首页 | 本学科首页   官方微博 | 高级检索  
相似文献
 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.
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
  相似文献   

6.
It is known that a complex — valued continuous functionS(x) as well as a Schwartz distribution on the real axis can be extended in the complex plane minus the support ofS to an analytic function?(z). In the case of a continuous function the jump of?(z) on the real axis represents exactlyS(x): $$\mathop {\lim }\limits_{\varepsilon \to 0 + } [\hat S(x + i\varepsilon ) - \hat S(x - i\varepsilon )] = S(x)$$ . We call regular a pointx on the support ofS such that \(\mathop {\lim }\limits_{\varepsilon \to 0 + } [\hat S(x + i\varepsilon ) - \hat S(x - i\varepsilon )]\) exists. Conditions are found for the existence of regular points on the support of a distribution. It is possible to call this limit (if this exists) the valueS(x) of the distributionS in the pointx. Properties of this type occur in the theory of dispersion relations.  相似文献   

7.
We derive an equation satisfied by the dissipation rate correlation function, for the homogeneous, isotropic state of fully-developed turbulence from the the Navier–Stokes equation. In the equal time limit we show that the equation leads directly to two intermittency exponents 1=2– 6 and 2=z4 4, where the 's are exponents of velocity structure functions and z4 is a dynamical exponent characterizing the fourth order structure function. We discuss the contributions of the pressure terms to the equation and the consequences of hyperscaling.  相似文献   

8.
For the Glimm scheme approximation \({u_\varepsilon}\) to the solution of the system of conservation laws in one space dimension
$$u_t + f(u)_x = 0, \qquad u(0, x) = u_0(x) \in \mathbb{R}^n,$$
with initial data u 0 with small total variation, we prove a quadratic (w.r.t. Tot. Var. (u 0)) interaction estimate, which has been used in the literature for stability and convergence results. No assumptions on the structure of the flux f are made (apart from smoothness), and this estimate is the natural extension of the Glimm type interaction estimate for genuinely nonlinear systems.
More precisely, we obtain the following results:
  • a new analysis of the interaction estimates of simple waves;
  • a Lagrangian representation of the derivative of the solution, i.e., a map \({\mathtt{x}(t, w)}\) which follows the trajectory of each wave w from its creation to its cancellation;
  • the introduction of the characteristic interval and partition for couples of waves, representing the common history of the two waves;
  • a new functional \({\mathfrak{Q}}\) controlling the variation in speed of the waves w.r.t. time.
This last functional is the natural extension of the Glimm functional for genuinely nonlinear systems.
The main result is that the distribution \({D_{t} \hat \sigma_k(t,w)}\) is a measure with total mass \({\leq \mathcal{O}(1) {\rm Tot. Var.} (u_0)^2}\) , where \({\hat{\sigma}_k(t, w)}\) is the speed given to the wave w by the Riemann problem at the grid point \({(i\varepsilon, \mathtt{x}(i\varepsilon, w)), t \in [i\varepsilon, (i + 1)\varepsilon)}\).  相似文献   

9.
We consider the Dirichlet Laplacian for astrip in with one straight boundary and a width , where $f$ is a smooth function of acompact support with a length 2b. We show that in the criticalcase, , the operator has nobound statesfor small .On the otherhand, a weakly bound state existsprovided . In thatcase, there are positive c 1,c 2 suchthat the corresponding eigenvalue satisfies for all sufficiently small.  相似文献   

10.
For real analytic expanding interval maps, a novel method is given for rigorously approximating the diffusion coefficient of real analytic observables. As a theoretical algorithm, our approximation scheme is shown to give quadratic exponential convergence to the diffusion coefficient. The method for converting this rapid convergence into explicit high precision rigorous bounds is illustrated in the setting of Lanford’s map \(x\mapsto 2x +\frac{1}{2}x(1-x) \pmod 1 \).  相似文献   

11.
The linear and nonlinear optical properties in non-centro-symmetric cubic semiconductor GaX (X=As, Sb, P) are studied by using the first-principle full potential linear augmented plane wave (FP-LAPW) and the linear muffin-tin orbital (LMTO) methods. We present calculations of the frequency-dependent complex dielectric function and it zero-frequency limit . A simple scissor operator is applied to adjust the band gap from the local-density calculations to match the experimental value. Calculations are reported for the frequency-dependent complex second-order non-linear optical susceptibilities up to 6 eV and it zero-frequency limit . Comparison with available experimental data shows good agreement. Our calculations show excellent agreement between the two methods.  相似文献   

12.
We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator \({\int\limits_{0} ^{\infty}{\rm e}^{iH_{0}t}\zeta(\frac{\vert x\vert }{R}) {\rm e}^{-iH_{0}t}{\rm d}t}\), as \({R \rightarrow \infty}\), is presented. Here, H0 is the free Dirac operator and \({\zeta\left(t\right)}\) is such that \({\zeta\left(t\right) = 1}\) for \({0 \leq t \leq 1}\) and \({\zeta\left(t\right) = 0}\) for \({t > 1}\). This approach allows us to obtain the time delay operator \({\delta \mathcal{T}\left(f\right)}\) for initial states f in \({\mathcal{H} _{2}^{3/2+\varepsilon}(\mathbb{R}^{3};\mathbb{C}^{4})}\), \({\varepsilon > 0}\), the Sobolev space of order \({3/2+\varepsilon}\) and weight 2. The relation between the time delay operator \({\delta\mathcal{T}\left(f\right)}\) and the Eisenbud–Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented.  相似文献   

13.
The effect of an external magnetic field on the nonlinear interaction of S-polarized electromagnetic radiation incident on a S-polarized surface wave in a plasma layer was studied analytically. We have calculated the amplitudes of generated waves at combination frequencies. The generated waves are of P-polarization and can be either electromagnetic or surface waves, depending on the signal of the value=\documentclass{article}\pagestyle{empty}\begin{document}$ ^{\chi '^2 = \frac{{k'^2 }}{{\varepsilon '}} - \frac{{\omega '^2 }}{{c^2 }} + k'\frac{\partial }{{\partial x}}\frac{{\varepsilon '_2 }}{{\varepsilon '\varepsilon '_1 }}} $\end{document}.  相似文献   

14.
A surface plasma wave (SPW) over bismuth-vacuum interface has a signature of mass anisotropy of free electrons. For SPW propagation along the trigonal axis there is no birefringence. The frequency cutoff of SPW lies in the far infrared region and can be accessed using free electron laser. The damping rate of waves at low temperatures is low. The surface plasma wave may be excited by an electron beam of current ∼100 mA propagating parallel to the interface in its close proximity.  相似文献   

15.
Given n2, we put r=min . Let be a compact, C r -smooth surface in n which contains the origin. Let further be a family of measurable subsets of such that as . We derive an asymptotic expansion for the discrete spectrum of the Schrödinger operator in L 2( n ), where is a positive constant, as . An analogous result is given also for geometrically induced bound states due to a interaction supported by an infinite planar curve.  相似文献   

16.
For convex co-compact hyperbolic quotients , we analyze the long-time asymptotic of the solution of the wave equation u(t) with smooth compactly supported initial data f = (f 0, f 1). We show that, if the Hausdorff dimension δ of the limit set is less than n/2, then where and . We explain, in terms of conformal theory of the conformal infinity of X, the special cases , where the leading asymptotic term vanishes. In a second part, we show for all the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip . As a byproduct we obtain a lower bound on the remainder R(t) for generic initial data f.  相似文献   

17.
We prove that if the initial condition of the Swift–Hohenberg equation $$\partial _t u(x,t) = (\varepsilon ^2 - (1 + \partial _x^2 )^2 ){\text{ }}u(x,t) - u^3 (x,t)$$ is bounded in modulus by Ce ?βx as x→+∞, the solution cannot propagate to the right with a speed greater than $$\mathop {\sup }\limits_{0 < {\gamma } \leqslant \beta } {\gamma }^{ - 1} (\varepsilon ^2 + 4{\gamma }^2 + 8{\gamma }^4 ).$$ This settles a long-standing conjecture about the possible asymptotic propagation speed of the Swift–Hohenberg equation. The proof does not use the maximum principle and is simple enough to generalize easily to other equations. We illustrate this with an example of a modified Ginzburg–Landau equation, where the critical speed is not determined by the linearization alone.  相似文献   

18.
We consider the energy dependent Schrödinger operator , which we have previously shown to be associated with multi-Hamiltonian structures [2]. In this paper we use an unusual form of the Lax approach to derive by asingle construction the time evolutions of the eigenfunctions of , the associated Hamiltonian operators and the Hamiltonian functionals. We then generalise the well known factorisation of standard Lax operators to the case of energy-dependent operators. The simple product of linear factors is replaced by a -dependent quadratic form. We thus generalise the resulting construction of Miura maps and modified equations. We show that for some of our systems there exists a sequence ofN such modifications, ther th modification possessing (Nr+1) Hamiltonian structures.On leave of absence from Institute of Theoretical Physics, Warsaw University, Hoza 69, PL-00-681 Warsaw, Poland (present address)  相似文献   

19.
The incompressible limit in nonlinear elasticity is shown to fall under the theory of singular limits of quasilinear symmetric hyperbolic systems developed by Klainerman and Majda. Specifically, initial-value problems for a family of hyperelastic materials with stored energy functions $$W\left( {\frac{{\partial x}}{{\partial X}}} \right) = W_\infty \left( {\frac{{\partial x}}{{\partial X}}} \right) + \lambda ^2 w\left( {\det \frac{{\partial x}}{{\partial X}}} \right)$$ are considered, whereX andx are reference and deformed coordinates respectively. Under the assumption that the elasticity tensor $$A_{kl}^{ij} \equiv \frac{{\partial ^2 W_\infty }}{{\partial \left( {\frac{{\partial x^i }}{{\partial X^k }}} \right)\partial \left( {\frac{{\partial x^j }}{{\partial X^l }}} \right)}}$$ is positive definite near the identity matrix and thatw″(1)>0, the following results are proven for appropriate initial data: i) existence of solutions of the corresponding evolution equations on a time interval independent of λ as λ→∞, and ii) convergence as λ → ∞ of the solutions to a solution of the incompressible elastodynamics equations.  相似文献   

20.
Quantum scattering in the presence of a constant electric field (Stark effect) is considered. It is shown that the scattering matrix has a meromorphic continuation in the energy variable to the entire complex plane as an operator on L2(R n-1). The allowed potentials V form a general subclass of potentials that are short-range relative to the free Stark Hamiltonian: Roughly, the potential vanishes at infinity, and admits a decomposition , where is analytic in a sector with , and , for x1<0 and some >0. These potentials include the Coulomb potential. The wave operators used to define the scattering matrix are the two Hilbert space wave operators.  相似文献   

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号