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Register a new userFinding the jump rate for fastest decay in the Goldstein-Taylor model
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Helge Dietert
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For hypocoercive linear kinetic equations we first formulate an optimisation problem on a spatially dependent jump rate in order to find the fastest decay rate of perturbations. In the Goldstein-Taylor model we show (i) that for a locally optimal jump rate the spectral gap is determined by multiple, possible degenerate, eigenvectors and (ii) that globally the fastest decay is obtained with a spatially homogeneous jump rate. Our proofs rely on a connection to damped wave equations and a relationship to the spectral theory of Schr{o}dinger operators.
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We consider the development of implicit-explicit time integration schemes for optimal control problems governed by the Goldstein-Taylor model. In the diffusive scaling this model is a hyperbolic approximation to the heat equation. We investigate the relation of time integration schemes and the formal Chapman-Enskog type limiting procedure. For the class of stiffly accurate implicit-explicit Runge-Kutta methods (IMEX) the discrete optimality system also provides a stable numerical method for optimal control problems governed by the heat equation. Numerical examples illustrate the expected behavior.
In this article, we give an in-depth analysis of the problem of optimising the total population size for a standard logistic-diffusive model. This optimisation problem stems from the study of spatial ecology and amounts to the following question: assuming a species evolves in a domain, what is the best way to spread resources in order to ensure a maximal population size at equilibrium? {In recent years, many authors contributed to this topic.} We settle here the proof of two fundamental properties of optimisers: the bang-bang one which had so far only been proved under several strong assumptions, and the other one is the fragmentation of maximisers. Here, we prove the bang-bang property in all generality using a new spectral method. The technique introduced to demonstrate the bang-bang character of optimizers can be adapted and generalized to many optimization problems with other classes of bilinear optimal control problems where the state equation is semilinear and elliptic. We comment on it in a conclusion section.Regarding the geometry of maximisers, we exhibit a blow-up rate for the $BV$-norm of maximisers as the diffusivity gets smaller: if $O$ is an orthotope and if $m_mu$ is an optimal control, then $Vert m_muVert_{BV}gtrsim sqrt{mu}$. The proof of this results relies on a very fine energy argument.
The Schrodinger-Poisson-Newton equations for crystals with a cubic lattice and one ion per cell are considered. The ion charge density is assumed i) to satisfy the Wiener and Jellium conditions introduced in our previous paper [28], and ii) to be exponentially decaying at infinity. The corresponding examples are given. We study the linearized dynamics at the ground state. The dispersion relations are introduced via spectral resolution for the non-selfadjoint Hamilton generator using the positivity of the energy established in [28]. Our main result is the dispersion decay in the weighted Sobolev norms for solutions with initial states from the space of continuous spectrum of the Hamilton generator. We also prove the absence of singular spectrum and limiting absorption principle. The multiplicity of every eigenvalue is shown to be infinite. The proofs rely on novel exact bounds and compactness for the inversion of the Bloch generators and on uniform asymptotics for the dispersion relations. We derive the bounds by the energy positivity from [28]. We also use the theory of analytic sets.
Let $ngeq 3$, $alpha$, $betainmathbb{R}$, and let $v$ be a solution $Delta v+alpha e^v+beta xcdot abla e^v=0$ in $mathbb{R}^n$, which satisfies the conditions $lim_{Rtoinfty}frac{1}{log R}int_{1}^{R}rho^{1-n} (int_{B_{rho}}e^v,dx)drhoin (0,infty)$ and $|x|^2e^{v(x)}le A_1$ in $R^n$. We prove that $frac{v(x)}{log |x|}to -2$ as $|x|toinfty$ and $alpha>2beta$. As a consequence under a mild condition on $v$ we prove that the solution is radially symmetric about the origin.
This paper is devoted to establishing the optimal decay rate of the global large solution to compressible nematic liquid crystal equations when the initial perturbation is large and belongs to $L^1(mathbb R^3)cap H^2(mathbb R^3)$. More precisely, we show that the first and second order spatial derivatives of large solution $(rho-1, u, abla d)(t)$ converges to zero at the $L^2-$rate $(1+t)^{-frac54}$ and $L^2-$rate $(1+t)^{-frac74}$ respectively, which are optimal in the sense that they coincide with the decay rates of solution to the heat equation. Thus, we establish optimal decay rate for the second order derivative of global large solution studied in [12,18] since the compressible nematic liquid crystal flow becomes the compressible Navier-Stokes equations when the director is a constant vector. It is worth noticing that there is no decay loss for the highest-order spatial derivative of solution although the associated initial perturbation is large. Moreover, we also establish the lower bound of decay rates of $(rho-1, u, abla d)(t)$ itself and its spatial derivative, which coincide with the upper one. Therefore, the decay rates of global large solution $ abla^2(rho-1,u, abla d)(t)$ $(k=0,1,2)$ are actually optimal.
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