INTERFACE BEHAVIOR AND DECAY RATES OF COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND A VACUUM

In this paper,we study the one-dimensional motion of viscous gas near a vacuum,with the gas connecting to a vacuum state with a jump in density.The interface behavior,the pointwise decay rates of the density function and the expanding rates of the interface are obtained with the viscosity coefficientμ(ρ)=ρ^(α)for any 0<α<1;this includes the timeweighted boundedness from below and above.The smoothness of the solution is discussed.Moreover,we construct a class of self-similar classical solutions which exhibit some interesting properties,such as optimal estimates.The present paper extends the results in[Luo T,Xin Z P,Yang T.SIAM J Math Anal,2000,31(6):1175-1191]to the jump boundary conditions case with density-dependent viscosity.

GLOBAL WELL-POSEDNESS AND OPTIMAL TIME DECAY RATES FOR THE GENERALIZED PHAN-THIEN-TANNER MODEL IN R^(3)

In this paper, we consider the initial value problem for the incompressible generalized Phan-Thien-Tanner(GPTT) model. This model pertains to the dynamic properties of polymeric fluids. Under appropriate assumptions on smooth function f, we find a particular solution to the GPTT model. In dimension three, we establish the global existence and the optimal time decay rates of strong solutions provided that the initial data is close to the particular solution. The results which are presented here are generalizations of the network viscoelastic models.

DECAY RATES OF PLANAR VISCOUS RAREFACTION WAVE FOR MULTI-DIMENSIONAL SCALAR CONSERVATION LAW WITH DEGENERATE VISCOSITY ON HALF SPACE

We investigate the decay rates of the planar viscous rarefaction wave of the initial-boundary value problem to scalar conservation law with degenerate viscosity in several dimensions on the half-line space, where the corresponding one-dimensional problem admits the rarefaction wave as an asymptotic state. The analysis is based on the standard L2-energy method and L1-estimate.

THE TIME DECAY RATES OF THE CLASSICAL SOLUTION TO THE POISSON-NERNST-PLANCK-FOURIER EQUATIONS IN R^(3)

In this work,the Poisson-Nernst-Planck-Fourier system in three dimensions is considered.For when the initial data regards a small perturbation around the constant equilibrium state in a H^(3)∩■^(-s)(0≤s≤1/2)norm,we obtain the time convergence rate of the global solution by a regularity interpolation trick and an energy method.

Decay rates of higher-order norms of solutions to the Navier-Stokes-Landau-Lifshitz system

In this paper, we investigate a system of the incompressible Navier-Stokes equations coupled with Landau-Lifshitz equations in three spatial dimensions. Under the assumption of small initial data, we establish the global solutions with the help of an energy method. Furthermore, we obtain the time decay rates of the higher-order spatial derivatives of the solutions by applying a Fourier splitting method introduced by Schonbek(SCHONBEK, M. E. L2decay for weak solutions of the Navier-Stokes equations. Archive for Rational Mechanics and Analysis, 88, 209–222(1985)) under an additional assumption that the initial perturbation is bounded in L1(R3).

A solution of the complex Ginzburg-Landau equation with a continuum of decay rates

We prove the existence of a uniform initial datum whose solution decays, in var- ious Lp spaces, at different rates along different time sequences going to infinity, for complex Ginzburg-Landau equation on RN, of various parameters θ and γ.

The sharp time-decay rates for one-dimensional compressible isentropic Navier-Stokes and magnetohydrodynamic flows

In this paper, we investigate the large-time behavior of strong solutions to the Cauchy problem for one-dimensional compressible isentropic magnetohydrodynamic equations near a stable equilibrium. The difference between the one-dimensional and multi-dimensional cases is a feature for compressible flows and also brings new difficulties. In contrast to the multi-dimensional case, the decay rates of nonlinear terms may not be faster than linear terms in dimension one. To handle this, we shall present a new energy estimate in terms of a combination of the solutions with small initial data. We aim to establish the sharp upper and lower bounds on the L^(2)-decay rates of the solutions and all their spatial derivatives when the initial perturbation is small in L^(1)(R) ∩ H^(2)(R). It is worth noticing that there is no decay loss for the highest-order spatial derivatives of the solutions so that the large-time behavior for the hyperbolic-parabolic system is exactly sharp. As a byproduct, the above result is also valid for compressible Navier-Stokes equations. Our approach is based on various interpolation inequalities, energy estimates, spectral analysis, and Fourier time-splitting and high-low frequency decomposition methods.

Global existence and time decay rates for the 3D bipolar compressible Navier-Stokes-Poisson system with unequal viscosities

We consider the Cauchy problem for the three-dimensional bipolar compressible Navier-Stokes-Poisson system with unequal viscosities.Under the assumption that the H_(3) norm of the initial data is small but its higher order derivatives can be arbitrarily large,the global existence and uniqueness of smooth solutions are obtained by an ingenious energy method.Moreover,if additionally,the H^(−s)(1/2≤s<3/2)or B^(−s)_(2,∞)(1/2<s≤3/2)norm of the initial data is small,the optimal decay rates of solutions are also established by a regularity interpolation trick and delicate energy methods.

Environmental effects on nuclear decay rates

The possible change of nuclear decay rates in different environments has long been an interesting topic due to its importance not only in nuclear physics but also in astrophysics, geological dating, condensed matter physics, etc. The progress in the investigation of variations in nuclear decay rates are reviewed.

THE OPTIMAL LARGE TIME BEHAVIOR OF 3D QUASILINEAR HYPERBOLIC EQUATIONS WITH NONLINEAR

We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping.The main novelty of this paper is two-fold.First,we prove the optimal decay rates of the second and third order spatial derivatives of the solution,which are the same as those of the heat equation,and in particular,are faster than ones of previous related works.Second,for well-chosen initial data,we also show that the lower optimal L^(2) convergence rate of the k(∈[0,3])-order spatial derivatives of the solution is(1+t)^(-(2+2k)/4).Therefore,our decay rates are optimal in this sense.The proofs are based on the Fourier splitting method,low-frequency and high-frequency decomposition,and delicate energy estimates.

DECAY ESTIMATES OF PLANAR STATIONARY WAVES FOR DAMED WAVE EQUATIONS WITH NONLINEAR CONVECTION IN MULTI-DIMENSIONAL HALF SPACE

This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space R n ＋ ： u tt u ＋ u t ＋ divf （u） = 0, t 〉 0, x = （x 1 , x ′ ） ∈ R n ＋ ：= R ＋ × R n 1 , u（0, x） = u 0 （x） → u ＋ , as x 1 → ＋ ∞ , u t （0, x） = u 1 （x）, u（t, 0, x ′ ） = u b , x ′ = （x 2 , x 3 , ··· , x n ） ∈ R n 1 . （I） For the non-degenerate case f ′ 1 （u ＋ ） 〈 0, it was shown in [10] that the above initialboundary value problem （I） admits a unique global solution u（t, x） which converges to the corresponding planar stationary wave φ（x 1 ） uniformly in x 1 ∈ R ＋ as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. And in [10] Ueda, Nakamura, and Kawashima proved the algebraic decay estimates of the tangential derivatives of the solution u（t, x） for t → ＋ ∞ by using the space-time weighted energy method initiated by Kawashima and Matsumura [5] and improved by Nishihkawa [7]. Moreover, by using the same weighted energy method, an additional algebraic convergence rate in the normal direction was obtained by assuming that the initial perturbation decays algebraically. We note, however, that the analysis in [10] relies heavily on the assumption that f ′ （u） 〈 0. The main purpose of this paper isdevoted to discussing the case of f ′ 1 （u b ） ≥ 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.

THE GLOBAL EXISTENCE AND ANALYTICITY OF A MILD SOLUTION TO THE 3D REGULARIZED MHD EQUATIONS

In this paper,we study the three-dimensional regularized MHD equations with fractional Laplacians in the dissipative and diffusive terms.We establish the global existence of mild solutions to this system with small initial data.In addition,we also obtain the Gevrey class regularity and the temporal decay rate of the solution.

THE DECAY ESTIMATES FOR MAGNETOHYDRODYNAMIC EQUATIONS WITH COULOMB FORCE

In this article we consider the compressible viscous magnetohydrodynamic equations with Coulomb force.By spectral analysis and energy methods,we obtain the optimal time decay estimate of the solution.We show that the global classical solution converges to its equilibrium state at the same decay rate as the solution of the linearized equations.

THE GLOBAL EXISTENCE AND A DECAY ESTIMATE OF SOLUTIONS TO THE PHAN-THEIN-TANNER MODEL

In this paper,we study the global existence and decay rates of strong solutions to the three dimensional compressible Phan-Thein-Tanner model.By a refined energy method,we prove the global existence under the assumption that the H^(3) norm of the initial data is small,but that the higher order derivatives can be large.If the initial data belong to homogeneous Sobolev spaces or homogeneous Besov spaces,we obtain the time decay rates of the solution and its higher order spatial derivatives.Moreover,we also obtain the usual L^(p)-L^(2)(1≤p≤2)type of the decay rate without requiring that the Lpnorm of initial data is small.

The decay kinetics of residual chlorine in cooling seawater simulation experiments

To find out the decay character of residual chlorine （RC） in the sea water, the concentration of RC was analyzed by N, N-diethyl-p-phenylenediamine （DPD） method under different simulation experimental conditions, in which salinity, temperature, and Chemical Oxygen Demand （COD） were selected. The water used in the experiment was the mixture of aging ocean water, coastal water and extracting solution of coastal sediment at appropriate level. Results are shown as follows： （1） Piecewise function can well reflect the decay dynamics of RC in the cooling seawater. Concretely, the decay dynamics of first 1 min is too rapid to ascertain using a specific kinetic function, and that of the time from 1 to 30 min is fit for the first-order kinetic model. （2） The results could be the foundation of the chemical behavior of RC in seawater, and be used as not only the guidance of the coastal power plants production and sea water desalting companies, but also the establishment of the correlative trade standard.

DECAY RATE FOR DEGENERATE CONVECTION DIFFUSION EQUATIONS IN BOTH ONE AND SEVERAL SPACE DIMENSIONS

We consider degenerate convection-diffusion equations in both one space dimension and several space dimensions. In the first part of this article, we are concerned with the decay rate of solutions of one dimension convection diffusion equation. On the other hand, in the second part of this article, we are concerned with a decay rate of derivatives of solution of convection diffusion equation in several space dimensions.

TIME DECAY RATE OF SOLUTIONS TO THE HYPERBOLIC MHD EQUATIONS IN R3

In this paper, we first show the global existence, uniqueness and regularity of weak solutions for the hyperbolic magnetohydrodynamics(MHD) equations in R^3. Then we establish that the solutions with initial data belonging to H^m(R^3) ∩ L^1(R^3) have the following time decay rate:║▽~mu(x, t) ║~2+║ ▽~mb(x, t)║~ 2+ ║▽^(m+1)u(x, t)║~ 2+ ║▽^(m+1)b(x, t) ║~2≤ c(1 + t)^(-3/2-m)for large t, where m = 0, 1.

Decay rate of Larix gmelinii coarse woody debris on burned patches in the Greater Khingan Mountains

The decomposition of coarse woody debris(CWD)affects the energy flow and nutrient cycling in forest ecosystems.Previous studies on CWD have focused on the input,decomposition,reserve dynamics,and CWD functions,but coarse woody debris decomposition is complex and the results from different regions vary considerably.It is not clear which factors affect decay rate(k),especially at different decomposition stages.In this study,a single-exponential decay model was used to analyze the characteristics of CWD decomposition in Larix gmelinii forests over the 33 years following a fire in the Greater Khingan Mountains.The results show that the decay rate of coarse woody debris was positively correlated to decay class.The average decomposition rate was 0.019,and 41 years and 176 years are needed for a 50%and 95%mass loss,respectively.CWD nutrient content,density,and water content could explain the variance in the decay rate(~42%)of the decay factors such as amount of leaching,degree of fragmentation,respiration of the debris,and biotransformation,and varied significantly between different decay classes.Using the space-time substitution method,this study arranged the coarse woody debris of different mortality times to form a 33 year chronosequence which revealed the decomposition process.It was concluded that the decay rate was mainly explained by structural component of the debris and its nitrogen and water contents.This paper quantifies the indicators affecting CWD decay to explain the decomposition process.

DECAY RATE OF FOURIER TRANSFORMS OF SOME SELF-SIMILAR MEASURES

This paper is concerned with the Diophantine properties of the sequence {ξθn}, where 1 ≤ξ 〈 θ and θ is a rational or an algebraic integer. We establish a combinatorial proposition which can be used to study such two cases in the same manner. It is shown that the decay rate of the Fourier transforms of self-similar measures μλ with λ = θ-1 as the uniform contractive ratio is logarithmic. This generalizes some results of Kershner and Bufetov-Solomyak, who consider the case of Bernoulli convolutions. As an application, we prove that μλ ahaost every x is normal to any base b ≥ 2, which implies that there exist infinitely many absolute normal numbers on the corresponding self-similar set. This can be seen as a complementary result of the well-known Cassels-Schmidt theorem.

Improved spatial decay bounds in the plane Stokes flow

This paper investigates spatial decay bounds and a decay rate for the time- dependent Stokes flow of a viscous fluid in a semi-infinite channel. We show how to obtain a near optimal decay rate that is independent of the Reynolds number. We also show that a modification to an analysis given in the literature and a better choice of arbitrary constants yield a decay rate 1.328, which is clearly an improvement compared to 0.91 obtained in the referenced work.