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Register a new userWell-posedness and longtime behavior for the modified phase-field crystal equation
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We consider a modification of the so-called phase-field crystal (PFC) equation introduced by K.R. Elder et al. This variant has recently been proposed by P. Stefanovic et al. to distinguish between elastic relaxation and diffusion time scales. It consists of adding an inertial term (i.e. a second-order time derivative) into the PFC equation. The mathematical analysis of the resulting equation is more challenging with respect to the PFC equation, even at the well-posedness level. Moreover, its solutions do not regularize in finite time as in the case of PFC equation. Here we analyze the modified PFC (MPFC) equation endowed with periodic boundary conditions. We first prove the existence and uniqueness of a solution with initial data in a bounded energy space. This solution satisfies some uniform dissipative estimates which allow us to study the global longtime behavior of the corresponding dynamical system. In particular, we establish the existence of an exponential attractor. Then we demonstrate that any trajectory originating from the bounded energy phase space does converge to a unique equilibrium. This is done by means of a suitable version of the {L}ojasiewicz-Simon inequality. A convergence rate estimate is also given.
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Considering the Cauchy problem for the modified finite-depth-fluid equation $partial_tu-G_delta(partial_x^2u)mp u^2u_x=0, u(0)=u_0$, where $G_delta f=-i ft ^{-1}[coth(2pi delta xi)-frac{1}{2pi delta xi}]ft f$, $deltages 1$, and $u$ is a real-valued function, we show that it is uniformly globally well-posed if $u_0 in H^s (sgeq 1/2)$ with $ orm{u_0}_{L^2}$ sufficiently small for all $delta ges 1$. Our result is sharp in the sense that the solution map fails to be $C^3$ in $H^s (s<1/2)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the modified Benjamin-Ono equation if $delta$ tends to $+infty$.
We consider the modified phase-field crystal (MPFC) equation that has recently been proposed by P. Stefanovic et al. This is a variant of the phase-field crystal (PFC) equation, introduced by K.-R. Elder et al., which is characterized by the presence of an inertial term $betaphi_{tt}$. Here $phi$ is the phase function standing for the number density of atoms and $betageq 0$ is a relaxation time. The associated dynamical system for the MPFC equation with respect to the parameter $beta$ is analyzed. More precisely, we establish the existence of a family of exponential attractors $mathcal{M}_beta$ that are Holder continuous with respect to $beta$.
We study well-posedness of the complex-valued modified KdV equation (mKdV) on the real line. In particular, we prove local well-posedness of mKdV in modulation spaces $M^{2,p}_{s}(mathbb{R})$ for $s ge frac14$ and $2leq p < infty$. For $s < frac 14$, we show that the solution map for mKdV is not locally uniformly continuous in $M^{2,p}_{s}(mathbb{R})$. By combining this local well-posedness with our previous work (2018) on an a priori global-in-time bound for mKdV in modulation spaces, we also establish global well-posedness of mKdV in $M^{2,p}_{s}(mathbb{R})$ for $s ge frac14$ and $2leq p < infty$.
We prove that the complex-valued modified Benjamin-Ono (mBO) equation is locally wellposed if the initial data $phi$ belongs to $H^s$ for $sgeq 1/2$ with $ orm{phi}_{L^2}$ sufficiently small without performing a gauge transformation. Hence the real-valued mBO equation is globally wellposed for those initial datas, which is contained in the results of C. Kenig and H. Takaoka cite{KenigT} where the smallness condition is not needed. We also prove that the real-valued $H^infty$ solutions to mBO equation satisfy a priori local in time $H^s$ bounds in terms of the $H^s$ size of the initial data for $s>1/4$.
In this paper we consider the hyperbolic-elliptic Ishimori initial-value problem. We prove that such system is locally well-posed for small data in $H^{s}$ level space, for $s> 3/2$. The new ingredient is that we develop the methods of Ionescu and Kenig cite{IK} and cite{IK2} to approach the problem in a perturbative way.
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