinearity. Moreover, we find that the oscillation part of the solution decays exponentially to zero. Finally, our result mathematically verifies that the stabilization effect of a background magnetic field on magneto-micropolar fluids. [ABSTRACT FROM AUTHOR]

In this paper, we study the following chemotaxis system with signal-dependent motility arising from tumor invasion u t = Δ (φ (v) u) + ρ u - μ u l , x ∈ Ω , t > 0 , v t = Δ v + w z , x ∈ Ω , t > 0 , w t = - w z , x ∈ Ω , t > 0 , z t = Δ z - z + u , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n (n ≥ 1) , where the parameters ρ ≥ 0 , μ ≥ 0 and l > 1 are constants, the motility function φ (v) satisfies φ (v) ∈ C 3 ([ 0 , + ∞)) , φ 1 ≤ φ (v) ≤ φ 2 and | φ ′ (v) | ≤ φ 3 with φ 1 , φ 2 , φ 3 > 0. The purpose of this paper is to prove that the existence of global bounded solution for 1 ≤ n ≤ 3 . For n ≥ 4 , we prove that the nonnegative classical solution (u, v, w, z) is globally bounded if l > n 2 . In addition, we also show that all the global bounded solution will converge to the non-trivial constant steady state exponentially. [ABSTRACT FROM AUTHOR]

This paper is concerned with the global existence and large-time behavior of strong solutions to an initial boundary value problem of the one-dimensional isentropic compressible Navier-Stokes equations with degenerate density-dependent viscosity on the half line $ \mathbb{R}^+ = (0,+\infty) $. We focus on the impermeable wall problem where the velocity $ u(t,x) $ of the fluid is zero on the boundary $ x = 0 $. The case when the pressure $ p(v) = v^{-\gamma} $ and the viscosity coefficient $ \mu(v) = v^{-\alpha} $ for the specific volume $ v(t,x)>0 $ is considered, where $ \alpha,\gamma\in \mathbb{R} $ are some parameters. Under some suitable assumptions on the parameters $ \alpha,\gamma $ and the initial data, we prove that the impermeable wall problem of the one-dimensional compressible Navier-Stokes equations admits a unique global strong non-vacuum solution, which tends to the constant equilibrium state as time goes to infinity. Here the initial perturbation can be arbitrarily large. Our analysis is based on a new approach to derive the uniform-in-time positive lower and upper bounds on the specific volume and delicate estimates to control the boundary terms. [ABSTRACT FROM AUTHOR]

We consider the compressible Navier–Stokes equations for a reacting mixture describing the dynamic combustion in multidimensional unbounded domains. We prove the global existence, uniqueness, and large-time asymptotic behavior of spherically and cylindrically symmetric solutions with large initial data. The key point in the proof is to establish uniform-in-time bounds for the specific volume and temperature. Several new estimates are derived in order to handle the reacting terms and the unboundedness of the domain. [ABSTRACT FROM AUTHOR]

We investigate an n -dimensional free boundary problem modeling tumor growth in radial form with angiogenesis process. This process is reflected in the Robin boundary condition where time-dependent absorption rate β (t) is considered. The system consists of a reaction-diffusion equation for the nutrient u (r , t) with nonlinear consumption and an integro-differential equation for the tumor radius R (t) with nonlinear cell proliferation. It is shown that the large-time behavior of the tumor is greatly tied to the properties of β (t). We prove that for any sufficiently small c > 0 , the tumor radius R (t) will remain bounded or shrink to zero when β (t) is uniformly bounded or tends to zero, respectively; and if β (t) → β ⁎ , the solution (u , R) will converge to the unique steady state (u ⁎ , R ⁎) associating with β ⁎. We also give examples of R (t) blowing up as t → ∞ even if β (t) → 0 when c is not small. [ABSTRACT FROM AUTHOR]

The large-time behavior of a nonlinearly coupled pair of measure-valued transport equations with discontinuous boundary conditions, parameterized by a positive real-valued parameter λ, is considered. These equations describe the hydrodynamic or fluid limit of many-server queues with reneging (with traffic intensity λ), which model phenomena in diverse disciplines, including biology and operations research. For a broad class of reneging distributions with finite mean, and service distributions with finite mean and hazard rate function that is either nonincreasing or bounded away from zero and infinity, it is shown that if the fluid equations have a unique invariant state, then the Dirac measure at this invariant state is the unique invariant distribution of the fluid equations. In particular, this implies that the stationary distributions of scaled N-server systems converge to the unique invariant state of the corresponding fluid equations. Moreover, when the mean arrival rate is not equal to the mean service rate, that is, when \lambda \not= 1, it is shown that the solution to the fluid equation starting from any initial condition converges to this unique invariant state in the large-time limit. The proof techniques are different under the two sets of assumptions on the service distribution, as well as under the two regimes λ < 1 and λ > 1. When the hazard rate function is nonincreasing, a reformulation of the dynamics in terms of a certain renewal equation is used, in conjunction with recursive asymptotic estimates. When the hazard rate function is bounded away from zero and infinity, the proof uses an extended relative entropy functional as a Lyapunov function. Analogous large-time convergence results are also established for a system of coupled measure-valued equations modeling a multiclass queue. [ABSTRACT FROM AUTHOR]

In this article, we study the asymptotic behavior and decay of the solution of the fully discrete heat problem. We show basic properties of its solutions, such as the mass conservation principle and their moments, and we compare them to the known ones for the continuous analogue problems. We present the fundamental solution, which is given in terms of spherical harmonics, and we state pointwise and ℓ p estimates for that. Such considerations allow to prove decay and large-time behavior results for the solutions of the fully discrete heat problem, giving the corresponding rates of convergence on ℓ p spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study a free-boundary problem of the liquid-gas two-phase drift-flux model with constant viscosity, where the initial liquid and gas masses are assumed to be connected with vacuum continuously, and the pressure function takes a singular form. We utilize the flow map to reformulate the problem and prove the global existence and decay rates of strong solutions with general initial data that involve transition to pure single-phase points or regions. In particular, both the optimal decay rates of the mass functions and the decay estimates of the velocity towards its asymptotic profile as time goes to infinity are studied. As far as we are concerned, this is the first result on the global existence of strong solutions to the two-phase drift-flux model with singular pressure function and with general initial data that allow for transition to single-phase flow. It is also worth noting that our method in recovering the non-weighted estimates of the solution by utilizing the Poincaré type inequality can apply to the single-phase problem studied in Zeng (2015) [37] , so as to derive the decay estimates for the velocity and relax the constraints on the initial data. [ABSTRACT FROM AUTHOR]

To uncover that the magnetic field mechanism can stabilize electrically conducting turbulent fluids, we investigate the stability of a special two dimensional anisotropic MHD system with vertical dissipation in the horizontal velocity component and partial magnetic damping near a background magnetic field. Since the MHD system has only vertical dissipation in the horizontal velocity and vertical magnetic damping, the stability issue and large time behavior problem of the linearized magneto-hydrodynamic system is not trivial. By performing refined energy estimates on the linear system coupled with a careful analysis of the nonlinearities, the stability of a MHD-type system near a background magnetic field is justified for the initial data belonging to H 3 (R 2) space. The authors also build the explicit decay rates of the linearized system. [ABSTRACT FROM AUTHOR]

In this paper we mainly deal with the global well-posedness and large-time behavior of the 2D tropical climate model with small initial data. We first establish the global well-posedness of solution in the Besov space, then we obtain the optimal decay rates of solutions by virtue of the frequency decomposition method. Specifically, for the low frequency part, we use the Fourier splitting method of Schonbek and the spectrum analysis method, and for the high frequency part, we use the global energy estimate and the behavior of exponentially decay operator. [ABSTRACT FROM AUTHOR]

This paper studies the stability of large data solutions to the 2D fully/partially dissipative Boussinesq systems on the unit square subject to various types of boundary conditions, including dynamic Couette flow. It is shown that under suitable conditions on the boundary data, solutions starting in H 3 exist globally in time and the differences between the solutions and their corresponding boundary data converge to zero in certain topology as time goes to infinity. There is no smallness restrictions on initial data. [ABSTRACT FROM AUTHOR]

Abstract We consider the two-phase flow model in 3D exterior domains with slip boundary conditions. We establish the global existence of classical solutions of this system, provided that the initial energy is suitably small. Furthermore, we prove that the pressure has large oscillations and contains vacuum states when the initial pressure allows large oscillations and a vacuum. Finally, we also obtain the large-time behavior of the classical solutions.

A tumor invasion system, which is proposed in [2] as a modified tumor invasion system of Chaplain-Anderson type in [1], is considered in this paper. The system consists of three PDEs and one ODE. As a main characteristic, it is given that one of the PDEs has not only a quasi-variational structural degenerate diffusion but also a chemotaxis effect. Moreover, two kinds of quasi-variational structures are mixed in the diffusion. One is in the permeability of the tissue, which is regarded as a porous medium, and the other is in the constraint condition, which plays a role as double obstacles for the density of tumor cells. Actually, these quasi-variational structures depend on the density of extracellular matrix, which is also unknown in the system. The main purpose of this paper is to investigate the large-time behavior of a strong global-in-time solution to the tumor invasion system. In order to do this, it is a key point to find a suitable function space so that we can repeatedly apply the Ascoli-Arzelà compactness theorem for each component of the orbit associated with the strong global-in-time solution to the system under consideration. [ABSTRACT FROM AUTHOR]

We study a logarithmic Keller-Segel system proposed by Rodríguez for crime modeling as follows: $ \begin{equation*} \left\{ \begin{split} &u_t = \Delta u-\chi\nabla\cdot\left(u\nabla\ln v\right)- \kappa uv+ h_1,\\ &v_t = \Delta v- v+ u+h_2, \end{split} \right. \end{equation*} $ in a bounded and smooth spatial domain $ \Omega\subset \mathbb R^n $ with $ n\geq3 $, with the parameters $ \chi > 0 $ and $ \kappa > 0 $, and with the nonnegative functions $ h_1 $ and $ h_2 $. For the case that $ \kappa = 0 $, $ h_1\equiv0 $ and $ h_2\equiv0 $, recent results showed that the corresponding initial-boundary value problem admits a global generalized solution provided that $ \chi < \chi_0 $ with some $ \chi_0 > 0 $. In the present work, our first result shows that for the case of $ \kappa > 0 $ such problem possesses global generalized solutions provided that $ \chi < \chi_1 $ with some $ \chi_1 > \chi_0 $, which seems to confirm that the mixed-type damping $ -\kappa uv $ has a regularization effect on solutions. Besides the existence result for generalized solutions, a statement on the large-time behavior of such solutions is derived as well.

We consider the two-phase flow model in 3D exterior domains with slip boundary conditions. We establish the global existence of classical solutions of this system, provided that the initial energy is suitably small. Furthermore, we prove that the pressure has large oscillations and contains vacuum states when the initial pressure allows large oscillations and a vacuum. Finally, we also obtain the large-time behavior of the classical solutions. [ABSTRACT FROM AUTHOR]

This paper investigates the global well-posedness and the stability of perturbations near a background magnetic field on the 2D incompressible anisotropic magnetohydrodynamic equations. More precisely, we consider the system with partial mixed velocity dissipations and horizontal magnetic diffusion. Two goals are achieved. First, we obtain the global well-posedness and the H 2 -stability for the nonlinear MHD equations. The approach is to apply the bootstrapping argument. Efforts are devoted to the a priori estimate of a energy functional. Second, we establish the long-time behavior of explicit decay rates of the solution in homogeneous Sobolev spaces H ˙ s for the linear system under the suitable assumptions for the initial data. Our work proves that any perturbations near a background magnetic field remain asymptotically stable. [ABSTRACT FROM AUTHOR]

This paper aims as the stability and large-time behavior of 3D incompressible magnetohydrodynamic (MHD) equations with fractional horizontal dissipation and magnetic diffusion. By using the energy methods, we obtain that if the initial data are small enough in H 3 (R 3) , then this system possesses a global solution, and whose horizontal derivatives decay at least at the rate of (1 + t) - 1 2 . Moreover, if we control the initial data further small in H 3 (R 3) ∩ H h - 1 (R 3) , the sharp decay of this solution and its first-order derivatives is established. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability problem and large‐time behavior for the 3D Boussinesq equations with mixed partial kinematic viscosity and thermal diffusivity near one equilibrium, i.e., (U(0),Θ(0))=((0,0,0),βx3)$$ \left({U}&#x0005E;{(0)},{\Theta}&#x0005E;{(0)}\right)&#x0003D;\left(\left(0,0,0\right),\beta {x}_3\right) $$ with β=1$$ \beta &#x0003D;1 $$ (where β$$ \beta $$ is chosen to be large, e.g., satisfying the well‐known Miles‐Howard criterion). These problems can be extremely difficult due to the equations lack of full dissipation. Our results are twofold. In the first place, we establish the global H2$$ {H}&#x0005E;2 $$ and H3$$ {H}&#x0005E;3 $$ stability for the nonlinear system with seven different partially dissipated systems of Boussinesq equations. In the second place, we derive precise large‐time behavior for the linearized system. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the weakly dissipative peakon equations with higher-order nonlinearity. Under the effect of dissipation, we first derive the infinite propagation speed if the initial datum has a nonnegative compact support. Furthermore, we obtain the large-time behavior of the support of momentum density with the initial data compactly supported. The corresponding results are obtained by using some prior estimates and the energy method. It is worth noting that we need to overcome the difficulties caused by the high-order nonlinear structure and the dissipative effect of the equation. The obtained results generalize the previous results to a certain degree. [ABSTRACT FROM AUTHOR]

This paper concerns with the Cauchy problem of the 1-D bipolar hydrodynamic model for semiconductors, a system of Euler-Poisson equations with time-dependent damping effects − J (1 + t) − λ and − K (1 + t) − λ for − 1 < λ < 1. Here, we consider a more physical case that allows the two pressure functions can be different and the doping profile can be non-zero. Different from the previous study [Li HT, Li JY, Mei. M, et al. Asymptotic behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping. J Math Anal Appl. 2019;437:1081-1121] which considered two identical pressure functions and zero doping profile, the asymptotic profiles of the solutions to this model are constant states rather than the nonlinear diffusion waves. When the initial perturbation around the constant states are sufficiently small in the sense of L 2 , by means of the time-weighted energy method, we prove the global existence and uniqueness of the smooth solutions to the Cauchy problem, and obtain the optimal convergence rates of the solutions toward the constant states. [ABSTRACT FROM AUTHOR]

This paper concerns the one-dimensional compressible Navier–Stokes system with temperature-dependent heat conductivity in R with large initial data. We prove that velocity and temperature are uniformly bounded from below and above in time and space when the heat conductivity coefficient takes κ = κ ¯ (1 + θ b) for all b > 5 2 . In addition, we show that the global solution is asymptotically stable as time tends to infinity. [ABSTRACT FROM AUTHOR]

We study the decay/growth rates in all $ L^p $ norms of solutions to an inhomogeneous nonlocal heat equation in $ \mathbb{R}^N $ involving a Caputo $ \alpha $-time derivative and a power $ \beta $ of the Laplacian when the dimension is large, $ N > 4\beta $. Rates depend strongly on the space-time scale and on the time behavior of the spatial $ L^1 $ norm of the forcing term.

In this article, we study the global well‐posedness and large‐time behaviors of solutions to the two‐dimensional tropical climate system with zero thermal diffusion for small initial data in the whole space. The main approaches include high‐ and low‐frequency decomposition method and exploiting the structure of system (1) to obtain the estimates of thermal dissipation. We utilize the time decay properties of the kernels to a linear differential equation to obtain the decay rates of solutions of the low‐frequency part and the decay property of exponential operator for the high‐frequency part. The key ingredient here is the explicit large‐time decay rate of solutions. [ABSTRACT FROM AUTHOR]

We shall investigate the large-time behavior of solutions to the Cauchy problem for the one-dimensional bipolar quantum Euler-Poisson system with critical time-dependent over-damping. By means of the time-weighted energy method, we prove that the smooth solutions to the Cauchy problem exist uniquely and globally, and time-asymptotically converge to the nonlinear diffusion waves when the initial perturbation around the nonlinear diffusion waves are small enough. Particularly, we show the optimal decay rates of solutions toward the nonlinear diffusion waves. [ABSTRACT FROM AUTHOR]

This paper is concerned with the large-time behavior of smooth solutions to the Cauchy problem for the three-dimensional bipolar quantum Navier–Stokes–Poisson equations. First, we show the existence of the stationary wave under the small assumption of the doping profile. Next, based on elaborate energy estimates, we prove that the smooth solutions to the Cauchy problem exist uniquely and globally, and time-asymptotically converge to the stationary wave when the prescribed initial data are close to the stationary wave. Finally, we obtain the L 2 -decay rate of solutions toward the stationary wave by means of a detailed analysis of the Green function associated to the corresponding linear problem and the energy estimates for the nonlinear system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic behavior and decay of the solution of the discrete in time N-dimensional heat equation. We give a convergence rate with which the solution tends to the discrete fundamental solution, and the asymptotic decay, both in L p (R N). Furthermore, we prove optimal L 2 -decay of solutions. Since the technique of energy methods is not applicable, we follow the approach of estimates based on the discrete fundamental solution which is given by an original integral representation and also by MacDonald's special functions. As a consequence, the analysis is different to the continuous in time heat equation and the calculations are rather involved. [ABSTRACT FROM AUTHOR]

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In this paper, we study the global existence and large–time behavior of strong solutions to a two–phase flow model with magnetic field in three dimensions. The system consists of compressible isothermal Euler equations coupled with compressible magnetohydrodynamic system through a drag forcing term. The coupled system can be derived as the hydrodynamic limit of the Vlasov–Fokker–Planck/compressible magnetohydrodynamic equations with strong local alignment forces. We establish the global existence of strong solutions of the coupled system by standard energy method under suitable assumptions on the initial data. We also show that the strong solution converges to its associated equilibrium with the optimal rate which is the same as that of the heat equation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the large-time behavior for the non-isentropic compressible Navier–Stokes equations with capillarity in the whole space R d ( d ≥ 3 ). Under an additional smallness assumption of the low frequencies of initial data, the time-decay estimates of L q – L r type for global strong solutions near constant equilibrium (away from vacuum) can be deduced by establishing the time-weighted energy inequality. On the other hand, a pure energy approach (without the spectral analysis) different from the time-weighted energy method is performed, which allows us not only to get the time-decay rates but also to remove the smallness condition of low frequencies of initial data. The treatment of new nonlinear terms arising from capillary mainly depends on non classical Besov product estimates and the refined use of Sobolev embeddings and interpolations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the well-posedness results of the three-dimensional incompressible Hall-magnetohydrodynamic equations with fractional dissipation. More precisely, we provide a direct proof of the local well-posedness of smooth solutions for the Hall-magnetohydrodynamic equations with the diffusive term for the magnetic field consisting of the fractional Laplacian with its power bigger than or equal to one half. Furthermore, the small data global well-posedness results are also derived. In addition, we obtain the optimal decay rate when the fractional powers are further restricted to a certain range. [ABSTRACT FROM AUTHOR]

In this paper, we consider the two-species chemotaxis-Navier-Stokes system with Lotka-Volterra competitive kinetics and fractional diffusion of order s ∈ (1 2 , 1) { (n 1) t + u ⋅ ∇ n 1 = − (− Δ) s n 1 − χ 1 ∇ ⋅ (n 1 ∇ c) + μ 1 n 1 (1 − n 1 − a 1 n 2) , (n 2) t + u ⋅ ∇ n 2 = − (− Δ) s n 2 − χ 2 ∇ ⋅ (n 2 ∇ c) + μ 2 n 2 (1 − a 2 n 1 − n 2) , c t + u ⋅ ∇ c = Δ c − (α n 1 + β n 2) c , u t + (u ⋅ ∇) u = Δ u + ∇ P + (γ n 1 + δ n 2) ∇ ϕ , ∇ ⋅ u = 0 on the three dimensional periodic torus T 3 . Due to the influence of the Navier-Stokes equation, we cannot get the existence of the classical solution to above system in the three-dimensional case. Instead, we investigate the global existence of the weak solution of the system with weaker cell diffusion and arrive at the eventual smoothness of the solution. Furthermore, by constructing different Lyapunov functionals, we arrive at the following asymptotic stability: • When a 1 , a 2 ∈ (0 , 1) , the solution components (n 1 , n 2 , c , u) → (1 − a 1 1 − a 1 a 2 , 1 − a 2 1 − a 1 a 2 , 0 , 0) as t → ∞ ; • When a 1 ≥ 1 > a 2 > 0 , the solution components (n 1 , n 2 , c , u) → (0 , 1 , 0 , 0) as t → ∞ ; • When a 2 ≥ 1 > a 1 > 0 , the solution components (n 1 , n 2 , c , u) → (1 , 0 , 0 , 0) as t → ∞ . [ABSTRACT FROM AUTHOR]

This paper focuses on a special 2D Boussinesq equation with partial dissipation, for which the velocity equation involves no dissipation and there is only damping in the horizontal component equation. Without buoyancy force, the corresponding vorticity equation is a 2D Euler-like equation with an extra Calderon–Zygmund-type term. Its stability is an open problem. Our results reveal that the buoyancy force exactly stabilizes the fluids by the coupling and interaction between the velocity and temperature. In addition, we prove the solution decays exponentially to zero in Sobolev norm. [ABSTRACT FROM AUTHOR]

In this paper, we address the existence, uniqueness, decay estimates, and the large-time behavior of the Radon measure-valued solutions for a class of nonlinear strongly degenerate parabolic equations involving a source term under Neumann boundary conditions with bounded Radon measure as initial data. $ \begin{equation*} \begin{cases} u_{t} = \Delta\psi(u)+h(t)f(x, t) \ \ &\text{in} \ \ \Omega\times(0, T), \\ \frac{\partial\psi(u)}{\partial\eta} = g(u) \ \ &\text{on} \ \ \partial\Omega\times(0, T), \\ u(x, 0) = u_{0}(x) \ \ &\text{in} \ \ \Omega, \end{cases} \end{equation*} $ where $ T > 0 $, $ \Omega\subset \mathbb{R}^{N}(N\geq2) $ is an open bounded domain with smooth boundary $ \partial\Omega $, $ \eta $ is an outward normal vector on $ \partial\Omega $. The initial value data $ u_{0} $ is a nonnegative bounded Radon measure on $ \Omega $, the function $ f $ is a solution of the linear inhomogeneous heat equation under Neumann boundary conditions with measure data, and the functions $ \psi $, $ g $ and $ h $ satisfy the suitable assumptions.