共查询到20条相似文献,搜索用时 468 毫秒
1.
We consider the narrow escape problem in two-dimensional Riemannian manifolds (with a metric g) with corners and cusps, in an annulus, and on a sphere. Specifically, we calculate the mean time it takes a Brownian particle
diffusing in a domain Ω to reach an absorbing window when the ratio
between the absorbing window and the otherwise reflecting boundary is small. If the boundary is smooth, as in the cases of
the annulus and the sphere, the leading term in the expansion is the same as that given in part I of the present series of
papers, however, when it is not smooth, the leading order term is different. If the absorbing window is located at a corner
of angle α, then
if near a cusp, then
grows algebraically, rather than logarithmically. Thus, in the domain bounded between two tangent circles, the expected lifetime
is
, where
is the ratio of the radii. For the smooth boundary case, we calculate the next term of the expansion for the annulus and
the sphere. It can also be evaluated for domains that can be mapped conformally onto an annulus. This term is needed in real
life applications, such as trafficking of receptors on neuronal spines, because
is not necessarily large, even when
is small. In these two problems there are additional parameters that can be small, such as the ratio δ of the radii of the
annulus. The contributions of these parameters to the expansion of the mean escape time are also logarithmic. In the case
of the annulus the mean escape time is
. 相似文献
2.
We consider self-similar solutions to Smoluchowski’s coagulation equation for kernels \(K=K(x,y)\) that are homogeneous of degree zero and close to constant in the sense that $$\begin{aligned} -\varepsilon \le K(x,y)-2 \le \varepsilon \Big ( \Big (\frac{x}{y}\Big )^{\alpha } + \Big (\frac{y}{x}\Big )^{\alpha }\Big ) \end{aligned}$$ for \(\alpha \in [0,1)\) . We prove that self-similar solutions with given mass are unique if \(\varepsilon \) is sufficiently small which is the first such uniqueness result for kernels that are not solvable. Our proof relies on a contraction argument in a norm that measures the distance of solutions with respect to the weak topology of measures. 相似文献
3.
The measure of the splitting of the separatrices of the rapidly forced pendulum
相似文献
4.
Claudio Cacciapuoti Domenico Finco Diego Noja Alessandro Teta 《Letters in Mathematical Physics》2014,104(12):1557-1570
In the present paper, we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension: $$ i\frac{\rm d}{{\rm d}t}\psi^{\varepsilon}(t)=-\Delta\psi^{\varepsilon}(t) +\frac{1}{\varepsilon}V\left(\frac{x}{\varepsilon} \right)|\psi^{\varepsilon}(t)|^{2\mu}\psi^{\varepsilon}(t)\quad \varepsilon > 0\,\quad V\in L^1(\mathbb{R},(1+|x|){\rm d}x) \cap L^\infty(\mathbb{R}).$$ This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit \({\varepsilon\to 0}\) the weak (integral) dynamics converges in \({H^1(\mathbb{R})}\) to the weak dynamics of the NLS with point-concentrated nonlinearity: $$ i\frac{{\rm d}}{{\rm d}t} \psi(t) =H_{\alpha} \psi(t) .$$ where H α is the Laplacian with the nonlinear boundary condition at the origin \({\psi'(t,0+)-\psi'(t,0-)=\alpha|\psi(t,0)|^{2\mu}\psi(t,0)}\) and \({\alpha=\int_{\mathbb{R}}V{\rm d}x}\) . The convergence occurs for every \({\mu\in \mathbb{R}^+}\) if V ≥ 0 and for every \({\mu\in (0,1)}\) otherwise. The same result holds true for a nonlinearity with an arbitrary number N of concentration points. 相似文献
5.
The Radiative Transfer Equation is the nonlinear transport equation
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