Time-weighted Energy Research Articles

Global well-posedness and decay estimates for the one-dimensional models of blood flow with a general parabolic velocity profile

In this paper, we study the one-dimensional models of blood flow arising from the hemodynamics of aorta, which are derived from the averaging of the Navier–Stokes equations. We establish the global well-posedness and long-time behavior of the viscid 1D models of blood flow in the Sobolev space framework, where a general parabolic velocity profile is considered. Precisely speaking, we prove the global existence of smooth solution when the initial data is sufficiently small. Moreover, by combining the time-weighted energy estimates with the Green function method, we obtain the optimal time decay rate in Lp (2≤p≤∞) norm. In addition, one can see the area-averaged axial velocity decays faster than the cross-sectional area of vessel from the Green function of the linearized system. This observation is essential to study the decay rates of our nonlinear system.

Convergence rates to the Barenblatt solutions for the compressible Euler equations with time-dependent damping

In this paper, we are concerned with the asymptotic behavior of L∞ weak entropy solutions for the compressible Euler equations with time-dependent damping and vacuum for any large initial data. This model describes the motion for the compressible fluid through a porous medium, and the friction force is time-dependent. We obtain that the density converges to the Barenblatt solution of a well-known porous medium equation with the same finite initial mass in L1 decay rate when 1+52<γ≤2,0≤λ<γ2−γ−1γ2+γ−1 or γ≥2,0≤λ<12γ+1 which partially improves and extends the previous work [14,6]. The proof is mainly based on the detailed analysis of the relative weak entropy, time-weighted energy estimates and the iterative method.

Global well-posedness of 2D incompressible Oldroyd-B model with only velocity dissipation

The small data global well-posedness in Sobolev setting for the incompressible Oldroyd-B equations with only velocity dissipation (namely there is no damping or dissipation of stress tensor) on R2 remains open by now. The approach for the similar problem on R3 becomes invalid due to the fewer decay rates. In this paper, we will give a positive answer to this problem. Namely, we shall focus on the Cauchy problem for the incompressible Oldroyd-B equations with only velocity dissipation on R2 and derive the small data global well-posedness under Sobolev setting. We also give the time decay estimate for the solutions. To achieve this goal, we explore the dissipative structure of system and develop the fractional time-weighted energy framework. Moreover, some delicate commutator and bilinear estimates are proposed to deal with the wildest nonlinear terms.

Large-time behavior of solutions to three-dimensional bipolar Euler–Poisson equations with time-dependent damping

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 for . 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.

Time-weighted estimates for the Blackstock equation in nonlinear ultrasonics

High frequencies at which ultrasonic waves travel give rise to nonlinear phenomena. In thermoviscous fluids, these are captured by Blackstock’s acoustic wave equation with strong damping. We revisit in this work its well-posedness analysis. By exploiting the parabolic-like character of this equation due to strong dissipation, we construct a time-weighted energy framework for investigating its local solvability. In this manner, we obtain the small-data well-posedness on bounded domains under less restrictive regularity assumptions on the initial conditions compared to the known results. Furthermore, we prove that such initial boundary-value problems for the Blackstock equation are globally solvable and that their solution decays exponentially fast to the steady state.

On the Global Well-Posedness for a Hyperbolic Model Arising from Chemotaxis Model with Fractional Laplacian Operator

Several cells and microorganisms, such as bacteria and somatic, have many essential features, one of which can be modeled by the chemotaxis system, which we consider to be our main interest in this article. More precisely, we studied the hyperbolic system derived from the chemotaxis model with fractional dissipation, which is a generalization for the hyperbolic system with classical dissipation. The results of this article are divided into two parts. In the first part, we used energy methods to obtain the existence of small solutions in the Besov spaces. The second one deals with the optimal decay of perturbed solutions using a refined time-weighted energy combined with the Littlewood-Paley decomposition technique. To the authors’ best knowledge, this type of system (with fractional dissipation) has not been studied in the literature.

Optimal Decay Rates of the Compressible Euler Equations with Time-Dependent Damping in \({\mathbb {R}}^n\): (II) Overdamping Case

This paper is concerned with the large time behavior of the multidimensional compressible Euler equations with time-dependent overdamping of the form in , where , , and . This continues our previous work dealing with the underdamping case for . We show the optimal decay estimates of the solutions such that for and , and , which indicates that a stronger damping gives rise to solutions decaying optimally slower. For the critical case of , we prove the optimal logarithmical decay of the perturbation of density for the damped Euler equations such that and for . The overdamping effect reduces the decay rates of the solutions to be slow, which causes us some technical difficulty in obtaining the optimal decay rates by the Fourier analysis method and the Green function method. Here, we propose a new idea to overcome such a difficulty by artfully combining the Green function method and the time-weighted energy method.

Asymptotic behavior of solutions to the Euler-Korteweg equations with time-dependent damping

This paper is concerned with the asymptotic behavior of solutions to the Cauchy problem for the one-dimensional Euler-Korteweg equations with time-dependent damping effect $ -\frac{\mu }{(1+t)^{\lambda}}u $ for $ 0&lt;\lambda&lt;1 $, $ \mu&gt;0 $. For the case that $ v_+\neq v_- $, by means of the time-weighted energy method, we prove that the solutions to the Cauchy problem exist uniquely and globally, and tend to the nonlinear diffusion waves time-asymptotically when the initial perturbations around the diffusion waves are small enough, and we also obtain the algebraic time-convergence-rates for the solutions toward the nonlinear diffusion waves. Moreover, for the case that $ v_+ = v_- $, we show that the solutions of the Cauchy problem of this model converge to the constant states in time algebraically provided the initial data are close to the constant states. This study generalizes the results in [H.-B. Cui, J. Du, J. Evol. Equ., 18(2018), 29-47] which considered the Euler-Korteweg equations with constant coefficient damping.

Long-time behavior of solutions to the $ M_1 $ model with boundary effect

In this paper, we are concerned with the asymptotic behavior of solutions of $ M_1 $ model on quadrant $ (x,t) \in \mathbb{R}^{+} \times \mathbb{R}^{+} $. From this model, combined with damped compressible Euler equations, a more general system is introduced. We show that the solutions to the initial boundary value problem of this system globally exist and tend time-asymptotically to the corresponding nonlinear parabolic equation governed by the related Darcy's law. Compared with previous results on compressible Euler equations with damping obtained by Nishihara and Yang in [26], and Marcati, Mei and Rubino in [17], the better convergence rates are obtained. The approach adopted is based on the technical time-weighted energy estimates together with the Green's function method.

The sharp time decay rates and stability of large solutions to the two-dimensional Phan-Thien–Tanner system with magnetic field

In this paper, we consider the magnetohydrodynamic (MHD) flow of an incompressible Phan-Thien–Tanner (PTT) fluid in two space dimensions. We focus upon the sharp time decay rates (upper and lower bounds) and global-in-time stability of large strong solutions for the PTT system with magnetic field. Firstly, the convergence of large solutions to the equilibrium have been investigated and these convergence rates are shown to be sharp. We then show that two large solutions converge globally in time as long as two initial data are close to each other. One of the main objectives of this paper is to develop a way to capture L 2 -convergence result via auxiliary logarithmic time decay estimates with the initial data in L p ( R 2 ) ∩ L 2 ( R 2 ). Improving time decay rates for the high-order derivatives of large solutions by using interpolation inequalities. In addition, time-weighted energy estimate, Fourier time-splitting method, semigroup method and iterative scheme have also been utilized.

Asymptotic behavior of solutions to a chemotaxis-logistic model with transitional end-states

We study Cauchy problem of a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate. Our Cauchy data connect two different end-states for the chemical signal while the cell density takes its typical carrying capacity at the far fields. We are interested in the time-asymptotic behavior of the solution. We show that in the borderline, the component representing the chemical signal converges to a permanent, diffusive background wave, which connects the two end-states monotonically. On the other hand, the cell component converges to the spatial derivative of a heat kernel. The asymptotic solution has explicit formulation and is common to all solutions sharing the same end-states. Optimal L2 and L∞ convergence rates are obtained. We first convert the model into a 2×2 hyperbolic-parabolic system via inverse Hopf-Cole transformation. Then we apply Chapman-Enskog expansion to identify the asymptotic solution. After extracting the asymptotic solution, we use a variety of analytic tools to study the remainder and obtain optimal rates. These include time-weighted energy method, spectral analysis, Green's function estimate and iterations. Our results apply to a general class of Cauchy data for the model and for its transformed system. In particular, our results apply to large data solutions.

SN 2019va: a Type IIP Supernova with Large Influence of Nickel-56 Decay on the Plateau-phase Light Curve

ABSTRACT We present multiband photometric and spectroscopic observations of the type II supernova, (SN) 2019va, which shows an unusually flat plateau-phase evolution in its V-band light curve. Its pseudo-bolometric light curve even shows a weak brightening towards the end of the plateau phase. These uncommon features are related to the influence of 56Ni decay on the light curve during the plateau phase, when the SN emission is usually dominated by cooling of the envelope. The inferred 56Ni mass of SN 2019va is 0.088 ± 0.018 M⊙, which is significantly larger than most SNe II. To estimate the influence of 56Ni decay on the plateau-phase light curve, we calculate the ratio (dubbed as ηNi) between the integrated time-weighted energy from 56Ni decay and that from envelope cooling within the plateau phase, obtaining a value of 0.8 for SN 2019va, which is the second largest value among SNe II that has been measured. After removing the influence of 56Ni decay on the plateau-phase light curve, we found that the progenitor/explosion parameters derived for SN 2019va are more reasonable. In addition, SN 2019va is found to have weaker metal lines in its spectra compared to other SNe IIP at similar epochs, implying a low-metallicity progenitor, which is consistent with the metal-poor environment inferred from the host-galaxy spectrum. We further discuss the possible reasons that might lead to SN 2019va-like events.

Convergence to nonlinear diffusion waves for solutions of M1 model

In this paper, we are concerned with the asymptotic behavior of solutions of M1 model proposed in the radiative transfer fields. Starting from this model, combined with the compressible Euler equation with damping, we introduce a more general system. We rigorously prove that the solutions to the Cauchy problem of this system globally exist and time-asymptotically converge to the shifted nonlinear diffusion waves whose profile is self-similar solution to the corresponding parabolic equation governed by the classical Darcy's law. Moreover, the optimal convergence rates are also obtained. Compared with previous results obtained by Nishihara, Wang and Yang in [29], we have a weaker and more general condition on the initial data, and the conclusions are more sharper. The approach adopted in the paper is the technical time-weighted energy estimates with the Green function method together.

Optimal convergence rate to nonlinear diffusion waves for bipolar Euler–Poisson equations with critical overdamping

This paper is concerned with the large time behaviors of smooth solutions to the Cauchy problem of the one dimensional bipolar Euler-Poisson equations with the time dependent critical overdamping. We show that in this critical overdamping case the bipolar Euler-Poisson system admits a unique global smooth solution that asymptotically converges to the nonlinear diffusion wave. In particular, the optimal convergence rate in logarithmic form is derived when the initial perturbations are L 2 sense by using the technical time-weighted energy method.

Optimal convergence rates to diffusion waves for solutions of p-system with damping on quadrant

This paper is concerned with the asymptotic behavior of solutions to the initial-boundary value problem for the p-system with linear damping. We show that the solutions to this system globally exist and converge time-asymptotically to nonlinear diffusion wave whose profile is self-similar solution to the corresponding parabolic equation governed by the classical Darcy's law. Compared with the results obtained by Nishihara and Yang [15], the better convergence rates are obtained. The proof is based on time-weighted energy estimates together with Green's function method.

Convergence to nonlinear diffusion waves for a hyperbolic-parabolic chemotaxis system modelling vasculogenesis

In this paper, we are concerned with a quasi-linear hyperbolic-parabolic system of persistence and endogenous chemotaxis modelling vasculogenesis. Under some suitable structural assumption on the pressure function, we first predict and derive the system admits a nonlinear diffusion wave in R driven by the damping effect. Then we show that the solution of the concerned system will locally and asymptotically converge to this nonlinear diffusion wave if the wave strength is small. By using the time-weighted energy estimates, we further prove that the convergence rate of the nonlinear diffusion wave is algebraic.

Large-time behavior of solutions to the bipolar quantum Euler-Poisson system with critical time-dependent over-damping

&lt;p style='text-indent:20px;'&gt;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.&lt;/p&gt;

Global well-posedness of 2D incompressible Magnetohydrodynamic equations with horizontal dissipation

&lt;p style='text-indent:20px;'&gt;This paper focuses on two-dimensional incompressible non-resistive MHD equations with only horizontal dissipation in &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathbb{T}\times\mathbb{R} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. Invoking three Poincaré-type inequalities about the horizontal derivative, we study the global well-posedness of the system near a background magnetic via the structure of the perturbation MHD system and the symmetry condition imposed on the initial data. By a precise time-weighted energy estimate, we also establish the global well-posedness of the system with only horizontal magnetic damping. Here we overcome the difficulties brought by the absence of magnetic diffusion and the appearance of the boundary. We note that the stability of MHD equations with one-directional dissipation in &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \mathbb{R}^2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; or a bounded domain appears to be unknown.&lt;/p&gt;

Large-time behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping in the half space

This paper is concerned with the large-time behavior of solutions to an initial boundary value problem for the one-dimensional bipolar Euler-Poisson equations with time-dependent damping effects Ji(1+t)λ(i=1,2) for −1<λ<1. We first show the decay rates of the corresponding asymptotic profiles, the so-called nonlinear diffusion waves, then by means of the time-weighted energy method, we prove that the smooth solutions to the initial-boundary value problem exist uniquely and globally, and time-asymptotically converge to the nonlinear diffusion waves, provided that the initial perturbation around the diffusion wave is small enough. The convergence rates are in the forms that O(t−34(1+λ)) for −1<λ<35 and O(tλ−32) for 35<λ<1, respectively, where λ=35 is the critical point, and the convergence rate at the critical point is O(t65lnt). The results are different from those of the Cauchy problem in Li et al. (2019) [20].

On the global existence and time-decay rates for a parabolic–hyperbolic model arising from chemotaxis

In this paper, we are concerned with the study of the Cauchy problem for a parabolic–hyperbolic model arising from chemotaxis in any dimension [Formula: see text]. We first prove the global existence of the model in [Formula: see text] critical regularity framework with respect to the scaling of the associated equations. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we also establish the time-decay rates for the constructed global solutions. Our analyses mainly rely on Fourier frequency localization technology and on a refined time-weighted energy inequalities in different frequency regimes.