Continuity and analyticity for a cross-coupled Camassa–Holm equation with waltzing peakons and compacton pairs

In this paper, we mainly study the well-posedness in the sense of Hadamard, non-uniform dependence, Hölder continuity and analyticity of the data-to-solution map for a cross-coupled Camassa–Holm equation with waltzing peakons and compacton pairs on both the periodic and the nonperiodic case. Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces \(H^{s} \times H^{s},s>5/2\) in the sense of Hadamard, that is, the data-to-solution mapis continuous. In conjunction with the well-posedness estimate, it is also proved that this dependence is sharp by showing that the solution map is not uniformly continuous. Furthermore, the Hölder continuous in the \(H^r \times H^r\) topology when \(0\le r< s\) with Hölder exponent \(\alpha \) depending on both s and r are shown. Finally, applying generalized Ovsyannikov type theorem and the basic properties of Sobolev-Gevrey spaces, we prove the Gevrey regularity and analyticity of the CCCH system. Moreover, we obtain a lower bound of the lifespan and the continuity of the data-to-solution map     

The Regularity Problems with Data in Hardy–Sobolev Spaces for Singular Schrödinger Equation in Lipschitz Domains

Let the nonnegative singular potential V belong to the reverse Hölder class \({\mathcal B}_n\) on \({\mathbb R}^n\), and let (n???1)/n?p?≤?2, we establish the solvability and derivative estimates for the solutions to the Neumann problem and the regularity problem of the Schrödinger equation ??Δu?+?Vu?=?0 in a connected Lipschitz domain Ω, with boundary data in the Hardy space \(H^p(\partial \Omega)\) and the modified Hardy–Sobolev space \(H_{1, V}^p(\partial \Omega)\) related to the potential V. To deal with the H ^p regularity problem, we construct a new characterization of the atomic decomposition for \(H_{1, V}^p(\partial \Omega)\) space. The invertibility of the boundary layer potentials on Hardy spaces and Hölder spaces are shown in this paper.     

Localized Anisotropic Regularity Conditions for the Navier–Stokes Equations

We establish a sufficient regularity condition for local solutions of the Navier–Stokes equations. For a suitable weak solution (u, p) on a domain D we prove that if \(\partial _3 u\) belongs to the space \(L_t^{s_0}L_x^{r_0}(D)\) where \(2/s_0 + 3/r_0 \le 2 \) and \(9/4 \le r_0\le 5/2\), then the solution is Hölder continuous in D.     

Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations

We investigate quantitative properties of nonnegative solutions \(u(x)\ge 0\) to the semilinear diffusion equation \(\mathcal {L}u= f(u)\), posed in a bounded domain \(\Omega \subset \mathbb {R}^N\) with appropriate homogeneous Dirichlet or outer boundary conditions. The operator \(\mathcal {L}\) may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian \((-\Delta )^s\) (\(0<s<1\)) in a bounded domain with zero Dirichlet conditions, and a number of other nonlocal versions. The nonlinearity f is increasing and looks like a power function \(f(u)\sim u^p\), with \(p\le 1\). The aim of this paper is to show sharp quantitative boundary estimates based on a new iteration process. We also prove that, in the interior, solutions are Hölder continuous and even classical (when the operator allows for it). In addition, we get Hölder continuity up to the boundary. Particularly interesting is the behaviour of solution when the number \(\frac{2s}{1-p}\) goes below the exponent \(\gamma \in (0,1]\) corresponding to the Hölder regularity of the first eigenfunction \(\mathcal {L}\Phi _1=\lambda _1 \Phi _1\). Indeed a change of boundary regularity happens in the different regimes \(\frac{2s}{1-p} \gtreqqless \gamma \), and in particular a logarithmic correction appears in the “critical” case \(\frac{2s}{1-p} = \gamma \). For instance, in the case of the spectral fractional Laplacian, this surprising boundary behaviour appears in the range \(0<s\le (1-p)/2\).     

On the Hölder regularity of the weak solution to a drift–diffusion system with pressure

In this paper we address the regularity issue of weak solution for the following linear drift–diffusion system with pressure

$$\begin{aligned} \partial _t u + b\cdot \nabla u -\Delta u + \nabla p = 0,\quad \mathrm {div}\,u=0,\quad u|_{t=0}(x)=u_0(x), \end{aligned}$$

where \(x\in \mathbb {R}^n\) and b is a given divergence-free vector field. Under some assumptions of the drift field b in the critical sense, and for the initial data \(u_0\in (L^2(\mathbb {R}^n))^n\), we prove that there exists a weak solution u(t) to this system such that u(t) for any time \(t>0\) is \(\alpha \)-Hölder continuous with \(\alpha \in (0,1)\). The proof of the Hölder regularity result utilizes a maximum-principle type method to improve the regularity of weak solution step by step.

     

Construction of sign-changing solutions for a harmonic equation with subcritical exponent

In this paper, we study the harmonic equation involving subcritical exponent \((P_{\varepsilon })\): \( \Delta u = 0 \), in \(\mathbb {B}^n\) and \(\displaystyle \frac{\partial u}{\partial \nu } + \displaystyle \frac{n-2}{2}u = \displaystyle \frac{n-2}{2} K u^{\frac{n}{n-2}-\varepsilon }\) on \( \mathbb {S}^{n-1}\) where \(\mathbb {B}^n \) is the unit ball in \(\mathbb {R}^n\), \(n\ge 5\) with Euclidean metric \(g_0\), \(\partial \mathbb {B}^n = \mathbb {S}^{n-1}\) is its boundary, K is a function on \(\mathbb {S}^{n-1}\) and \(\varepsilon \) is a small positive parameter. We construct solutions of the subcritical equation \((P_{\varepsilon })\) which blow up at two different critical points of K. Furthermore, we construct solutions of \((P_{\varepsilon })\) which have two bubbles and blow up at the same critical point of K.     

Reverse Hölder Property for Strong Weights and General Measures

We present dimension-free reverse Hölder inequalities for strong \(A^*_p\) weights, \(1\le p < \infty \). We also provide a proof for the full range of local integrability of \(A_1^*\) weights. The common ingredient is a multidimensional version of Riesz’s “rising sun” lemma. Our results are valid for any nonnegative Radon measure with no atoms. For \(p=\infty \), we also provide a reverse Hölder inequality for certain product measures. As a corollary we derive mixed \(A_p^*-A_\infty ^*\) weighted estimates.     

Some Estimates of Higher Order Riesz Transform Related to Schrödinger Type Operators

In this paper we consider the Schrödinger type operators \(H_2=(-\Delta)^2 +V^2\), where the nonnegative potential V belongs to the reverse Hölder class \(B_{q_{_1}}\) for \(q_{_1}\geq \frac{n}{2}, n\geq 5\). The L ^p and weak type (1, 1) estimates of higher order Riesz transform \(\nabla^2H^{-\frac{1}{2}}_2 \) related to Schrödinger type operators H ₂ are obtained. In particular, \(\nabla^2H^{-\frac{1}{2}}_2 \) is a Calderón-Zygmund operator if V?∈?B _2n or \(V\in B_\frac{n}{2}\) and there exists a constant C such that V(x)?≤?Cm(x,V)².     

Sharp nonremovability examples for Hölder continuous quasiregular mappings in the plane

Let α ∈ 2 (0, 1), K ≥ 1, and \(d = 2\frac{{1 + \alpha K}}{{1 + K}}\). Given a compact set E ? ?, it is known that if \(\mathcal{H}^d (E) = 0\), then E is removable for α-Hölder continuous K-quasiregular mappings in the plane. The sharpness of the index d is shown with the construction, for any t > d, of a set E of Hausdorff dimension dim(E) = t which is not removable. In this paper, we improve this result and construct compact nonremovable sets E such that \(0 < \mathcal{H}^d (E) < \infty \). For the proof, we give a precise planar K-quasiconformal mapping whose Hölder exponent is strictly bigger than \(\frac{1}{K}\) and which exhibits extremal distortion properties.     

A general low-order partial regularity theory for asymptotically convex functionals with asymptotic dependence on the minimizer

We consider a minimizer \(u\in W^{1,p}(\Omega )\), where \(\Omega \subseteq \mathbb {R}^{n}\) is a bounded open set, of the integral functional

$$\begin{aligned} u\mapsto \int _{\Omega }f(x,u,Du)\ dx \end{aligned}$$

in the case when \(f : \Omega \subseteq \mathbb {R}^{n}\times \mathbb {R}^{N}\times \mathbb {R}^{N\times n}\rightarrow \mathbb {R}\) is asymptotically related to a more regular function; since we assume that \(N\ge 2\), we study here the vectorial case. The asymptotical relatedness condition is such that dependence on u is allowed even as \(|\xi |\rightarrow +\infty \). Unlike in previous work f is allowed to satisfy a more general growth condition, which permits a coupling between the x and u variables. Due to the generality of the asymptotical relatedness condition this coupling induces a subtle restriction on the regularity required of the various functions in the growth condition, and we study these restrictions in this paper. In addition, here we do not obtain the full spectrum of Hölder continuity for the minimizer, but rather a restricted range of Hölder exponents that depend on the initial data.

     

Rainbow Spanning Subgraphs of Small Diameter in Edge-Colored Complete Graphs

Let s(n, t) be the maximum number of colors in an edge-coloring of the complete graph \(K_n\) that has no rainbow spanning subgraph with diameter at most t. We prove \(s(n,t)={\left( {\begin{array}{c}n-2\\ 2\end{array}}\right) }+1\) for \(n,t\ge 3\), while \(s(n,2)={\left( {\begin{array}{c}n-2\\ 2\end{array}}\right) }+\left\lfloor {\frac{n-1}{2}}\right\rfloor \) for \(n\ne 4\) (and \(s(4,2)=2\)).     

Lengths of words in transformation semigroups generated by digraphs

Given a simple digraph D on n vertices (with \(n\ge 2\)), there is a natural construction of a semigroup of transformations \(\langle D\rangle \). For any edge (a, b) of D, let \(a\rightarrow b\) be the idempotent of rank \(n-1\) mapping a to b and fixing all vertices other than a; then, define \(\langle D\rangle \) to be the semigroup generated by \(a \rightarrow b\) for all \((a,b) \in E(D)\). For \(\alpha \in \langle D\rangle \), let \(\ell (D,\alpha )\) be the minimal length of a word in E(D) expressing \(\alpha \). It is well known that the semigroup \(\mathrm {Sing}_n\) of all transformations of rank at most \(n-1\) is generated by its idempotents of rank \(n-1\). When \(D=K_n\) is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate \(\ell (K_n,\alpha )\), for any \(\alpha \in \langle K_n\rangle = \mathrm {Sing}_n\); however, no analogous non-trivial results are known when \(D \ne K_n\). In this paper, we characterise all simple digraphs D such that either \(\ell (D,\alpha )\) is equal to Howie–Iwahori’s formula for all \(\alpha \in \langle D\rangle \), or \(\ell (D,\alpha ) = n - \mathrm {fix}(\alpha )\) for all \(\alpha \in \langle D\rangle \), or \(\ell (D,\alpha ) = n - \mathrm {rk}(\alpha )\) for all \(\alpha \in \langle D\rangle \). We also obtain bounds for \(\ell (D,\alpha )\) when D is an acyclic digraph or a strong tournament (the latter case corresponds to a smallest generating set of idempotents of rank \(n-1\) of \(\mathrm {Sing}_n\)). We finish the paper with a list of conjectures and open problems.     

Improved upper bounds for partial spreads

A partial \((k-1)\)-spread in \({\text {PG}}(n-1,q)\) is a collection of \((k-1)\)-dimensional subspaces with trivial intersection. So far, the maximum size of a partial \((k-1)\)-spread in \({\text {PG}}(n-1,q)\) was known for the cases \(n\equiv 0\pmod k\), \(n\equiv 1\pmod k\), and \(n\equiv 2\pmod k\) with the additional requirements \(q=2\) and \(k=3\). We completely resolve the case \(n\equiv 2\pmod k\) for the binary case \(q=2\).     

Spectral multiplier theorems and averaged <Emphasis Type="Italic">R</Emphasis>-boundedness

Let A be a 0-sectorial operator with a bounded \(H^\infty (\Sigma _\sigma )\)-calculus for some \(\sigma \in (0,\pi ),\) e.g. a Laplace type operator on \(L^p(\Omega ),\, 1< p < \infty ,\) where \(\Omega \) is a manifold or a graph. We show that A has a \(\mathcal {H}^\alpha _2(\mathbb {R}_+)\) Hörmander functional calculus if and only if certain operator families derived from the resolvent \((\lambda - A)^{-1},\) the semigroup \(e^{-zA},\) the wave operators \(e^{itA}\) or the imaginary powers \(A^{it}\) of A are R-bounded in an \(L^2\)-averaged sense. If X is an \(L^p(\Omega )\) space with \(1 \le p < \infty \), R-boundedness reduces to well-known estimates of square sums.     

Weak $$A_\infty $$ Weights and Weak Reverse Hölder Property in a Space of Homogeneous Type

In the Euclidean setting, the Fujii–Wilson-type \(A_\infty \) weights satisfy a reverse Hölder inequality (RHI), but in spaces of homogeneous type the best-known result has been that \(A_\infty \) weights satisfy only a weak reverse Hölder inequality. In this paper, we complement the results of Hytönen, Pérez and Rela and show that there exist both \(A_\infty \) weights that do not satisfy an RHI and a genuinely weaker weight class that still satisfies a weak RHI. We also show that all the weights that satisfy a weak RHI have a self-improving property, but the self-improving property of the strong reverse Hölder weights fails in a general space of homogeneous type. We prove most of these purely non-dyadic results using convenient dyadic systems and techniques.     

Mixed Wavelet Leaders Multifractal Formalism in a Product of Critical Besov Spaces

In this paper, we will prove (resp. study) the Baire generic validity of the upper-Hölder (resp. iso-Hölder) mixed wavelet leaders multifractal formalism on a product of two critical Besov spaces \(B_{t_{1}}^{\frac{m}{t_{1}},q_{1}}(\mathbb {R}^m) \times B_{t_{2}}^{\frac{m}{t_{2}},q_{2}}(\mathbb {R}^m)\), for \(t_1,t_2>0\), \(q_1 \le 1\) and \(q_2 \le 1\). Contrary to product spaces \(B_{t_{1}}^{s_{1},\infty }(\mathbb {R}^m) \times B_{t_{2}}^{s_{2},\infty }(\mathbb {R}^m) \) with \(s_{1} > \frac{m}{t_{1}}\) and \(s_{2} >\frac{m}{t_{2}}\) (Ben Slimane in Mediterr J Math, 13(4):1513–1533, 2016) and \((B_{t_{1}}^{s_{1},\infty }(\mathbb {R}^m) \cap C^{\gamma _{1}}(\mathbb {R}^m)) \times (B_{t_{2}}^{s_{2},\infty }(\mathbb {R}^m) \cap C^{\gamma _{2}}(\mathbb {R}^m)\) with \(0<\gamma _{1}<s_{1}<\frac{m}{t_{1}}\) and \(0<\gamma _{2}<s_{2}<\frac{m}{t_{2}}\) (Ben Abid et al. in Mediterr J Math, 13(6):5093–5118, 2016), all pairs of functions in the obtained generic set are not uniform Hölder. Nevertheless, the characterization of the upper bound of the Hölder exponent by decay conditions of local wavelet leaders suffices for our study.     

On Star–Wheel Ramsey Numbers

For two given graphs \(G_1\) and \(G_2\), the Ramsey number \(R(G_1,G_2)\) is the least integer r such that for every graph G on r vertices, either G contains a \(G_1\) or \(\overline{G}\) contains a \(G_2\). In this note, we determined the Ramsey number \(R(K_{1,n},W_m)\) for even m with \(n+2\le m\le 2n-2\), where \(W_m\) is the wheel on \(m+1\) vertices, i.e., the graph obtained from a cycle \(C_m\) by adding a vertex v adjacent to all vertices of the \(C_m\).     

Quantum MDS codes with relatively large minimum distance from Hermitian self-orthogonal codes

It has become common knowledge that constructing q-ary quantum MDS codes with minimum distance bigger than \(q/2+1\) is significantly more difficult than constructing those with minimum distance less than or equal to \(q/2+1\). Despite of various constructions of q-ary quantum MDS codes, all known q-ary quantum MDS codes have minimum distance bounded by \(q/2+1\) except for some lengths. The purpose of the current paper is to provide some new q-ary quantum MDS codes with minimum distance bigger than \(q/2+1\). In this paper, we provide several classes of quantum MDS codes with minimum distance bigger than \(q/2+1\). For instance, some examples in these classes include q-ary \([n,n-2k, k+1]\)-quantum MDS codes for cases: (i) \(q\equiv -1\bmod {5}, n=(q^2+4)/5\) and \(1\le k\le (3q-2)/5\); (ii) \(q\equiv -1\bmod {7}, n=(q^2+6)/7\) and \(1\le k\le (4q-3)/7\); (iii) \(2|q, q\equiv -1\bmod {3}, n=2(q^2-1)/3\) and \(1\le k\le (2q-1)/3\); and (iv) \(2|q, q\equiv -1\bmod {5}, n=2(q^2-1)/5\) and \(1\le k\le (3q-2)/5\).     

Couplings of Brownian Motions of Deterministic Distance in Model Spaces of Constant Curvature

We consider the model space \(\mathbb {M}^{n}_{K}\) of constant curvature K and dimension \(n\ge 1\) (Euclidean space for \(K=0\), sphere for \(K>0\) and hyperbolic space for \(K<0\)), and we show that given a function \(\rho :[0,\infty )\rightarrow [0, \infty )\) with \(\rho (0)=\mathrm {dist}(x,y)\) there exists a coadapted coupling (X(t), Y(t)) of Brownian motions on \(\mathbb {M}^{n}_{K}\) starting at (x, y) such that \(\rho (t)=\mathrm {dist}(X(t),Y(t))\) for every \(t\ge 0\) if and only if \(\rho \) is continuous and satisfies for almost every \(t\ge 0\) the differential inequality

$$\begin{aligned} -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) \le \rho '(t)\le -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) +\tfrac{2(n-1)\sqrt{K}}{\sin (\sqrt{K}\rho (t))}. \end{aligned}$$

In other words, we characterize all coadapted couplings of Brownian motions on the model space \(\mathbb {M}^{n}_{K}\) for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of \(\rho \) satisfying the above hypotheses.

     

Classifying First Extremal Points for a Fractional Boundary Value Problem with a Fractional Boundary Condition

Let \(n\in \mathbb {N}\), \(n\ge 2\), \(\beta >0\) fixed, and \(0<b\le \beta \). For \(n-1<\alpha \le n\), we look to classify extremal points for the fractional differential equation \(D_{0^+}^{\alpha }u+p(t) u=0\), satisfying the boundary conditions \(u^{(i)}(0)=0\), \(i=0,\ldots ,n-2\), \(D_{0^+}^\gamma u(b)=0\), where p(t) is a continuous nonnegative function on \([0,\beta ]\) which does not vanish identically on any nondegenerate compact subinterval of \([0,\beta ]\). Using the theory of Krein and Rutman, first extremal points of this boundary value problem are classified. As an application, the results are applied, along with a fixed-point theorem, to show the existence of a solution of a nonlinear fractional boundary value problem.