Global Well-posedness Of Solutions Research Articles

Global well-posedness of solutions for the epitaxy thin film growth model

In this paper, we consider the global well-posedness of solutions for the initial-boundary value problems of the epitaxy growth model. We first construct the local smooth solution, then by combining some a priori estimates, continuity argument, the local smooth solutions are extended step by step to all t > 0, provided that the initial datums sufficiently small and the smooth nonlinear functions satisfy certain local growth conditions.

A theoretical approach to understanding rumor propagation dynamics in a spatially heterogeneous environment

<p style='text-indent:20px;'>Most of the previous work on rumor propagation either focus on ordinary differential equations with temporal dimension or partial differential equations (PDE) with only consideration of spatially independent parameters. Little attention has been given to rumor propagation models in a spatiotemporally heterogeneous environment. This paper is dedicated to investigating a SCIR reaction-diffusion rumor propagation model with a general nonlinear incidence rate in both heterogeneous and homogeneous environments. In spatially heterogeneous case, the well-posedness of global solutions is established first. The basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula> is introduced, which can be used to reveal the threshold-type dynamics of rumor propagation: if <inline-formula><tex-math id="M2">\begin{document}$ R_0 &lt; 1 $\end{document}</tex-math></inline-formula>, the rumor-free steady state is globally asymptotically stable, while <inline-formula><tex-math id="M3">\begin{document}$ R_0 &gt; 1 $\end{document}</tex-math></inline-formula>, the rumor is uniformly persistent. In spatially homogeneous case, after introducing the time delay, the stability properties have been extensively studied. Finally, numerical simulations are presented to illustrate the validity of the theoretical analysis and the influence of spatial heterogeneity on rumor propagation is further demonstrated.

Long-time behavior of a class of viscoelastic plate equations

&lt;p style='text-indent:20px;'&gt;This paper is concerned with the initial-boundary value problem for a class of viscoelastic plate equations on an arbitrary dimensional bounded domain. Under certain assumptions on the memory kernel and the source term, the global well-posedness of solutions and the existence of global attractors are obtained.&lt;/p&gt;

Well-posedness for the Navier–Stokes equations in critical mixed-norm Lebesgue spaces

We study the Cauchy problem in n-dimensional space for the system of Navier–Stokes equations in critical mixed-norm Lebesgue spaces. Local well-posedness and global well-posedness of solutions are established in the class of critical mixed-norm Lebesgue spaces. Being in the mixed-norm Lebesgue spaces, both of the initial data and the solutions could be singular at certain points or decaying to zero at infinity with different rates in different spatial variable directions. Some of these singular rates could be very strong, and some of the decaying rates could be significantly slow. Besides other interests, the results of the paper demonstrate the persistence of the anisotropic behavior of the initial data under the evolution. To achieve the goals, fundamental analysis theory such as Young’s inequality, time decaying of solutions for heat equations, the boundedness of the Helmholtz–Leray projection, and the boundedness of the Riesz transform are developed in mixed-norm Lebesgue spaces. These analysis results are topics of independent interests, and they are potentially useful in other problems.

Derivation in Strong Topology and Global Well-Posedness of Solutions to the Gross-Pitaevskii Hierarchy

We derive the cubic defocusing GP hierarchy in ℝ3 from a bosonic N-particle Schrödinger equation as N → ∞, in the strong topology corresponding to the space of sequences of marginal density matrices. In particular, we thereby eliminate the requirement of regularity for the initial data used in previous work. Moreover, the marginal density matrices obtained in this strong limit are allowed to be of infinite rank. This contrasts previous results where weak-* limits were derived, and subsequently enhanced to strong limits based on the condition that the limiting density matrices have finite rank. Furthermore, we prove that positive semidefiniteness of marginal density matrices is preserved over time, which we combine with previous results Chen and Pavlović, to obtain the global well-posedness of solutions.

Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system

We consider a quasilinear PDE system which models nonlinear vibrationsof a thermoelastic plate defined on a bounded domain in $\mathbb{R}^n$. Global Well-posedness of solutions is shown by applying the theory of maximal parabolic regularity of type $L_p$. In addition, we prove exponential decay rates for strong solutions and their derivatives.

Global well-posedness and scattering for the derivative nonlinear Schrödinger equation with small rough data

We study the Cauchy problem for the generalized elliptical and non-elliptical derivative nonlinear Schrödinger equations (DNLS) and get the global well posedness of solutions with small data in modulation spaces M2,1s(Rn). Noticing that B2,1s+n/2⊂M2,1s⊂B2,1s are optimal inclusions, we have shown the global well posedness of DNLS with a class of rough data. As a by-product, the existence of the scattering operators in modulation spaces with small data is also obtained.

Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations

Sufficient and necessary conditions for the embeddings between Besov spaces B p , q s 1 and modulation spaces M p , q s 2 are obtained. Moreover, using the frequency-uniform decomposition method, we study the Cauchy problem for the generalized BO, KdV and NLS equations, for which the global well-posedness of solutions with the small rough data in certain modulation spaces M 2 , 1 s is shown.