By constructing a series of perturbation functions through localization in the Fourier domain and translation, we show that the data-to-solution map for the Euler-Poincaré equations is nowhere uniformly continuous in $B^s_{p,r}(\mathbb{R} ^d)$ with $s>\max\{1+\frac d2,\frac32\}$ and $(p,r)\in (1,\infty)\times [1,\infty)$. This improves our previous result which shows the data-to-solution map for the Euler-Poincaré equations is non-uniformly continuous on a bounded subset of $B^s_{p,r}(\mathbb{R} ^d)$ near the origin.
In this note, we study the mapping properties of global pseudo-differential operators with symbols in Ruzhansky-Turunen classes on Besov spaces $B^{s}_{\infty,\infty}(G).$ The considered classes satisfy Fefferman type conditions of limited regularity.
In this paper, we prove that the solution map of Fokas_Olver_Rosenau_Qiao equation (FORQ) is not uniformly continuous on the initial data in Besov spaces. Our result extends the previous non_uniform continuity in Sobolev spaces (Nonlinear Anal., 2014) to Besov spaces and is consistent with the present work (J. Math. Fluid Mech., 2020) on Novikov equation up to some coefficients when dropping the extra term $(\partial_xu)^3$ in FORQ.
In this paper, we consider the Cauchy problem for the generalized Camassa-Holm equation proposed by Hakkaev and Kirchev (2005) \cite{Hakkaev 2005}. We prove that the solution map of the generalized Camassa-Holm equation is not uniformly continuous on the initial data in Besov spaces. Our result include the present work (2020) \cite{Li 2020,Li 2020-1} on Camassa-Holm equation with $Q=1$ and extends the previous non-uniform continuity in Sobolev spaces (2015) \cite{Mi 2015} to Besov spaces. In addition, the non-uniform continuity in critical space $B_{2, 1}^{\frac{3}{2}}(\mathbb{R})$ is the firs…
In this paper, we obtain the local-in-time existence and uniqueness of solution to the Degasperis-Procesi equation in $B^1_{\infty,1}(\R)$. Moreover, we prove that the data-to-solution of this equation is continuous but not uniformly continuous in $B^1_{\infty,1}$.
For Besov spaces $B^s_{p,r}(\rr)$ with $s>\max\{ 2 + \frac1p , \frac52\} $, $p \in (1,\infty]$ and $r \in [1 , \infty)$, it is proved that the data-to-solution map for the FORQ equation is not uniformly continuous from $B^s_{p,r}(\rr)$ to $C([0,T]; B^s_{p,r}(\rr))$. The proof of non-uniform dependence is based on approximate solutions and the Littlewood-Paley decomposition.
In this paper, we consider the Cauchy problem of a two-component b-family system, which includes the two-component Camassa-Holm system and the two-component Degasperis-Procesi system. It is shown that the solution map of the two-component b-family system is not uniformly continuous on the initial data in Besov spaces $B_{p, r}^{s-1}(\mathbb{R})\times B_{p, r}^s(\mathbb{R})$ with $s>\max\{1+\frac{1}{p}, \frac{3}{2}\}$, $1\leq p, r< \infty$. Our result covers and extends the previous non-uniform continuity in Sobolev spaces $H^{s-1}(\mathbb{R})\times H^s(\mathbb{R})$ for $s>\frac{5}{2}$ (Nonline…
In this paper, we prove well-posedness of the Fornberg-Whitham equation in Besov spaces $B_{2,r}^s$ in both the periodic and non-periodic cases. This will imply the existence and uniqueness of solutions in the aforementioned spaces along with the continuity of the data-to-solution map provided that the initial data belongs to $B_{2,r}^s $. We also establish sharpness of continuity on the data-to-solution map by showing that it is not uniformly continuous from any bounded subset of $B_{2,r}^s$ to $C([-T,T]; B^s_{2,r})$. Furthermore, we prove a Cauchy-Kowalevski type theorem for this equation th…
We construct an example showing that the solution map of the Euler equations is not continuous in the Hölder space from $C^{1,α}$ to $L^\infty_tC^{1,α}_x$ for any $0<α<1$. On the other hand we show that it is continuous when restricted to the little Hölder subspace $c^{1,α}$. We apply the latter to prove an ill-posedness result for solutions of the vorticity equations in Besov spaces near the critical space $B^1_{2,1}$. As a consequence we show that a sequence of best constants of the Sobolev embedding theorem near the critical function space is not continuous.
We prove the non-uniform continuity of the data-to-solution map of the incompressible Euler equations in Besov spaces $B_{p,q}^{s}$, where the parameters $p, q$ and $s$ considered here are such that the local existence and uniqueness result holds.
In this paper, we prove the energy and cross-helicity conservation of weak solutions to the three-dimensional ideal MHD equations in bounded domain under the interior Besov regularity conditions which are exactly same as the three-dimensional periodic domain case in \cite{Caflisch 97}, and the boundedness and the Besov-type continuity for both the velocity and magnetic fields near the boundary, which seem crucial for the bounded domain case due to the boundary effect. Note that the Besov-type continuity condition near the boundary is consistent with the interior Besov regularity, which is a ne…
In this paper we introduce the notion of distributional $k$-Hessian associated with Besov type functions in Euclidean $n$-space, $k=2,\ldots,n$. Particularly, inspired by recent work of Baer and Jerison on distributional Hessian determinant, we show that the distributional $k$-Hessian is weak continuous on the Besov space $B(2-\frac{2}{k},k)$, and the result is optimal in the framework of the space $B(s,p)$, i.e., the distributional $k$-Hessian is well defined in $B(s,p)$ if and only if $B(s,p)\subset B_{loc}(2-\frac{2}{k},k)$.
In this paper we study the model of heat transfer in a porous medium with a critical diffusion. We obtain global existence and uniqueness of solutions to the equations of heat transfer of incompressible fluid in Besov spaces $\dot B^{3/p}_{p,1}(\mathbb{R}^3)$ with $1\le p\le\infty$ by the method of modulus of continuity and Fourier localization technique.
In this paper, we show that the solution map of the two-component Novikov system is not uniformly continuous on the initial data in Besov spaces $B_{p, r}^{s-1}(\mathbb{R})\times B_{p, r}^s(\mathbb{R})$ with $s>\max\{1+\frac{1}{p}, \frac{3}{2}\}$, $1\leq p< \infty$, $1\leq r<\infty$. Our result covers and extends the previous non-uniform continuity in Sobolev spaces $H^{s-1}(\mathbb{R})\times H^s(\mathbb{R})$ for $s>\frac{5}{2}$ (J. Math. Phys., 2017) to Besov spaces.
In this paper, we consider pseudodifferential operators on the torus with operator-valued symbols and prove continuity properties on vector-valued toroidal Besov spaces, without assumptions on the underlying Banach spaces. The symbols are of limited smoothness with respect to $x$ and satisfy a finite number of estimates on the discrete derivatives. The proof of the main result is based on a description of the operator as a convolution operator with a kernel representation which is related to the dyadic decomposition appearing in the definition of the Besov space.
This article studies strong A-infinity weights in Ahlfors Q-regular and geodesic metric spaces satisfying a weak (1,s)-Poincare inequality for some 1<s<=Q, where Q is finite. It is shown that whenever max(1,Q-1)<s<=Q, a function u yields a strong A-infinity weight of the form w=exp(Qu) if u has a minimal s-weak upper gradient with sufficiently small Morrey norm. Similarly, it is proved that if 1<Q<p for some finite p, then w=exp(Qu) is a strong A-infinity weight whenever u has sufficiently small Besov p-seminorm.
In this paper, we aim to solving the open question left in [Nie, Yuan: Nonlinear Anal 196 (2020); J. Math. Anal. Appl 505 (2022)) and Xiao, Fei: J. Math. Anal. Appl 514 (2022)]. We prove that a multidimensional chemotaxis system is ill-posedness in $\dot{B}_{2d, r}^{-\frac{3}{2}} \times\big(\dot{B}_{2d, r}^{-\frac{1}{2}}\big)^{d}$ when $1\leq r<d$ due to the lack of continuity of the solution.
In this paper we mainly investigate the initial value problem of the periodic Euler-Poincaré equations. We first present a new blow-up result to the system for a special class of smooth initial data by using the rotational invariant properties of the system. Then, we prove that the periodic Euler-Poincaré equations is ill-posed in critical Besov spaces by a contradiction argument. Finally, we verify the system possesses a class of peakon solutions in the sense of distributions.
We study Besov capacities in a compact Ahlfors regular metric measure space by means of hyperbolic fillings of the space. This approach is applicable even if the space does not support any Poincaré inequalities. As an application of the Besov capacity estimates we show that if a homeomorphism between two Ahlfors regular metric measure spaces preserves, under some additional assumptions, certain Besov classes, then the homeomorphism is necessarily a quasisymmetric map.
We establish optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on $\mathbb{R}^N$. In particular, inspired by recent work of Brezis and Nguyen on the distributional Jacobian determinant, we show that the action is continuous on the Besov space of fractional order $B(2-\frac{2}{N},N)$, and that all continuity results in this scale of Besov spaces are consequences of this result. A key ingredient in the argument is the characterization of $B(2-\frac{2}{N},N)$ as the space of traces of functions in the Sobolev space $W^{2,N…
Sharp Besov regularities in time and space variables are investigated for $\left(u(t,x),\; t\in [0,T],\; x\in \mathbb{R}\right)$, the mild solution to the stochastic heat equation driven by space-time white noise. Existence, Hölder continuity, and Besov regularity of local times are established for $u(t,x)$ viewed either as a process in the space variable or time variable. Hausdorff dimensions of their corresponding level sets are also obtained.
The Boutet de Monvel calculus of pseudo-differential boundary operators is generalised to the full scales of Besov and Triebel--Lizorkin spaces (though with finite integral exponents for the latter). The continuity and Fredholm properties proved here extend those previously obtained by Franke and Grubb, and the results on range complements of surjectively elliptic Green operators improve the earlier known, even for the classical spaces with $1<p<\infty$. The symbol classes treated are the uniformly estimated ones. Some precisions are given for the general definitions of trace and singular Gree…
Let $(X,d)$ be a pathwise connected metric space equipped with an Ahlfors $Q$-regular measure $μ$, $Q\in[1,\infty)$. Suppose that $(X,d,μ)$ supports a 2-Poincaré inequality and a Sobolev-Poincaré type inequality for the corresponding "Gaussian measure". The author uses the heat equation to study the Lipschitz regularity of solutions of the Poisson equation $Δu=f$, where $f\in L^p_\loc$. When $p>Q$, the local Lipschitz continuity of $u$ is established.
Nonexistence of Hölder continuous solution for the Camassa-Holm equation in Besov spaces
School of Mathematics and Statistics, Anhui Normal University（安徽师范大学数学与统计学院）；School of Mathematics and Computer Sciences, Gannan Normal University（赣南师范大学数学与计算机科学学院）；School of Mathematics, Foshan University（佛山大学数学学院）
We prove the existence and the Besov regularity of the density of the solution to a general parabolic SPDE which includes the stochastic Burgers equation on an unbounded domain. We use an elementary approach based on the fractional integration by parts.
We make use of the method of modulus of continuity \cite{K-N-S} and Fourier localization technique \cite{A-H} to prove the global well-posedness of the critical Burgers equation $\partial_{t}u+u\partial_{x}u+Λu=0$ in critical Besov spaces $\dot{B}^{\frac{1}{p}}_{p,1}(\mathbb{R})$ with $p\in[1,\infty)$, where $Λ=\sqrt{-\triangle}$.
We study generalized fractional $p$-Laplacian equations to prove local boundedness and Hölder continuity of weak solutions to such nonlocal problems by finding a suitable fractional Sobolev-Poincaré inquality.
For the trace of Besov spaces $B^s_{p,q}$ onto a hyperplane, the borderline case with $s=\frac{n}{p}-(n-1)$ and $0<p<1$ is analysed and a new dependence on the sum-exponent $q$ is found. Through examples the restriction operator defined for $s$ down to $1/p$, and valued in $L_p$, is shown to be distinctly different and, moreover, unsuitable for elliptic boundary problems. All boundedness properties (both new and previously known) are found to be easy consequences of a simple mixed-norm estimate, which also yields continuity with respect to the normal coordinate. The surjectivity for the classi…
In this paper we define Besov-Lipschitz and Triebel-Lizorkin spaces in the context of Gaussian harmonic analysis, the harmonic analysis of Hermite polynomial expansions. We study inclusion relations among them, some interpolation results and continuity results of some important operators (the Ornstein-Uhlenbeck and the Poisson-Hermite semigroups and the Bessel potentials) on them. We also prove that the Gaussian Sobolev spaces $L^p_α(γ_d)$ are contained in them. The proofs are general enough to allow extensions of these results to the case of Laguerre or Jacobi expansions and even further in t…
In 2016, Seregin and uSverák, conceived a notion of global in time solution (as well as proving existence of them) to the three dimensional Navier-Stokes equation with $L_3$ solenoidal initial data called 'global $L_3$ solutions'. A key feature of global $L_3$ solutions is continuity with respect to weak convergence of a sequence of solenoidal $L_3$ initial data. The first aim of this paper is to show that a similar notion of ' global $\dot{B}^{-\frac{1}{4}}_{4,\infty}$ solutions' exists for solenoidal initial data in the wider critical space $\dot{B}^{-\frac{1}{4}}_{4,\infty}$ and satisfies c…
We introduce first weighted function spaces on Rd using the Dunkl convolution that we call Besov-Dunkl spaces. We provide characterizations of these spaces by decomposition of functions. Next we obtain in the real line and in radial case on Rd weighted Lp-estimates of the Dunkl transform of a function in terms of an integral modulus of continuity which gives a quantitative form of the Riemann-Lebesgue lemma. Finally, we show in both cases that the Dunkl transform of a function is in L1 when this function belongs to a suitable Besov-Dunkl space.
We prove Sobolev-Poincaré and Poincaré inequalities in variable Lebesgue spaces $L^{p(\cdot)}(Ω)$, with $Ω\subset{\mathbb R}^n$ a bounded John domain, with weaker regularity assumptions on the exponent $p(\cdot)$ that have been used previously. In particular, we require $p(\cdot)$ to satisfy a new \emph{boundary $\log$-Hölder condition} that imposes some logarithmic decay on the oscillation of $p(\cdot)$ towards the boundary of the domain. Some control over the interior oscillation of $p(\cdot)$ is also needed, but it is given by a very general condition that allows $p(\cdot)$ to be discontinu…
In the setting of a metric space equipped with a doubling measure supporting a Poincaré inequality, we show that BV functions are, in the sense of multiple limits, continuous with respect to a 1-fine topology, at almost every point with respect to the codimension 1 Hausdorff measure.
We obtain new molecular decompositions and molecular synthesis estimates for Hermite Besov and Hermite Triebel--Lizorkin spaces and use such tools to prove boundedness properties of Hermite pseudo-multipliers on those spaces. The notion of molecule we develop leads to boundedness of pseudo-multipliers associated to symbols of Hörmander-type adapted to the Hermite setting on spaces with non-positive smoothness; in particular, we obtain continuity results for such operators on Lebesgue and Hermite local Hardy spaces. As a byproduct of our results on boundedness properties of pseudo-multipliers, …
We give a proof of the Hölder continuity of weak solutions of certain degenerate doubly nonlinear parabolic equations in measure spaces. We only assume the measure to be a doubling non-trivial Borel measure which supports a Poincaré inequality. The proof discriminates between large scales, for which a Harnack inequality is used, and small scales, that require intrinsic scaling methods.
We establish the local Hölder continuity for the nonnegative weak solutions of certain doubly nonlinear parabolic equations possessing a singularity in the time derivative part and a degeneracy in the principal part. The proof involves the method of intrinsic scaling and the setting is a measure space equipped with a doubling non-trivial Borel measure supporting a Poincaré inequality.
We complete the study of the regularity for Trudinger's equation by proving that weak solutions are Hölder continuous also in the singular case. The setting is that of a measure space with a doubling non-trivial Borel measure supporting a Poincaré inequality. The proof uses the Harnack inequality and intrinsic scaling.
We study the dependence of the continuity constants for the regularized Poincaré and Bogovski\uı integral operators acting on differential forms defined on a domain $Ω$ of $\mathbb{R}^n$. We, in particular, study the dependence of such constants on certain geometric characteristics of the domain when these operators are considered as mappings from (a subset of) $L^2(Ω,Λ^\ell)$ to $H^1(Ω,Λ^{\ell-1})$, $\ell \in \{1, \ldots, n\}$. For domains $Ω$ that are star shaped with respect to a ball $B$ we study the dependence of the constants on the ratio $diam(Ω)/diam(B)$. A program on how to develop es…
We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical ($α<1/2$) dissipation $(-Δ)^α$. This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical ($α= 1/2$) QG equation \cite{CV}. Their approach successively increases the regularity levels of Leray-Hopf weak solutions: from $L^2$ to $L^\infty$, from $L^\infty$ to Hölder ($C^δ$, $δ>0$), and from Hölder to classical solutions. In the supercritical case, Leray-Hopf weak solutions can still be shown to be $L^\infty$, but it do…
Here we investigate the Cauchy problem for the barotropic Navier-Stokes equations in R^n, in the critical Besov spaces setting. We improve recent results as regards the uniqueness condition: initial velocities in critical Besov spaces with (not too) \emph{negative} indices generate a unique local solution. Apart from (critical) regularity, the initial density just has to be bounded away from 0 and to tend to some positive constant at infinity. Density-dependent viscosity coefficients may be considered. Using Lagrangian coordinates is the key to our statements as it enables us to solve the syst…
In this paper we fix $1\le p<\infty$ and consider $(\Om,d,μ)$ be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure $μ$ supporting a $p$-Poincaré inequality such that $\Om$ is a uniform domain in its completion $\bar\Om$. We realize the trace of functions in the Dirichlet-Sobolev space $D^{1,p}(\Om)$ on the boundary $\partial\Om$ as functions in the homogeneous Besov space $HB^α_{p,p}(\partial\Om)$ for suitable $α$; here, $\partial\Om$ is equipped with a non-atomic Borel regular measure $ν$. We show that if $ν$ satisfies a $θ$-codimensional condit…
We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincaré inequality and a weak Bakry-Émery curvature type condition, this BV class is identified with the heat semigroup based Besov class $\mathbf{B}^{1,1/2}(X)$ that was introduced in our previous paper. Assuming furthermore a quasi Bakry-Émery curvature type condition, we identify the Sobolev class $W^{1,p}(X)$ with $\mathbf{B}^{p,1/2}(X)$ for $p>1$. Consequences of those identifications in terms of isoperimetric and Sobolev inequalities wit…
The main result is that every pseudo-differential operator of type 1,1 and order $d$ is continuous from the Triebel--Lizorkin space $F^d_{p,1}$ to $L_p$, $1\le p<\infty$, and that this is optimal within the Besov and Triebel--Lizorkin scales.The proof also leads to the known continuity for $s>d$, while for all real $s$ the sufficiency of Hörmander's condition on the twisted diagonal is carried over to the Besov and Triebel--Lizorkin framework. To obtain this, type 1,1-operators are extended to distributions with compact spectrum, and Fourier transformed operators of this type are on such distr…
We give a proof for the Hölder continuity of functions in the parabolic De Giorgi classes in metric measure spaces. We assume the measure to be doubling, to support a weak $(1,p)$-Poincaré inequality and to satisfy the annular decay property.
This paper finishes the goal of the authors started in two previous manuscripts dedicated to revisiting the continuity properties of toroidal pseudo-differential operators with symbols in the Hörmander classes. Here we prove pointwise estimates in terms of the Fefferman-Stein sharp maximal function and of the Hardy-Littlewood maximal function. Combining these estimates with the properties of Muckenhoupt's weight class $A_p$ we obtain boundedness theorems for pseudo-differential operators between weighted Lebesgue spaces on the torus $L^p(w)$. These results are given in the context of the globa…
We prove that the density of the law of any finite dimensional projection of solutions of the Navier--Stokes equations with noise in dimension $3$ is Hölder continuous in time with values in the natural space $L^1$. When considered with values in Besov spaces, Hölder continuity still holds. The Hölder exponents correspond, up to arbitrarily small corrections, to the expected diffusive scaling.
