Blow-up of solutions to critical semilinear wave equations with variable coefficients

We verify the critical case $p=p_0(n)$ of Strauss' conjecture [30] concerning the blow-up of solutions to semilinear wave equations with variable coefficients in ${\mathbf{R}}^n$, where $n\ge 2$. The perturbations of Laplace operator are assumed to be smooth and decay exponentially fast at infinity. We also obtain a sharp lifespan upper bound for solutions with compactly supported data when $p=p_0(n)$. The unified approach to blow-up problems in all dimensions combines several classical ideas in order to generalize and simplify the method of Zhou [43] and Zhou & Han [45]: exponential “eigenfunctions” of the Laplacian [37] are used to construct the test function ${?}_q$ for linear wave equation with variable coefficients and John's method of iterations [13] is augmented with the “slicing method” of Agemi, Kurokawa and Takamura [1] for lower bounds in the critical case.     

Local well-posedness of the incompressible Euler equations in B∞,11 and the inviscid limit of the Navier–Stokes equations

We prove the inviscid limit of the incompressible Navier–Stokes equations in the same topology of Besov spaces as the initial data. The proof is based on proving the continuous dependence of the Navier–Stokes equations uniformly with respect to the viscosity. To show the latter, we rely on some Bona–Smith type argument in the $L^p$ setting. Our obtained result implies a new result that the Cauchy problem of the Euler equations is locally well-posed in the borderline Besov space $B_{\infty ,1}^1({\mathbb{R}}^d)$, $d\ge 2$, in the sense of Hadmard, which is an open problem left in recent works by Bourgain and Li in [3], [4] and by Misio?ek and Yoneda in [12], [13], [14].     

Approximation locale précisée dans des problèmes multi-échelles avec défauts localisés

We proceed here with our systematic study, initiated in [3], of multiscale problems with defects, within the context of homogenization theory. The case under consideration here is that of a diffusion equation with a diffusion coefficient of the form of a periodic function perturbed by an $L^r({\mathbb{R}}^d)$, $1<r<+\infty$, function modelling a localized defect. We outline the proof of the following approximation result: the corrector function, the existence of which has been established in [3], [4], allows us to approximate the solution to the original multiscale equation with essentially the same accuracy as in the purely periodic case. The rates of convergence may however vary, and are made precise, depending upon the $L^r$ integrability of the defect. The generalization to an abstract setting is mentioned. Our proof exactly follows, step by step, the pattern of the original proof of Avellaneda and Lin in [1] in the periodic case, extended in the works of Kenig and collaborators [12], and borrows a lot from it. The details of the results announced in this Note are given in our publications [2], [11].     

A refined convergence result in homogenization of second order parabolic systems

We derive the sharp $O(\epsilon )$ convergence rate in $L^2(0,T;L^{q_0}(\mathrm{\Omega })),q_0=2d/(d?1)$ in periodic homogenization of second order parabolic systems with bounded measurable coefficients in Lipschitz cylinders. This extends the corresponding result for elliptic systems established in [20] to parabolic systems and improves the corresponding result in $L^2$ settings derived in [7], [28] for second order parabolic systems with time-dependent coefficients.     

Exact rates of convergence in some martingale central limit theorems

Renz [14], Ouchti [13], El Machkouri and Ouchti [3] and Mourrat [12] have established some tight bounds on the rate of convergence in the central limit theorem for martingales. In the present paper a modification of the methods, developed by Bolthausen [1] and Grama and Haeusler [7], is applied for obtaining exact rates of convergence in the central limit theorem for martingales with differences having conditional moments of order $2+\rho ,\rho >0$. Our results generalise and strengthen the bounds mentioned above.     

A uniform bound of reducibility index of good parameter ideals for certain class of modules

Let $(R,\mathfrak{m})$ be a Noetherian local ring and M a finitely generated R-module. The invariants $\mathrm{p}(M)$ and $\mathrm{sp}(M)$ of M were introduced in [3] and [17] in order to measure the non-Cohen–Macaulayness and the non-sequential-Cohen–Macaulayness of M, respectively. Let $M=D_0?D_1?\dots ?D_k$ be the filtration of M such that $D_i$ is the largest submodule of M of dimension less than $\dim ?D_{i?1}$ for all $i\le k$ and $\mathrm{p}(D_k)\le 1$. In this paper we prove that if $\mathrm{sp}(M)\le 1$, then there exists a constant c such that ${\mathrm{ir}}_M(\mathfrak{q}M)\le c$ for all good parameter ideals $\mathfrak{q}$ of M with respect to this filtration. Here ${\mathrm{ir}}_M(\mathfrak{q}M)$ is the reducibility index of $\mathfrak{q}$ on M. This is an extension of the main results of [19], [20], [24].     

Non-existence of extremals for the Adimurthi–Druet inequality

The Adimurthi–Druet [1] inequality is an improvement of the standard Moser–Trudinger inequality by adding a $L^2$-type perturbation, quantified by $\alpha \in [0,{\lambda }_1)$, where ${\lambda }_1$ is the first Dirichlet eigenvalue of Δ on a smooth bounded domain. It is known [3], [10], [14], [19] that this inequality admits extremal functions, when the perturbation parameter α is small. By contrast, we prove here that the Adimurthi–Druet inequality does not admit any extremal, when the perturbation parameter α approaches ${\lambda }_1$. Our result is based on sharp expansions of the Dirichlet energy for blowing sequences of solutions of the corresponding Euler–Lagrange equation, which take into account the fact that the problem becomes singular as $\alpha \to {\lambda }_1$.     

On modal logics arising from scattered locally compact Hausdorff spaces

For a topological space X, let $\mathsf{L}(X)$ be the modal logic of X where □ is interpreted as interior (and hence ◇ as closure) in X. It was shown in [3] that the modal logics S4, S4.1, S4.2, S4.1.2, S4.Grz, ${\mathsf{S4}.\mathsf{Grz}}_n$ ($n\ge 1$), and their intersections arise as $\mathsf{L}(X)$ for some Stone space X. We give an example of a scattered Stone space whose logic is not such an intersection. This gives an affirmative answer to [3, Question 6.2]. On the other hand, we show that a scattered Stone space that is in addition hereditarily paracompact does not give rise to a new logic; namely we show that the logic of such a space is either S4.Grz or ${\mathsf{S4}.\mathsf{Grz}}_n$ for some $n\ge 1$. In fact, we prove this result for any scattered locally compact open hereditarily collectionwise normal and open hereditarily strongly zero-dimensional space.     

Ill-posedness of the Camassa–Holm and related equations in the critical space

We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa–Holm equation, Degasperis–Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the Besov space $B_{p,r}^{1+1/p}$ for $p\in [1,\infty ],r\in (1,\infty ]$. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works ([5], [14], [16]).     

Energy identity and necklessness for a sequence of Sacks–Uhlenbeck maps to a sphere

Let u be a map from a Riemann surface M to a Riemannian manifold N and $\alpha >1$, the α energy functional is defined as

$E_{\alpha }(u)=\frac{1}{2}\underset{M}{\int }[{(1+|▽u{|}^2)}^{\alpha }?1]dV.$

We call $u_{\alpha }$ a sequence of Sacks–Uhlenbeck maps if $u_{\alpha }$ are critical points of $E_{\alpha }$ and

$\underset{\alpha >1}{\sup }?E_{\alpha }(u_{\alpha })<\infty .$

In this paper, we show the energy identity and necklessness for a sequence of Sacks–Uhlenbeck maps during blowing up, if the target N is a sphere $S^{K?1}$. The energy identity can be used to give an alternative proof of Perelman's result [15] that the Ricci flow from a compact orientable prime non-aspherical 3-dimensional manifold becomes extinct in finite time (cf. [3], [4]).     

Asymptotic spreading of a diffusive competition model with different free boundaries

Recently, the authors of [22] studied a diffusive prey–predator model with two different free boundaries. They first obtained the existence, uniqueness, regularity, uniform estimates and long time behaviors of global solution, and then established the conditions for spreading and vanishing. Especially, when spreading occurs, they provided accurate limits of two species as $t\to +\infty$, and gave some estimates of asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries. Motivated by the paper [22], in this paper we discuss the diffusive competition model with two different free boundaries, which had been investigated by [7], [11], [15], [21]. The main purpose of this paper is to establish much sharper estimates of asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries when spreading occurs. Furthermore, how the solution approaches the semi-wave when spreading happens is also described.     

Bases in the mod p Steenrod algebra, β included

In [3] well known results of Wall and Arnon on the monomial bases in the mod 2 Steenrod algebra (see [9], [1]) were generalized to the subalgebra ${\stackrel{￣}{A}}_p$ of the mod p Steenrod algebra, $p>2$, generated by the reduced powers. In the present paper we considered the case of the full Steenrod algebra $A_p$. We constructed βX-, βZ-, βC-, ZA-, and XC-bases. We proved extremal properties of the βX-, βZ-, ZA-, and XC-bases. Also we constructed a new polynomial generators of the ring $H^{?}(K(\mathbb{Z}/p,n),\mathbb{Z}/p)$ in terms of the βC-basis.     

Simple finite-dimensional modular noncommutative Jordan superalgebras

We classify the central simple finite-dimensional noncommutative Jordan superalgebras over an algebraically closed field of characteristic $p>2$. The case of characteristic 0 was considered by the authors in the previous paper [21]. In particular, we describe Leibniz brackets on all finite dimensional central simple Jordan superalgebras except mixed (nor vector neither Poisson) Kantor doubles of the supercommutative superalgebra $B(m,n)$.     

Interior regularity for fractional systems

We study the regularity of solutions of elliptic fractional systems of order 2s, $s\in (0,1)$, where the right hand side f depends on a nonlocal gradient and has the same scaling properties as the nonlocal operator. Under some structural conditions on the system we prove interior Hölder estimates in the spirit of [1]. Our results are stable in s allowing us to recover the classic results for elliptic systems due to S. Hildebrandt and K. Widman [11] and M. Wiegner [19].     

Cyclotomic p-adic multi-zeta values

The cyclotomic p-adic multi-zeta values are the p-adic periods of ${\pi }_1^{uni}({\mathbb{G}}_m?{\mu }_M,?)$, the unipotent fundamental group of the multiplicative group minus the M-th roots of unity. In this paper, we compute the cyclotomic p-adic multi-zeta values at all depths. This paper generalizes the results in [9] and [10]. Since the main result gives quite explicit formulas we expect it to be useful in proving non-vanishing and transcendence results for these p-adic periods and also, through the use of p-adic Hodge theory, in proving non-triviality results for the corresponding p-adic Galois representations.     

Bounds on multiplicities of spherical spaces over finite fields

Let G be a reductive group scheme of type A acting on a spherical scheme X. We prove that there exists a number C such that the multiplicity $\dim \mathrm{Hom}(\rho ,\mathbb{C}[X(F)])$ is bounded by C, for any finite field F and any irreducible representation ρ of $G(F)$. We give an explicit bound for C. We conjecture that this result is true for any reductive group scheme and when F ranges (in addition) over all local fields of characteristic 0.Different aspects of this conjecture were studied in [3], [11], [6], [7].     

Blowup analysis for integral equations on bounded domains

Consider the integral equation

$f^{q?1}(x)=\underset{\mathrm{\Omega }}{\int }\frac{f(y)}{|x?y{|}^{n?\alpha }}dy,f(x)>0,x\in \stackrel{￣}{\mathrm{\Omega }},$

where $\mathrm{\Omega }?{\mathbb{R}}^n$ is a smooth bounded domain. For $1<\alpha <n$, the existence of energy maximizing positive solution in the subcritical case $2<q<\frac{2n}{n+\alpha }$, and nonexistence of energy maximizing positive solution in the critical case $q=\frac{2n}{n+\alpha }$ are proved in [6]. For $\alpha >n$, the existence of energy minimizing positive solution in the subcritical case $0<q<\frac{2n}{n+\alpha }$, and nonexistence of energy minimizing positive solution in the critical case $q=\frac{2n}{n+\alpha }$ are also proved in [4]. Based on these, in this paper, the blowup behaviour of energy maximizing positive solution as $q\to {(\frac{2n}{n+\alpha })}^{+}$ (in the case of $1<\alpha <n$), and the blowup behaviour of energy minimizing positive solution as $q\to {(\frac{2n}{n+\alpha })}^{?}$ (in the case of $\alpha >n$) are analyzed. We see that for $1<\alpha <n$ the blowup behaviour obtained is quite similar to that of the elliptic equation involving the subcritical Sobolev exponent. But for $\alpha >n$, different phenomena appear.     

Existence and nonexistence of positive solutions for a static Schrödinger–Poisson–Slater equation

In this paper we study the following type of the Schrödinger–Poisson–Slater equation with critical growth

$?△u+(u^2?\frac{1}{|4\pi x|})u=\mu |u{|}^{p?1}u+|u{|}^{4}u,\text{in}{\mathbb{R}}^3,$

where $\mu >0$ and $p\in (11/7,5)$. For the case of $p\in (2,5)$. We develop a novel perturbation approach, together with the well-known Mountion–Pass theorem, to prove the existence of positive ground states. For the case of $p=2$, we obtain the nonexistence of nontrivial solutions by restricting the range of μ and also study the existence of positive solutions by the constrained minimization method. For the case of $p\in (11/7,2)$, we use a truncation technique developed by Brezis and Oswald [9] together with a measure representation concentration-compactness principle due to Lions [27] to prove the existence of radial symmetrical positive solutions for $\mu \in (0,{\mu }^{?})$ with some ${\mu }^{?}>0$. The above results nontrivially extend some theorems on the subcritical case obtained by Ianni and Ruiz [18] to the critical case.