Solutions In Besov Spaces Research Articles

Characterization of solutions in Besov spaces for fractional Rayleigh–Stokes equations

This paper considers fractional Rayleigh–Stokes equations with a power-type nonlinearity. The linear equation can be simulated a non-Newtonian fluid for a generalized second grade fluid and display a nonlocal behavior in time. Because the coexistence of fractional and classical derivatives leads to the lack of semigroup structure of the solution operator, we need to develop a suitable tool to establish some Lp−Lq estimates in the framework of Lp spaces and Besov spaces, respectively. Further, global existence of solutions is showed in spaces of Besov type.

Global well-posedness and large time behavior for the 3D magneto-micropolar equations

This paper focuses on the global well-posedness and optimal decay estimates to the 3D magneto-micropolar equations in Besov spaces. We obtain the global existence and uniqueness of solutions in Besov spaces. Moreover, the decay estimates of the global solutions are established provided that the initial data belongs to the negative Besov space.

Local and global existence of solutions to a time-fractional wave equation with an exponential growth

A time-fractional wave equation with an exponential growth source function is considered. This model can be regarded as a modified version of the one studied by Nakamura and Ozawa (1999). The main feature of this model is that the exponential growth nonlinearity is essentially different from the polynomial growth one, and harder in controlling. We first prove the local existence and uniqueness of mild solutions in an Orlicz space by deriving a nonlinear estimate and Lp−Lq estimates the source term and solution operators, respectively. Furthermore, by additionally using specific techniques for estimating integrals, we show that small data solutions exist globally over time in such Orlicz space. The last theoretical result is about the second global-in-time existence of solutions in Besov spaces. Such a result is based on a different nonlinear estimate which is derived from a logarithmic interpolation inequality. The main ingredients of proof are the efficient application of function spaces such as Lebesgue spaces, Orlicz space,Besov spaces and computational techniques involving generalized integrals. In addition, some numerical examples are provided to illustrate theoretical results.

On the initial value problem for the hyperbolic Keller-Segel equations in Besov spaces

In this paper, we first show by constructing a special initial data that the solution map for the one dimensional hyperbolic Keller-Segel equations (HKSE) starting from u0 is discontinuous at t=0 in the metric of B2,∞s(R), s>32. Then, we establish the Hadamard local well-posedness result for the high dimensional HKSE in the larger Besov spaces Bp,11+dp(Rd), 1≤p<∞, which improves the local theory proved by [Zhou, Zhang & Mu, J. Differ. Equ., 302(2021), pp.662-679]. Moreover, we investigate the inviscid limit of the Keller-Segel equations with small diffusivity ϵΔu as ϵ→0 in the same topology of Besov spaces as the initial data. Finally, we establish two kinds of blow-up criteria for strong solutions in Besov spaces by means of the Littlewood-Paley theory.

Global existence for the Jordan–Moore–Gibson–Thompson equation in Besov spaces

In this paper, we consider the Cauchy problem of a model in nonlinear acoustic, named the Jordan–Moore–Gibson–Thompson equation. This equation arises as an alternative model to the well-known Kuznetsov equation in acoustics. We prove global existence and optimal time decay of solutions in Besov spaces with a minimal regularity assumption on the initial data, lowering the regularity assumption required in Racke and Said-Houari (Commun Contemp Math. 1–39, 2019. https://doi.org/10.1142/S0219199720500698) for the proof of the global existence. Using a time-weighted energy method with the help of appropriate Lyapunov-type estimates, we also extend the decay rate in Racke and Said-Houari (2019) and show an optimal decay rate of the solution for initial data in the Besov space \(\dot{{B}}_{2,\infty }^{-3/2}(\mathbb {R}^3)\), which is larger than the Lebesgue space \(L^1(\mathbb {R}^3)\) due to the embedding \(L^1(\mathbb {R} ^3)\hookrightarrow \dot{{B}}_{2,\infty }^{-3/2}(\mathbb {R}^3)\). Hence, we removed the \(L^1\)-assumption on the initial data required in Racke and Said-Houari (2019) in order to prove the decay estimates of the solution.

Gevrey Regularity and Time Decay of Fractional Porous Medium Equation in Critical Besov Spaces

In this paper, we show the existence and regularity of mild solutions depending on the small initial data in Besov spaces to the fractional porous medium equation. When 1 α ≤ 2, we prove global well-posedness for initial data with 1 ≤ p q ≤ ∞, and analyticity of solutions with 1 p q ≤ ∞. In particular, we also proved that when α = 1, both u and belong to . We solve this equation through the contraction mapping method based on Littlewood-Paley theory and Fourier multiplier. Furthermore, we can get time decay estimates of global solutions in Besov spaces, which is as t → ∞.

Uniqueness of Dissipative Solutions to the Complete Euler System

Dissipative solutions have recently been studied as a generalized concept for weak solutions of the complete Euler system. Apparently, these are expectations of suitable measure valued solutions. Motivated from Feireisl et al. (Commun Partial Differ Equ 44(12):1285–1298, 2019), we impose a one-sided Lipschitz bound on velocity component as uniqueness criteria for a weak solution in Besov space $$B^{\alpha ,\infty }_{p}$$ with $$\alpha >1/2$$ . We prove that the Besov solution satisfying the above mentioned condition is unique in the class of dissipative solutions. In the later part of this article, we prove that the one sided Lipschitz condition gives uniqueness among weak solutions with the Besov regularity, $$B^{\alpha ,\infty }_{3}$$ for $$\alpha >1/3$$ . Our proof relies on commutator estimates for Besov functions and the relative entropy method.

Long-time solvability for the 2D dispersive SQG equation with improved regularity

In this paper, we study the long-time existence and uniqueness (solvability) for the initial value problem of the 2D inviscid dispersive SQG equation. First we obtain the local solvability with existence-time independent of the amplitude parameter \begin{document}$ A $\end{document} . Then, assuming more regularity and using a blow-up criterion of BKM type and a space-time estimate of Strichartz type, we prove long-time solvability of solutions in Besov spaces for large \begin{document}$ A $\end{document} and arbitrary initial data. In comparison with previous results, we are able to consider improved cases of the regularity and larger initial data classes.

Erratum to: The Space $${B^{-1}_{\infty, \infty}}$$ B ∞ , ∞ - 1 , Volumetric Sparseness, and 3D NSE

In the context of the $L^\infty$-theory of the 3D NSE, it is shown that suitable smallness of a solution in Besov space $B^{-1}_{\infty, \infty}$ suffices to prevent a possible blow-up.

The Space $${B^{-1}_{\infty, \infty}}$$ B ∞ , ∞ - 1 , Volumetric Sparseness, and 3D NSE

In the context of the \({L^\infty}\)-theory of the 3D NSE, it is shown that smallness of a solution in Besov space \({B^{-1}_{\infty, \infty}}\) suffices to prevent a possible blow-up. In particular, it is revealed that the aforementioned condition implies a particular local spatial structure of the regions of high velocity magnitude, namely, the structure of local volumetric sparseness on the scale comparable to the radius of spatial analyticity measured in \({L^\infty}\).

Gevrey analyticity of solutions to the 3D nematic liquid crystal flows in critical Besov space

We show that the solution to the Cauchy problem of the 3D nematic liquid crystal flows, with initial data belonging to a critical Besov space, belongs to a Gevrey class. More precisely, it is proved that for any (u0,d0−d¯0)∈Ḃp,13p−1(R3)×Ḃq,13q(R3) with some suitable conditions imposed on p,q∈(1,∞), there exists T∗>0 depending only on initial data, such that the nematic liquid crystal flows admit a unique solution (u,d) on R3×(0,T∗), and satisfies ‖etΛ1u(t)‖L˜T∗∞(Ḃp,13p−1)∩L˜T∗1(Ḃp,13p+1)+‖etΛ1(d(t)−d¯0)‖L˜T∗∞(Ḃq,13q)∩L˜T∗1(Ḃq,13q+2)<∞. Here, d¯0∈S2 is a constant unit vector, and Λ1 is the Fourier multiplier whose symbol is given by |ξ|1=|ξ1|+|ξ2|+|ξ3|. Moreover, if the initial data is sufficiently small, then T∗=∞. As a consequence of the results, decay estimates of higher-order derivatives of solutions in Besov spaces are deduced.

On the regularity of weak solutions for a modified dissipative quasi-geostrophic equation

In this paper, we are concerned with weak solutions of a modified critical dissipative quasi-geostrophic equation. By Fourier localization technique, we obtain a new regularity criterion of weak solutions in Besov spaces.

Local well-posedness for the chemotaxis-Navier–Stokes equations in Besov spaces

In this paper, we are concerned with the local well-posedness for the chemotaxis-Navier–Stokes equations in Rd, d=2,3. By fully using the advantage of weighted function generated by heat kernel and Fourier localization technique, we obtain the existence and uniqueness of smooth solutions in Besov spaces. More importantly, we show a Beale–Kato–Majda type blow-up criterion with the help of a logarithmic inequality.

Analyticity and Decay Estimates of the Navier–Stokes Equations in Critical Besov Spaces

In this paper, we establish analyticity of the Navier–Stokes equations with small data in critical Besov spaces \({\dot{B}^{\frac{3}{p}-1}_{p,q}}\) . The main method is Gevrey estimates, the choice of which is motivated by the work of Foias and Temam (Contemp Math 208:151–180, 1997). We show that mild solutions are Gevrey regular, that is, the energy bound \({\|e^{\sqrt{t}\Lambda}v(t)\|_{E_p}>\infty}\) holds in \({E_p:=\tilde{L}^{\infty}(0,T;\dot{B}^{\frac{3}{p}-1}_{p,q})\cap \tilde{L}^{1}(0,T;\dot{B}^{\frac{3}{p}+1}_{p,q})}\) , globally in time for p < ∞. We extend these results for the intricate limiting case p = ∞ in a suitably designed E∞ space. As a consequence of analyticity, we obtain decay estimates of weak solutions in Besov spaces. Finally, we provide a regularity criterion in Besov spaces.

Solutions in Besov spaces of a class of abstract parabolic equations of higher order in time

Solutions in Besov spaces of a class of abstract parabolic equations of higher order in time