Bootstrap Argument Research Articles

Eigensystem bootstrap multiscale analysis for the Anderson model

We use a bootstrap argument to enhance the eigensystem multiscale analysis, introduced by Elgart and Klein for proving localization for the Anderson model at high disorder. The eigensystem multiscale analysis studies finite volume eigensystems, not finite volume Green’s functions. It yields pure point spectrum with exponentially decaying eigenfunctions and dynamical localization. The starting hypothesis for the eigensystem bootstrapmultiscale analysis only requires the verification of polynomial decay of the finite volume eigenfunctions, at some sufficiently large scale, with some minimal probability independent of the scale. It yields exponential localization of finite volume eigenfunctions in boxes of side L , with the eigenvalues and eigenfunctions labeled by the sites of the box, with probability higher than 1-\mathrm{e}^{-L^\xi} , for any desired 0 <\xi <1 .

Uniform global strong solutions of the 2D magnetic Bénard problem in a bounded domain

In this paper, we use the bootstrap argument to prove the uniform-in-k global strong solutions of the 2D magnetic Bénard problem in a bounded domain. Here k is the heat conductivity coefficient.

Pattern formation for a nonlinear diffusion chemotaxis model with logistic source

This paper deals with a Neumann boundary value problem in a d-dimensional box mathbb{T}^{d}=(0,pi)^{d} (d=1, 2, 3) for a nonlinear diffusion chemotaxis model with logistic source. By using the embedding theorem, the higher-order energy estimates and bootstrap arguments, the condition of chemotaxis-driven instability and the nonlinear evolution near an unstable positive constant equilibrium for this chemotaxis model are proved. Our result provides a quantitative characterization for early spatial pattern formation on the positive constant equilibrium. Finally, numerical simulations are carried out to support our theoretical nonlinear instability results.

ОБРАТНЫЕ ЗАДАЧИ ОБ ОПРЕДЕЛЕНИИ ГРАНИЧНЫХ ДАННЫХ С ИНТЕГРАЛЬНЫМИ УСЛОВИЯМИ ПЕРЕОПРЕДЕЛЕНИЯ

In the present article we examine an inverse problem of recovering unknown functions being part of the Dirichlet boundary condition together solving an initial boundary problem for a parabolic second order equation. Such problems on recovering the boundary data arise in various tasks of mathematical physics: control of heat exchange prosesses and design of thermal protection systems, diagnostics and identification of heat transfer in supersonic heterogeneous flows, identification and modeling of heat transfer in heat-shielding materials and coatings, modeling of properties and heat regimes of reusable heat protection of spacecrafts, study of composite materials, etc. As the overdetrermination conditions we take the integrals of a solution over the spatial domain with weights. The problem is reduced to an operator equation of the Volterra-type. The existence and uniqueness theorem for solutions to this inverse problem is established in Sobolev spaces. A solution is regular, i. e., all generalized derivatives occuring into the equation exists and are summable to some power. The proof relies on the fixed point theorem and bootstrap arguments. Stability estimates for solutions are also given. The solvability conditions are close to necessary conditions

Solution to the Boltzmann equation in velocity-weighted Chemin-Lerner type spaces

In this paper the Boltzmann equation near global Maxwellians is studied in the $d$ -dimensional whole space. A unique global-in-time mild solution to the Cauchy problem of the equation is established in a Chemin-Lerner type space with respect to the phase variable $(x,v)$ . Both hard and soft potentials with angular cutoff are considered. The new function space for global well-posedness is introduced to essentially treat the case of soft potentials, and the key point is that the velocity variable is taken in the weighted supremum norm, and the space variable is in the $s$ -order Besov space with $s≥ d/2$ including the spatially critical regularity. The proof is based on the time-decay properties of solutions to the linearized equation together with the bootstrap argument. Particularly, the linear analysis in case of hard potentials is due to the semigroup theory, where the extra time-decay plays a role in coping with initial data in $L^2$ with respect to the space variable. In case of soft potentials, for the time-decay of linear equations we borrow the results based on the pure energy method and further extend them to those in $L^∞$ framework through the technique of $L^2-L^∞$ interplay. In contrast to hard potentials, $L^1$ integrability in $x$ of initial data is necessary for soft potentials in order to obtain global solutions to the nonlinear Cauchy problem.

On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals

This work is concerned with the solvability of a Navier-Stokes/Q-tensor coupled system modeling the nematic liquid crystal flow on a bounded domain in three dimensional Euclidian space with strong anchoring boundary condition for the order parameter. We prove the existence and uniqueness of local in time strong solutions to the system with an anisotropic elastic energy. The proof is based on mainly two ingredients: first, we show that the Euler-Lagrange operator corresponding to the Landau-de Gennes free energy with general elastic coefficients fulfills the strong Legendre condition. This result together with a higher order energy estimate leads to the well-posedness of the linearized system, and then a local in time solution of the original system which is regular in temporal variable follows via a fixed point argument. Secondly, the hydrodynamic part of the coupled system can be reformulated into a quasi-stationary Stokes type equation to which the regularity theory of the generalized Stokes system, and then a bootstrap argument can be applied to enhance the spatial regularity of the local in time solution.

Non-Hermitian random matrices with a variance profile (I): deterministic equivalents and limiting ESDs

For each $n$, let $A_n=(\sigma_{ij})$ be an $n\times n$ deterministic matrix and let $X_n=(X_{ij})$ be an $n\times n$ random matrix with i.i.d. centered entries of unit variance. We study the asymptotic behavior of the empirical spectral distribution $\mu_n^Y$ of the rescaled entry-wise product \[ Y_n = \left(\frac1{\sqrt{n}} \sigma_{ij}X_{ij}\right). \] For our main result we provide a deterministic sequence of probability measures $\mu_n$, each described by a family of Master Equations, such that the difference $\mu^Y_n - \mu_n$ converges weakly in probability to the zero measure. A key feature of our results is to allow some of the entries $\sigma_{ij}$ to vanish, provided that the standard deviation profiles $A_n$ satisfy a certain quantitative irreducibility property. An important step is to obtain quantitative bounds on the solutions to an associate system of Schwinger--Dyson equations, which we accomplish in the general sparse setting using a novel graphical bootstrap argument.

Global strong solutions to the 3D full compressible Navier–Stokes system with vacuum in a bounded domain

In this short paper we establish the global well-posedness of strong solutions to the 3D full compressible Navier–Stokes system with vacuum in a bounded domain Ω ⊂ R 3 by the bootstrap argument provided that the viscosity coefficients λ and μ satisfy that 7 λ > 9 μ and the initial data ρ 0 and u 0 satisfy that ‖ ρ 0 ‖ L ∞ ( Ω ) and ‖ ρ 0 | u 0 | 5 ‖ L 1 ( Ω ) are sufficiently small.

Positive ground state of coupled systems of Schrödinger equations in involving critical exponential growth

In this paper, we study the existence of a positive ground state solution to the following coupled system of nonlinear Schrödinger equations: urn:x-wiley:mma:media:mma4498:mma4498-math-0003 where the nonlinearities f1(x,s) and f2(x,s) are superlinear at infinity and have exponential critical growth of the Trudinger‐Moser type. The potentials V1(x) and V2(x) are nonnegative and satisfy a condition involving the coupling term λ(x), namely, λ(x)2<δ2V1(x)V2(x) for some 0<δ<1. For this purpose, we use the minimization technique over the Nehari manifold and strong maximum principle to get a positive ground state solution. Moreover, by using a bootstrap argument and Lq‐estimates, we get regularity and asymptotic behavior.

Regularity of Milne problem with geometric correction in 3D

Consider the Milne problem with geometric correction in a 3D convex domain. Via bootstrapping arguments, we establish [Formula: see text]-regularity for its solutions. Combined with a uniform [Formula: see text]-estimate, such regularity leads to the validity of diffusive expansion for the neutron transport equation with diffusive boundary conditions.

Mathematical modeling and its analysis for instability of the immune system induced by chemotaxis.

In this paper, we study how chemotaxis affects the immune system by proposing a minimal mathematical model, a reaction-diffusion-advection system, describing a cross-talk between antigens and immune cells via chemokines. We analyze the stability and instability arising in our chemotaxis model and find their conditions for different chemotactic strengths by using energy estimates, spectral analysis, and bootstrap argument. Numerical simulations are also performed to the model, by using the finite volume method in order to deal with the chemotaxis term, and the fractional step methods are used to solve the whole system. From the analytical and numerical results for our model, we explain not only the effective attraction of immune cells toward the site of infection but also hypersensitivity when chemotactic strength is greater than some threshold.

Smooth Solutions to a Class of Nonlocal Fully Nonlinear Elliptic Equations

We show that a certain class of fully nonlinear nonlocal equations have smooth solutions as long as the right-hand side is nice and the boundary datum is bounded. To this end we follow the classical strategy. We first show that solutions are $C^{\sigma+\alpha}$ continuous, then develop a bootstrap argument robust enough for operators that may depend on the solution and its \sigma-derivatives.

The additive structure of elliptic homogenization

One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this paper, we address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the weak convergence of the gradients, fluxes and energy densities of the first-order correctors (under blow-down) which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument, completing the program initiated in \cite{AKM}: using the regularity theory recently developed for stochastic homogenization, we reduce the error in additivity as we pass to larger and larger length scales. In the second part of the paper, we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables. In particular, we prove that the first-order correctors converge, in the large-scale limit, to a variant of the Gaussian free field.

Bootstrapping and dogmatism

Dogmatists claim that having a perceptual experience as of p can provide one with immediate and defeasible warrant to believe that p. A persistent complaint against this position is that it sanctions an intuitively illicit form of reasoning: bootstrapping. I argue that dogmatism has no such commitments. Dogmatism is compatible with a principle that disallows the final non-deductive inference in the bootstrapping procedure. However, some authors have maintained that such strategy is doomed to failure because earlier stages of in the bootstrapping inference are already problematic. I argue that insofar as these inferences appear problematic, the dogmatist is not committed to sanctioning them. I conclude that the bootstrapping argument presents no significant objection to the claim that perceptual experiences can provide immediate and defeasible warrant to believe their contents.

Nonlinear Hartree equation in high energy-mass

This paper is concerned with the Cauchy problem of the nonlinear Hartree equation. By constructing a constrained variational problem, we get a refined Gagliardo–Nirenberg inequality and the best constant for this inequality. We thus derive two conclusions. Firstly, by establishing and analyzing the invariant manifolds, we obtain a new criteria for global existence and blowup of the solutions. Secondly, we get other sufficient condition for global existence with the discussing of the Bootstrap argument. And based on these two conclusions, we also deduce so-called energy-mass control maps, which expose the relationship between the initial data and the solutions.

The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential

Boundary effects are central to the dynamics of the dilute particles governed by Boltzmann equation. In this paper, we study both the diffuse reflection and the specular reflection boundary value problems for Boltzmann equation with soft potential, in which the collision kernel is ruled by the inverse power law. For the diffuse reflection boundary condition, based on an $L^2$ argument and its interplay with intricate $L^\infty$ analysis for the linearized Boltzmann equation, we first establish the global existence and then obtain the exponential decay in $L^\infty$ space for the nonlinear Boltzmann equation in general classes of bounded domain. It turns out that the zero lower bound of the collision frequency and the singularity of the collision kernel lead to some new difficulties for achieving the {\it a priori} $L^\infty$ estimates and time decay rates of the solution. In the course of the proof, we capture some new properties of the probability integrals along the stochastic cycles and improve the $L^2-L^\infty$ theory to give a more direct approach to overcome those difficulties. As to the specular reflection condition, our key contribution is to develop a new time-velocity weighted $L^\infty$ theory so that we could deal with the greater difficulties stemmed from the complicated velocity relations among the specular cycles and the zero lower bound of the collision frequency. From this new point, we are also able to prove the solutions of the linearized Boltzmann equation tend to equilibrium exponentially in $L^\infty$ space with the aid of the $L^2$ theory and a bootstrap argument. These methods in the latter case can be applied to the Boltzmann equation with soft potential for all other types of boundary condition.

Weak solutions to the two-dimensional steady compressible Navier–Stokes equations

We study motions of the steady compressible viscous isothermal fluids in a bounded two dimensional domain governed by the Navier–Stokes equations. We obtain that there exists at least one weak solution to such a problem satisfying some integrability. The proof of this result is based on delicate estimates for the momentum equations and a bootstrapping argument.

L p $L^{p}$ -Estimates for quasilinear subelliptic equations with VMO coefficients under the controllable growth

We prove an interior $L^{p}$ -estimate of X-gradient of weak solutions to a class of quasilinear subelliptic equations with VMO coefficients under controllable growth. Here, we use a reverse Holder inequality and De Giorgi’s iteration to establish the boundedness of their weak solutions. Then a local $L^{p}$ -estimate of the X-gradient of the weak solutions is derived by way of the bootstrap argument.

Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization

We introduce a new method for obtaining quantitative results in stochastic homogenization for linear elliptic equations in divergence form. Unlike previous works on the topic, our method does not use concentration inequalities (such as Poincar\'e or logarithmic Sobolev inequalities in the probability space) and relies instead on a higher ($C^{k}$, $k \geq 1$) regularity theory for solutions of the heterogeneous equation, which is valid on length scales larger than a certain specified mesoscopic scale. This regularity theory, which is of independent interest, allows us to, in effect, localize the dependence of the solutions on the coefficients and thereby accelerate the rate of convergence of the expected energy of the cell problem by a bootstrap argument. The fluctuations of the energy are then tightly controlled using subadditivity. The convergence of the energy gives control of the scaling of the spatial averages of gradients and fluxes (that is, it quantifies the weak convergence of these quantities) which yields, by a new "multiscale" Poincar\'e inequality, quantitative estimates on the sublinearity of the corrector.

Random Attractor Associated with the Quasi-Geostrophic Equation

We study the long time behavior of the solutions to the 2D stochastic quasi-geostrophic equation on $\mathbb{T}^2$ driven by additive noise and real linear multiplicative noise in the subcritical case (i.e. $\alpha>1/2$) by proving the existence of a random attractor. The key point for the proof is the exponential decay of the $L^p$-norm and a boot-strapping argument. The upper semicontinuity of random attractors is also established. Moreover, if the viscosity constant is large enough, the system has a trivial random attractor.