Framework Of Besov Spaces Research Articles

On the stability of stationary compressible Navier–Stokes flows in 3D

AbstractThis paper studies the stability of the stationary solution of the compressible Navier–Stokes equation in the 3D whole space with an external force which decays at spatial infinity. We obtain the global existence result of the non-stationary problem under the smallness assumptions on the initial perturbation around the small stationary solution. We also derive the time decay rates of the perturbations under the smallness assumption on the initial perturbations, and show the optimality of the time decay rates. The proofs are based on the combination of the spectral analysis and energy method in the framework of Besov spaces. The time-space integral estimate for the linearized semigroup around the constant state in some Besov spaces plays a crucial role in the proofs.

Free boundary problems of the incompressible Navier–Stokes equations with non-flat initial surface in the critical Besov space

Global well-posedness of the Navier–Stokes equations with a free boundary condition is considered in the scaling critical homogeneous Besov spaces B˙p,1-1+n/p(R+n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\dot{B}}_{p,1}^{-1+n/p}({\\mathbb {R}}^n_+)$$\\end{document} with n-1<p<2n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-1< p< 2n-1$$\\end{document}. To show the global well-posedness, we establish end-point maximal L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-regularity for the initial-boundary value problem of the Stokes equations. Such an estimate is obtained via related estimate for the initial-boundary value problem of the heat equation with the inhomogeneous Neumann data as well as the pressure estimate in the critical Besov space framework. The proof heavily depends on the explicit expression of the fundamental integral kernel of the Lagrange transformed linearized Stokes equations and the almost orthogonal estimates with the space-time Littlewood–Paley dyadic decompositions. Our result here improves the initial space and boundary state than previous results by Danchin–Hieber–Mucha–Tolksdorf (Free boundary problems via Da Prato–Grisvard theory. arXiv:2011.07918v2) and ourselves (Ogawa and Shimizu in J Evol Equ 22(30):67, 2022; Ogawa and Shimizu in J Math Soc Jpn. arXiv:2211.06952v3).

On the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations

In this paper, we are concerned with the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations, which is a generalization of the Camassa-Holm equation and the Fokas-Olver-Rosenau-Qiao equation. We explore how high-order nonlinearities affect the dispersive dynamics and breakdown mechanism of solutions. Firstly, we established the local well-posedness for the Cauchy problem in the framework of Besov spaces. Then, we derive the precise blow-up mechanism for strong solutions by means of the transport equation theory. Finally, a sufficient condition on initial data that leads to the finite time blow-up of the second-order derivative of the solutions is described in detail.

An application of spectral localization to the critical SQG on a ball

We study the Cauchy problem for the quasi-geostrophic equations in a unit ball of the two-dimensional space with the homogeneous Dirichlet boundary condition. We show existence and uniqueness of the strong solution in the framework of Besov spaces. We also establish a spectral localization technique and commutator estimates.

Besov spaces with generalized smoothness and summability of multiple Fourier series

The paper deals with summation methods for multiple Fourier series in the framework of Besov spaces with generalized smoothness. The main focus lies on approximation with respect to the (quasi-) metric in Lp, where 0<p<1 as well as to refined smoothness near zero.

Leibniz rules and Gauss–Green formulas in distributional fractional spaces

We apply the results established in [12] to prove some new fractional Leibniz rules involving BVα,p and Sα,p functions, following the distributional approach adopted in the previous works [8,13,14]. In order to achieve our main results, we revise the elementary properties of the fractional operators involved in the framework of Besov spaces and we rephraze the Kenig–Ponce–Vega Leibniz-type rule in our fractional context. We apply our results to prove the well-posedness of the boundary-value problem for a general 2α-order fractional elliptic operator in divergence form.

Cancer detection through Electrical Impedance Tomography and optimal control theory: theoretical and computational analysis.

The Inverse Electrical Impedance Tomography (EIT) problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the $ m $ electrodes for a given electrode current is analyzed. A PDE constrained optimal control framework in Besov space is developed, where the electrical conductivity tensor and boundary voltages are control parameters, and the cost functional is the norm difference of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. The novelty of the control-theoretic model is its adaptation to the clinical situation when additional "voltage-to-current" measurements can increase the size of the input data from $ m $ up to $ m! $ while keeping the size of the unknown parameters fixed. The existence of the optimal control and Fréchet differentiability in the Besov space along with optimality condition is proved. Numerical analysis of the simulated model example in the 2D case demonstrates that by increasing the number of input boundary electrode currents from $ m $ to $ m^2 $ through additional "voltage-to-current" measurements the resolution of the electrical conductivity of the body identified via gradient method in Besov space framework is significantly improved.

On the ranges of bilinear pseudo-differential operators of S0,0-type on L2 × L2

In this paper, the ranges of bilinear pseudo-differential operators of S0,0-type on L2×L2 are determined in the framework of Besov spaces. Our result improves the L2×L2→L1 boundedness of those operators with symbols in the bilinear Hörmander class BS0,0m.

Global existence and optimal time-decay estimates of solutions to the generalized double dispersion equation on the framework of Besov spaces

We investigate the initial value problem for the generalized double dispersion equation in any dimensions. Inspired by [28] for the hyperbolic system of first order PDEs, we develop Littlewood-Paley pointwise energy estimates for the dissipative wave equation of high-order. Furthermore, with aid of the frequency-localization Duhamel principle, we establish the global existence and optimal decay estimates of solutions in spatially critical Besov spaces. Our results could hold true for any dimensions (n≥1). Indeed, the proofs are different in case of high dimensions and low dimensions owing to interpolation tricks.

Sharp well-posedness and ill-posedness of the three-dimensional primitive equations of geophysics in Fourier–Besov spaces

We study the well-posedness and ill-posedness for the Cauchy problem of the three-dimensional primitive equations describing the large-scale oceanic and atmospheric circulations. By using the Littlewood–Paley analysis technique, we prove that the Cauchy problem of the three-dimensional primitive equations with the Prandtl number P=1 is locally well-posed in the Fourier–Besov spaces FḂp,r2−3p(R3) for 1<p≤∞,1≤r<∞ and FḂ1,r−1(R3) for 1≤r≤2, and is globally well-posed in these spaces when the initial data are small. We also verify that such problem is ill-posed in FḂ1,r−1(R3) for 2<r≤∞, which implies that our work completes a dichotomy of well-posedness and ill-posedness for the three-dimensional primitive equations in the Fourier–Besov space framework.

On temporal decay of solution to the three‐dimensional compressible flow of nematic liquid crystal in Besov space

In this paper, we investigate large time behavior of global‐in‐time strong solution to the three‐dimensional compressible flow of nematic liquid crystal with low regularity assumptions on initial datum. More precisely, we show that the negative Besov space ‐norms (s ≥ 0) of solution are preserved along time evolution; by using this fact together with the conventional energy estimates in Besov space framework and the interpolation inequalities, we establish that, for the initial perturbation just small in homogeneous Besov space , the global‐in‐time strong solution to the Cauchy problem of the compressible flow of nematic liquid crystal has the following optimal temporal decay rate: urn:x-wiley:mma:media:mma5176:mma5176-math-0004 provided that we further assume that still belongs to . Here, is a constant unite vector. To illustrate our methods clearly, we also revisit the optimal temporal decay of solutions to the heat equation in the framework of Besov space.

Non-isothermal General Ericksen–Leslie System: Derivation, Analysis and Thermodynamic Consistency

We derive a model describing the evolution of a nematic liquid-crystal material under the action of thermal effects. The first and second laws of thermodynamics lead to an extension of the general Ericksen-Leslie system where the Leslie stress tensor and the Oseen-Frank energy density are considered in their general forms. The work postulate proposed by Ericksen-Leslie is traduced in terms of entropy production. We finally analyze the global-in-time well-posedness of the system for small initial data in the framework of Besov spaces.

Global solution to the 3D inhomogeneous nematic liquid crystal flows with variable density

In this paper, we investigate the global existence and uniqueness of solution to the 3D inhomogeneous incompressible nematic liquid crystal flows with variable density in the framework of Besov spaces. It is proved that there exists a global and unique solution to the nematic liquid crystal flows if the initial data (ρ0−1,u0,n0−e3)∈M(B˙p,13p−1(R3))×B˙p,13p−1(R3)×B˙p,13p(R3) with 1≤p<6, and satisfies‖ρ0−1‖M(B˙p,13p−1)+‖u0‖B˙p,13p−1+‖n0−e3‖B˙p,13p≤cfor some small c>0 depending only on p. Here M(B˙p,13p−1(R3)) is the multiplier space of Besov space B˙p,13p(R3). Using the Lagrangian approach in Danchin and Mucha (2012, 2013) [7,8] is the key to our results.

Optimal Time Decay of Navier–Stokes Equations with Low Regularity Initial Data

In this paper, we study the optimal time decay rate of isentropic Navier-Stokes equations under the low regularity assumptions about initial data. In the previous works about optimal time decay rate, the initial data need to be small in $H^{[N/2]+2}(\mathbb{R}^{N})$. Our work combined negative Besov space estimates and the conventional energy estimates in Besov space framework which is developed by R. Danchin. Though our methods, we can get optimal time decay rate with initial data just small in $\dot{B}^{N/2-1, N/2+1} \cap \dot{B}^{N/2-1, N/2}$ and belong to some negative Besov space(need not to be small). Finally, combining the recent results in \cite{zhang2014} with our methods, we can only need the initial data to be small in homogeneous Besov space $\dot{B}^{N/2-2, N/2} \cap \dot{B}^{N/2-1}$ to get the optimal time decay rate in space $L^{2}$.

Regularity criteria for the three dimensional Ericksen–Leslie system in homogeneous Besov spaces

In this paper, we establish some regularity criteria involving homogeneous Besov spaces for both the simplified and the general three dimensional Ericksen–Leslie system. This improves many previous results, and can be viewed as the ultimate optimal regularity criterion in the Besov space framework.

Weak invariance principle in some Besov spaces for stationary martingale differences

We extend the classical Donsker weak invariance principle to some Besov space framework. We consider polygonal line processes built from partial sums of stationary martingale differences and of independent and identically distributed random variables. The results obtained are shown to be optimal.

Well-posedness for compressible MHD systems with highly oscillating initial data

In this paper, a unique local solution for compressible magnetohydrodynamics systems has been constructed in the critical Besov space framework by converting the system in Euler coordinates to a system in Lagrange coordinates. Our results improve the range of the Lebesgue exponent in the Besov space from [2, N) to [2, 2N), where N denotes the space dimension. Then, we give a lower bound for the maximal existence time, which is important for our construction of global solutions. Based on the lower bound, we use the effective viscous flux and Hoff’s energy method to obtain the unique global solution, which allows the initial velocity field and the magnetic field to have large energies and allows the initial density to exhibit large oscillations on a set of small measure.

Optimal time decay rate for compressible viscoelastic equations in critical spaces

In this paper, we are concerned with the optimal time convergence rate of the global strong solution to some constant equilibrium states for the compressible viscoelastic fluids in the whole space. Green’s matrix method and energy estimate method are used to obtain the optimal time decay rate under the critical Besov space framework. Our result implies the optimal -time decay rate and only need the initial datum to be small in some critical Besov space which have very low regularity compared with the classical Sobolev space.

On decay estimates of the 3D nematic liquid crystal flows in critical Besov spaces

In this paper, we develop the energy argument in homogeneous Besov space framework to study the large time behavior of global‐in‐time strong solutions to the Cauchy problem of the three‐dimensional incompressible nematic liquid crystal flows with low regularity assumptions on initial data. More precisely, if the small initial data with 1 &lt; p &lt; ∞ and further assume that with 1 &lt; q≤p and , then the global‐in‐time strong solution (u,d) to the nematic liquid crystal flows admits the following temporal decay rate: urn:x-wiley:mma:media:mma3846:mma3846-math-0004 Here, is a constant unit vector. The highlight of our argument is to show that the ‐norms (with ) of solution are preserved along time evolution. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations

This paper is devoted to the investigation of stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations. For a Leray weak solution of the Navier–Stokes equations in a critical Besov space, it is shown that the Leray weak solution is uniformly stable with respect to a small perturbation of initial velocity and external forcing. If the perturbation is not small, the perturbed weak solution converges asymptotically to the original weak solution as the time tends to the infinity. Additionally, an energy equality and weak–strong uniqueness for the three-dimensional Navier–Stokes equations are derived. The findings are mainly based on the estimations of the nonlinear term of the Navier–Stokes equations in a Besov space framework, the use of special test functions and the energy estimate method.