We develop a general energy method for proving the optimal time decay rates of the solutions to the dissipative equations in the whole space. Our method is applied to classical examples such as the heat equation, the compressible Navier-Stokes equations and the Boltzmann equation. In particular, the optimal decay rates of the higher-order spatial derivatives of solutions are obtained. The negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. We use a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis.
We study the well-posedness in weighted Sobolev spaces, for the initial value problem (IVP) associated with the dissipative Benjamin-Ono (dBO) equation. We establish persistence properties of the solution flow in the weighted Sobolev spaces $\z_{s,r}=H^s(\R)\cap L^2(|x|^{2r}dx)$, $s\geq r>0$. We also prove some unique continuation properties in these spaces. In particular, such results of unique continuation show that our results of well posedness are sharp.
In this note, we consider the ill-posedness issue for the cubic nonlinear Schrödinger equation. In particular, we prove norm inflation based at every initial condition in negative Sobolev spaces below or at the scaling critical regularity.
We continue the study on the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrödinger equation. By considering the renormalized equation, we extend the quasi-invariance results in [30, 27] to Sobolev spaces of negative regularity. Our proof combines the approach introduced by Planchon, Tzvetkov, and Visciglia [35] with the normal form approach in [30, 27].
In this paper, we study the Cauchy problem for the linear and semilinear Moore-Gibson-Thompson (MGT) equation in the dissipative case. Concerning the linear MGT model, by utilizing WKB analysis associated with Fourier analysis, we derive some $L^2$ estimates of solutions, which improve those in the previous research [46]. Furthermore, asymptotic profiles of the solution and an approximate relation in a framework of the weighted $L^1$ space are derived. Next, with the aid of the classical energy method and Hardy's inequality, we get singular limit results for an energy and the solution itself. …
Global Well-posedness of the 1D Dirac-Klein-Gordon system in Sobolev spaces of negative index
证明了一维Dirac Klein Gordon方程组的Cauchy问题在Dirac旋量为负指数、标量场为正指数的索伯列夫空间范围内是整体适定的.证明的主要内容是Colliander，Keel，Staffilani，Takaoka和Tao提出的几乎守恒律理论和I-方法。我们的证明也依赖于系统中的零结构，以及Klainerman-Machedon型的双线性时空估计。
We study semilinear damped wave equations with power nonlinearity $|u|^p$ and initial data belonging to Sobolev spaces of negative order $\dot{H}^{-γ}$. In the present paper, we obtain a new critical exponent $p=p_{\mathrm{crit}}(n,γ):=1+\frac{4}{n+2γ}$ for some $γ\in(0,\frac{n}{2})$ and low dimensions in the framework of Soblev spaces of negative order. Precisely, global (in time) existence of small data Sobolev solutions of lower regularity is proved for $p>p_{\mathrm{crit}}(n,γ)$, and blow-up of weak solutions in finite time even for small data if $1<p<p_{\mathrm{crit}}(n,γ)$. Furthermore, …
In this work, we study the extremal functions of the log-Sobolev functional on compact metric measure spaces satisfying the $\mathrm{RCD}^*(K,N)$ condition for $K$ in $\mathbb{R}$ and $N$ in $(2,\infty)$. We show the existence, regularity and positivity of non-negative extremal functions. Based on these results, we prove a Li-Yau type estimate for the logarithmic transform of any non-negative extremal functions of the log-Sobolev functional. As applications, we show a Harnack type inequality as well as lower and upper bounds for the non-negative extremal functions.
We study the three-dimensional cubic nonlinear wave equation (NLW) with random initial data below $L^2(\mathbb{T}^3)$. By considering the second order expansion in terms of the random linear solution, we prove almost sure local well-posedness of the renormalized NLW in negative Sobolev spaces. We also prove a new instability result for the defocusing cubic NLW without renormalization in negative Sobolev spaces, which is in the spirit of the so-called triviality in the study of stochastic partial differential equations. More precisely, by studying (un-renormalized) NLW with given smooth determi…
The three-dimensional incompressible Boussinesq system is one of the important equations in fluid dynamics. The system describes the motion of temperature-dependent incompressible flows. And the temperature naturally has diffusion. Recently, Elgindi, Ghoul and Masmoudi constructed a $C^{1,α}$ finite time blow-up solutions for Euler systems with finite energy. Inspired by their works, we constructed $C^{1,α}$ finite time blow-up solution for Boussinesq equations where the temperature has diffusion and finite energy. Generally speaking, the diffusion of temperature smooths the solution of the sy…
The Optimal Temporal Decay Estimates for the Fractional Power Dissipative Equation in Negative Besov Spaces
在本文中，我们首先推广了一个新的能量方法，由Y。Guo和Y. Wang \cite{GW12}等人在齐次Besov空间的框架下证明了分数阶幂耗散方程解的最优时间衰减率，然后我们将这种方法应用于超临界和临界准地转方程以及临界Keller-Segel系统。我们证明了解的负Besov范数在时间演化过程中是保持不变的，并通过Fourier分裂方法和插值技术得到了解的空间导数的最优时间衰减率.
Let $Ω= \mathbb R^3 \setminus \bar{K}$, where $K$ is an open bounded domain with smooth boundary $Γ$. Let $V(t) = e^{tG_b},\: t \geq 0,$ be the semigroup related to Maxwell's equations in $Ω$ with dissipative boundary condition $ν\wedge (ν\wedge E)+ γ(x) (ν\wedge H) = 0, γ(x) > 0, \forall x \in Γ.$ We study the case when $γ(x) \neq 1, \: \forall x \in Γ,$ and we establish a Weyl formula for the counting function of the eigenvalues of $G_b$ in a polynomial neighbourhood of the negative real axis.
