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Register a new userExistence of invariant measures for the stochastic damped KdV equation
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We address the long time behavior of solutions of the stochastic Korteweg-de Vries equation $ du + (partial^3_x u +upartial_x u +lambda u)dt = f dt+Phi dW_t$ on ${mathbb R}$ where $f$ is a deterministic force. We prove that the Feller property holds and establish the existence of an invariant measure. The tightness is established with the help of the asymptotic compactness, which is carried out using the Aldous criterion.
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In this paper, we address the long time behaviour of solutions of the stochastic Schrodinger equation in $mathbb{R}^d$. We prove the existence of an invariant measure and establish asymptotic compactness of solutions, implying in particular the existence of an ergodic measure.
In this paper we investigate the long-time behavior of stochastic reaction-diffusion equations of the type $du = (Au + f(u))dt + sigma(u) dW(t)$, where $A$ is an elliptic operator, $f$ and $sigma$ are nonlinear maps and $W$ is an infinite dimensional nuclear Wiener process. The emphasis is on unbounded domains. Under the assumption that the nonlinear function $f$ possesses certain dissipative properties, this equation is known to have a solution with an expectation value which is uniformly bounded in time. Together with some compactness property, the existence of such a solution implies the existence of an invariant measure which is an important step in establishing the ergodic behavior of the underlying physical system. In this paper we expand the existing classes of nonlinear functions $f$ and $sigma$ and elliptic operators $A$ for which the invariant measure exists, in particular, in unbounded domains. We also show the uniqueness of the invariant measure for an equation defined on the upper half space if $A$ is the Shr{o}dinger-type operator $A = frac{1}{rho}(text{div} rho abla u)$ where $rho = e^{-|x|^2}$ is the Gaussian weight.
We consider a quasilinear KdV equation that admits compactly supported traveling wave solutions (compactons). This model is one of the most straightforward instances of degenerate dispersion, a phenomenon that appears in a variety of physical settings as diverse as sedimentation, magma dynamics and shallow water waves. We prove the existence and uniqueness of solutions with sufficiently smooth, spatially localized initial data.
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A mathematical model for collision-induced breakage is considered. Existence of weak solutions to the continuous nonlinear collision-induced breakage equation is shown for a large class of unbounded collision kernels and daughter distribution functions, assuming the collision kernel $K$ to be given by $K(x,y)= x^{alpha} y^{beta} + x^{beta} y^{alpha}$ with $alpha le beta le 1$. When $alpha + beta in [1,2]$, it is shown that there exists at least one weak mass-conserving solution for all times. In contrast, when $alpha + beta in [0,1)$ and $alpha ge 0$, global mass-conserving weak solutions do not exist, though such solutions are constructed on a finite time interval depending on the initial condition. The question of uniqueness is also considered. Finally, for $alpha <0$ and a specific daughter distribution function, the non-existence of mass-conserving solutions is also established.
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