Data In Critical Besov Spaces Research Articles

Well-posedness for the one-dimensional inviscid Cattaneo-Christov system

This paper is devoted to studying the inviscid compressible Cattaneo-Christov system in one-dimensional space. The iterative method will be used to establish the local well-posedness of this system for large data in critical Besov spaces based on the L2 framework. Moreover, by using the renormalized energy method, one can prove the global existence of a strong solution when the initial perturbation around a constant state is sufficiently small.

Low Mach number limit of the global solution to the compressible Navier–Stokes system for large data in the critical Besov space

In this paper, we consider the compressible Navier–Stokes system around the constant equilibrium states and prove the existence of a unique global solution for arbitrarily large initial data in the scaling critical Besov space provided that the Mach number is sufficiently small, and the incompressible part of the initial velocity generates the global solution of the incompressible Navier–Stokes equation. Moreover, we consider the low Mach number limit and show that the compressible solution converges to the solution of the incompressible Navier–Stokes equation in some space time norms.

Global well-posedness for the Hall-magnetohydrodynamics system in larger critical Besov spaces

We prove the global well-posedness of the Cauchy problem to the 3D incompressible Hall-magnetohydrodynamic system supplemented with initial data in critical Besov spaces that allowing for different integrability indices for the velocity field u and magnetic field b (and its current J), which generalize the result in [13]. Meanwhile, we analyze the long-time behavior of the solutions and get some decay estimates. Finally, a stability theorem for global solutions is established.

Existence and blow up of solutions to the $ 2D $ Burgers equation with supercritical dissipation

This paper is concerned with the Cauchy problem for a fractal Burgers equation in two dimensions. When $ \alpha\in (0, 1) $, the same problem has been studied in one dimensions, we can refer to [1, 17, 24]. In this paper, we study well-posedness of solutions to the Burgers equation with supercritical dissipation. We prove the local existence with large initial data and global existence with small initial data in critical Besov space by energy method. Furthermore, we show that solutions can blow up in finite time if initial data is not small by contradiction method.

On the continuity of the solutions to the Navier–Stokes equations with initial data in critical Besov spaces

It is well known that there exists a unique local-in-time strong solution u of the initial boundary value problem for the Navier–Stokes system in a three-dimensional smooth bounded domain when the initial velocity $$u_0$$ belongs to critical Besov spaces. A typical space is $$B=B^{-1+3/q}_{q,s}$$ with $$3<q<\infty $$, $$2<s<\infty $$ satisfying $$2/s+3/q \le 1$$ or $$B=\mathring{B}^{-1+3/q}_{q,\infty }$$. In this paper, we show that the solution u is continuous in time up to initial time with values in B. Moreover, the solution map $$u_0\mapsto u$$ is locally Lipschitz from B to $$C\left( [0,T];B\right) $$. This implies that in the range $$3<q<\infty $$, $$2<s\le \infty $$ with $$3/q +2/s \le 1$$ the problem is well posed which is in strong contrast to norm inflation phenomena in the space $$B^{-1}_{\infty ,s}$$, $$1\le s <\infty $$ proved in the last years for the whole space case.

Existence for a class of chemotaxis models with fractional diffusion in Besov spaces

In this paper, we are interested in the Cauchy problem for the fractional diffusion equation with a chemotaxis term, which generalizes the Keller-Segel model, for the small enough initial data in critical Besov spaces. Using some estimates of the linear dissipative equation in the frame of mixed time-space spaces, we establish the existence of the solution for such kind of chemotaxis system in $\\mathbb{R}^n$.

Global existence in critical Besov spaces for the coupled chemotaxis–fluid equations

In this paper, we are concerned with a system coupling the parabolic–parabolic Keller–Segel system to the viscous incompressible Navier–Stokes equations in spatial dimensions three, and prove the global existence of solutions with small initial data in critical Besov spaces with minimal regularity order by using the elementary paradifferential calculus and a special type of iteration scheme.

Blow-up criteria for the Navier–Stokes equations in non-endpoint critical Besov spaces

We obtain an improved blow-up criterion for solutions of the Navier-Stokes equations in critical Besov spaces. If a mild solution $u$ has maximal existence time $T^* < \infty$, then the non-endpoint critical Besov norms must become infinite at the blow-up time: $$ \lim_{t \uparrow T^*} \lVert{u(\cdot,t)}\rVert_{\dot B^{-1+3/p}_{p,q}(\mathbb{R}^3)} = \infty, \quad 3 < p,q < \infty. $$ In particular, we introduce a priori estimates for the solution based on elementary splittings of initial data in critical Besov spaces and energy methods. These estimates allow us to rescale around a potential singularity and apply backward uniqueness arguments. The proof does not use profile decomposition.

Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system

In this paper, we are concerned with the Cauchy problem for a new two-component Camassa-Holm system with the effect of the Coriolis force in the rotating fluid, which is a model in the equatorial water waves. We first investigate the local well-posedness of the system in \begin{document}$ B_{p,r}^s× B_{p,r}^{s-1}$\end{document} with \begin{document}$s>\max\{1+\frac{1}{p},\frac{3}{2},2-\frac{1}{p}\}$\end{document} , \begin{document}$p,r∈ [1,∞]$\end{document} by using the transport theory in Besov space. Then by means of the logarithmic interpolation inequality and the Osgood's lemma, we establish the local well-posedness in the critical Besov space \begin{document}$ B_{2,1}^{3/2}× B_{2,1}^{1/2}$\end{document} , and we present a blow-up result with the initial data in critical Besov space by virtue of the conservation law. Finally, we study the Gevrey regularity and analyticity of solutions to the system in a range of Gevrey-Sobolev spaces in the sense of Hardamard. Moreover, a precise lower bound of the lifespan is obtained.

Well-posedness and Gevrey analyticity of the generalized Keller–Segel system in critical Besov spaces

In this paper, we study the Cauchy problem for the generalized Keller–Segel system with the cell diffusion being ruled by fractional diffusion: $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}u+\Lambda ^{\alpha }u+\nabla \cdot (u\nabla \psi )=0 &{}\quad \text{ in \mathbb {R}^n\times (0,\infty ), -\Delta \psi =u &{}\quad \text{ in \mathbb {R}^n\times (0,\infty ), u(x,0)=u_0(x) &{}\quad \text{ in \mathbb {R}^n. \end{array}\right. } \end{aligned}$$ In the case $$1<\alpha \le 2$$ , we prove local well-posedness for any initial data and global well-posedness for small initial data in critical Besov spaces $$\dot{B}^{-\alpha +\frac{n}{p}}_{p,q}(\mathbb {R}^{n})$$ with $$1\le p<\infty $$ , $$1\le q\le \infty $$ , and analyticity of solutions for initial data $$u_{0}\in \dot{B}^{-\alpha +\frac{n}{p}}_{p,q}(\mathbb {R}^{n})$$ with $$1< p<\infty $$ , $$1\le q\le \infty $$ . Moreover the global existence and analyticity of solutions with small initial data in critical Besov spaces $$\dot{B}^{-\alpha }_{\infty ,1}(\mathbb {R}^{n})$$ is also established. In the limit case $$\alpha =1$$ , we prove global well-posedness for small initial data in critical Besov spaces $$\dot{B}^{-1+\frac{n}{p}}_{p,1}(\mathbb {R}^{n})$$ with $$1\le p<\infty $$ and $$\dot{B}^{-1}_{\infty ,1}(\mathbb {R}^{n})$$ and show analyticity of solutions for small initial data in $$\dot{B}^{-1+\frac{n}{p}}_{p,1}(\mathbb {R}^{n})$$ with $$1<p<\infty $$ and $$\dot{B}^{-1}_{\infty ,1}(\mathbb {R}^{n})$$ , respectively.

Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data

In this paper, we mainly study the Cauchy problem of the Chemo-taxis-Navier-Stokes equations with initial data in critical Besov spaces. We first get the local wellposedness of the system in $\mathbb{R}^d \, (d≥2)$ by the Picard theorem, and then extend the local solutions to be global under the only smallness assumptions on $\|u_0^h\|_{\dot{B}_{p, 1}^{-1+\frac{d}{p}}}$, $\|n_0\|_{\dot{B}_{q, 1}^{-2+\frac{d}{q}}}$ and $\|c_0\|_{\dot{B}_{r, 1}^{\frac{d}{r}}}$. This obtained result implies the global wellposedness of the equations with large initial vertical velocity component. Moreover, by fully using the global wellposedness of the classical 2D Navier-Stokes equations and the weighted Chemin-Lerner space, we can also extend the obtained local solutions to be global in $\mathbb{R}^2$ provided the initial cell density $n_0$ and the initial chemical concentration $c_0$ are doubly exponential small compared with the initial velocity field $u_0$.

Gevrey regularity of mild solutions to the parabolic–elliptic system of drift-diffusion type in critical Besov spaces

This paper is devoted to studying analyticity of mild solutions to the parabolic–elliptic system of drift-diffusion type with small initial data in critical Besov spaces. More precisely, by using multilinear singular integrals and Fourier localization argument, we show that global-in-time mild solutions are Gevrey regular, i.e., (etΛ1v,etΛ1w)∈L˜t∞B˙p,r−2+np(Rn)∩L˜t1B˙p,rnp(Rn) for all 1<p<2n and 1≤r≤∞, where Λ1 is the Fourier multiplier whose symbol is given by |ξ|1=∑i=1n|ξi|. As a corollary, we obtain decay estimates of mild solutions in critical Besov spaces without using cumbersome recursive estimation of higher-order derivatives.

Commutator estimate in terms of partial derivatives of solutions for the dissipative quasi-geostrophic equation

Through Littlewood-Paley decomposition argument, a commutator estimate in terms of partial derivatives of solutions for the critical and supercritical dissipative 2D Quasi-Geostrophic equation is established. As an application of this estimate, we obtain some new a priori estimates and prove the existence and uniqueness of solution for the small initial data in critical Besov spaces.

Global strong solution for the Korteweg system with quantum pressure in dimension $$N\ge 2$$ N ≥ 2

This work is devoted to prove the existence of global strong solution in dimension \(N\ge 2\) for a isothermal model of capillary fluids derived by Dunn and Serrin (see Arch. Ration. Mech. Anal. 88(2):95–133, 1985), which can be used as a phase transition model. We will restrict us to the case of the so called compressible Navier-Stokes system with quantum pressure (which corresponds to the Korteweg system with capillary coefficient such that \(\kappa (\rho )=\frac{\kappa _1}{\rho }\) with \(\kappa _1>0\) and \(\rho \) the density). In a first part we prove the existence of strong solution in finite time for large initial data in critical Besov spaces with precise estimate on the life span \(T^*\). The second part consists in proving the existence of global strong solution with large initial data for a specific choice on the capillary coefficient \(\kappa _1 = \mu ^2\) with \(\mu \) the viscosity coefficient. To do this we derive different energy estimates on the density and the effective velocity v which allow us to extend the strong solution beyond \(T^*\). The main difficulty consists in estimating the \(L^\infty \) norm of \(\frac{1}{\rho }\). The proof relies mostly on a method introduced by De Giorgi (Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3(3):25–43, 1957) [see also Ladyzhenskaya et al. (Linear and quasilinear equations of parabolic type. AMS translations, Providence, 1968) for the parabolic case] to obtain regularity results for elliptic equations with discontinuous diffusion coefficients and a suitable bootstrap argument.

On Gevrey regularity of the supercritical SQG equation in critical Besov spaces

In this paper, we show that the solution of the supercritical surface quasi-geostrophic (SQG) equation, with initial data in a critical Besov space, belongs to a subanalytic Gevrey class. In order to prove this, a suitable estimate on the nonlinear term, in the form of a commutator, is required. We express the commutator as a bilinear multiplier operator and obtain single-scale estimates for its symbol. In particular, we show that the localized symbol is of Marcinkiewicz type, and show that due to the localizations inherited from working in the Besov spaces, this condition implies the requisite boundedness of the corresponding operator. This result strengthens previous ones which showed that solutions starting from initial data in critical Besov spaces are classical. As a direct consequence of our method, decay estimates of higher-order derivatives are easily deduced.

On the well-posedness for a multi-dimensional compressible viscous liquid–gas two-phase flow model in critical spaces

This paper is dedicated to study of the Cauchy problem for a multi-dimensional (\({N \geq 2}\)) compressible viscous liquid–gas two-phase flow model. We prove the local well-posedness of the system for large data in critical Besov spaces based on the L p framework under the sole assumption that the initial liquid mass is bounded away from zero.

On the well-posedness for the compressible nematic liquid crystal flow

The Cauchy problem for the compressible flow of nematic liquid crystals in dimension ( N ≧ 2) is considered. Employing the Littlewood-Paley theory and Schauder-Tychonoff fixed point theorem, we prove the local well-posedness of the system for large initial data in critical Besov spaces based on the L p framework under the sole natural assumption that the initial density is bounded away from zero.

Sharp Global Regularity for the 2 + 1-Dimensional Equivariant Faddeev Model

The aim of this article is to prove that for the |$2+1$|-dimensional equivariant Faddeev model, which is a quasilinear generalization of the corresponding nonlinear |$\sigma$| model, small initial data in critical Besov spaces evolve into global solutions which scatter.

Convergence of the compressible isentropic magnetohydrodynamic equations to the incompressible magnetohydrodynamic equations in critical spaces

We study the convergence of the compressible isentropic magnetohydrodynamic equations to the incompressible model with ill-prepared initial data in critical Besov spaces. Under the condition that the initial data is small in some norm, we show that the convergence holds globally as the Mach number goes to zero. Moreover, we also obtain the convergence rate.

On the Cauchy problem for the fractional drift-diffusion system in critical Besov spaces

In this paper, we establish global well-posedness and asymptotic stability of mild solutions for the Cauchy problem of the fractional drift-diffusion system with small initial data in critical Besov spaces. The regularizing-decay rate estimates of mild solutions are also proved, which imply that mild solutions are analytic in space variables.