Fermat-type configurations of lines in P3 and the containment problem

The purpose of this note is to show a new series of examples of homogeneous ideals I in $\mathbb{K}[x,y,z,w]$ for which the containment $I^{(3)}?I^2$ fails. These ideals are supported on certain arrangements of lines in ${\mathbb{P}}^3$, which resemble Fermat configurations of points in ${\mathbb{P}}^2$, see [14]. All examples exhibiting the failure of the containment $I^{(3)}?I^2$ constructed so far have been supported on points or cones over configurations of points. Apart from providing new counterexamples, these ideals seem quite interesting on their own.     

Cardinal characteristics at κ in a small u(κ) model

We provide a model where $\mathfrak{u}(\kappa )<2^{\kappa }$ for a supercompact cardinal κ. [10] provides a sketch of how to obtain such a model by modifying the construction in [6]. We provide here a complete proof using a different modification of [6] and further study the values of other natural generalizations of classical cardinal characteristics in our model. For this purpose we generalize some standard facts that hold in the countable case as well as some classical forcing notions and their properties.     

Stability of the sum of two solitary waves for (gDNLS) in the energy space

In this paper, we continue the study in [18]. We use the perturbation argument, modulational analysis and the energy argument in [15], [16] to show the stability of the sum of two solitary waves with weak interactions for the generalized derivative Schrödinger equation (gDNLS) in the energy space. Here (gDNLS) hasn't the Galilean transformation invariance, the pseudo-conformal invariance and the gauge transformation invariance, and the case $\sigma >1$ we considered corresponds to the $L^2$-supercritical case.     

Convergence to nonlinear diffusion waves for solutions of Euler equations with time-depending damping

In this paper, we are concerned with the asymptotic behavior of solutions to the system of Euler equations with time-depending damping, in particular, include the constant coefficient damping. We rigorously prove that the solutions time-asymptotically converge to the diffusion wave whose profile is self-similar solution to the corresponding parabolic equation, which justifies Darcy's law. Compared with previous results about Euler equations with constant coefficient damping obtained by Hsiao and Liu (1992) [2], and Nishihara (1996) [9], we obtain a general result when the initial perturbation belongs to the same space, i.e. $H^3(\mathbb{R})\times H^2(\mathbb{R})$. Our proof is based on the classical energy method.     

Curved fronts in the Belousov–Zhabotinskii reaction–diffusion systems in R2

In this paper we consider a diffusion system with the Belousov–Zhabotinskii (BZ for short) chemical reaction. Following Brazhnik and Tyson [4] and Pérez-Muñuzuri et al. [45], who predicted V-shaped fronts theoretically and discovered V-shaped fronts by experiments respectively, we give a rigorous mathematical proof of their results. We establish the existence of V-shaped traveling fronts in ${\mathbb{R}}^2$ by constructing a proper supersolution and a subsolution. Furthermore, we establish the stability of the V-shaped front in ${\mathbb{R}}^2$.     

Dispersive estimates for the wave equation inside cylindrical convex domains: A model case

In this work, we will establish local in time dispersive estimates for solutions to the model-case Dirichlet wave equation inside a cylindrical convex domain $\mathrm{\Omega }?{\mathbb{R}}^3$ with a smooth boundary $?\mathrm{\Omega }\ne ?$. Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Nonoptimal Strichartz estimates for waves inside an arbitrary domain Ω have been proved by Blair–Smith–Sogge [1], [2]. Better estimates in strictly convex domains have been obtained in [4]. Our case of cylindrical domains is an extension of the result of [4] in the case where the curvature radius ≥0 depends on the incident angle and vanishes in some directions.     

Finite-time blow-up for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper deals with the quasilinear degenerate Keller–Segel systems of parabolic–parabolic type in a ball of ${\mathbb{R}}^N$ ($N\ge 2$). In the case of non-degenerate diffusion, Cie?lak–Stinner [3], [4] proved that if $q>m+\frac{2}{N}$, where m denotes the intensity of diffusion and q denotes the nonlinearity, then there exist initial data such that the corresponding solution blows up in finite time. As to the case of degenerate diffusion, it is known that a solution blows up if $q>m+\frac{2}{N}$ (see Ishida–Yokota [13]); however, whether the blow-up time is finite or infinite has been unknown. This paper gives an answer to the unsolved problem. Indeed, the finite-time blow-up of energy solutions is established when $q>m+\frac{2}{N}$.     

Proof of the Hénon–Lane–Emden conjecture in R3

We study the Hénon–Lane–Emden conjecture, which states that there is no non-trivial non-negative solution for the Hénon–Lane–Emden elliptic system whenever the pair of exponents is subcritical. By scale invariance of the solutions and Sobolev embedding on $S^{N?1}$, we prove this conjecture is true for space dimension $N=3$; which also implies the single elliptic equation has no positive classical solutions in ${\mathbb{R}}^3$ when the exponent lies below the Hardy–Sobolev exponent, this covers the conjecture of Phan–Souplet [22] for ${\mathbb{R}}^3$.     

Regularity of the Eikonal equation with two vanishing entropies

Let $\mathrm{\Omega }?{\mathbb{R}}^2$ be a bounded simply-connected domain. The Eikonal equation $\left|\mathrm{?}u\right|=1$ for a function $u:\mathrm{\Omega }?{\mathbb{R}}^2\to \mathbb{R}$ has very little regularity, examples with singularities of the gradient existing on a set of positive $H^1$ measure are trivial to construct. With the mild additional condition of two vanishing entropies we show ?u is locally Lipschitz outside a locally finite set. Our condition is motivated by a well known problem in Calculus of Variations known as the Aviles–Giga problem. The two entropies we consider were introduced by Jin, Kohn [26], Ambrosio, DeLellis, Mantegazza [2] to study the Γ-limit of the Aviles–Giga functional. Formally if u satisfies the Eikonal equation and if

(1)$\mathrm{?}?({\stackrel{?}{\mathrm{\Sigma }}}_{e_1{e}_2}(\mathrm{?}{u}^{\perp }))=0\text{ and }\mathrm{?}?({\stackrel{?}{\mathrm{\Sigma }}}_{{?}_1{?}_2}(\mathrm{?}{u}^{\perp }))=0\text{ distributionally in }\mathrm{\Omega },$

where ${\stackrel{?}{\mathrm{\Sigma }}}_{e_1{e}_2}$ and ${\stackrel{?}{\mathrm{\Sigma }}}_{{?}_1{?}_2}$ are the entropies introduced by Jin, Kohn [26], and Ambrosio, DeLellis, Mantegazza [2], then ?u is locally Lipschitz continuous outside a locally finite set.Condition (1) is motivated by the zero energy states of the Aviles–Giga functional. The zero energy states of the Aviles–Giga functional have been characterized by Jabin, Otto, Perthame [25]. Among other results they showed that if ${\lim }_{n\to \infty }?I_{{?}_n}(u_n)=0$ for some sequence $u_n\in W_0^{2,2}(\mathrm{\Omega })$ and $u={\lim }_{n\to \infty }?u_n$ then ?u is Lipschitz continuous outside a finite set. This is essentially a corollary to their theorem that if u is a solution to the Eikonal equation $\left|\mathrm{?}u\right|=1$ a.e. and if for every “entropy” Φ (in the sense of [18], Definition 1) function u satisfies $\mathrm{?}?[\mathrm{\Phi }(\mathrm{?}{u}^{\perp })]=0$ distributionally in Ω then ?u is locally Lipschitz continuous outside a locally finite set. In this paper we generalize this result in that we require only two entropies to vanish.The method of proof is to transform any solution of the Eikonal equation satisfying (1) into a differential inclusion $DF\in K$ where $K?M^{2\times 2}$ is a connected compact set of matrices without Rank-1 connections. Equivalently this differential inclusion can be written as a constrained non-linear Beltrami equation. The set K is also non-elliptic in the sense of Sverak [32]. By use of this transformation and by utilizing ideas from the work on regularity of solutions of the Eikonal equation in fractional Sobolev space by Ignat [23], DeLellis, Ignat [15] as well as methods of Sverak [32], regularity is established.     

Trace formulas for relative Schatten class perturbations

We derive trace formulas for a pair of self-adjoint operators $H+V$ and H under the assumption that ${(H?\mathrm{i})}^{?1}V$ is in a Schatten class. This extends the trace formulas of [8], where V alone is assumed to be in a Schatten class. Our trace formulas apply, in particular, in the setting of differential operators and are based on Taylor-like approximations of operator functions. This significantly improves non-Taylor based trace formulas of [10].     

Blow-up for a three dimensional Keller–Segel model with consumption of chemoattractant

We investigate blow-up properties for the initial-boundary value problem of a Keller–Segel model with consumption of chemoattractant when the spatial dimension is three. Through a kinetic reformulation of the Keller–Segel system, we first derive some higher-order estimates and obtain certain blow-up criteria for the local classical solutions. These blow-up criteria generalize the results in [4], [5] from the whole space ${\mathbb{R}}^3$ to the case of bounded smooth domain $\mathrm{\Omega }?{\mathbb{R}}^3$. Lower global blow-up estimate on ${6n6}_{L^{\infty }(\mathrm{\Omega })}$ is also obtained based on our higher-order estimates. Moreover, we prove local non-degeneracy for blow-up points.     

On the irreducible action of PSL(2,R) on the 3-dimensional Einstein universe

In this paper, we study the irreducible representation of $\text{PSL}(2,\mathbb{R})$ in $\text{PSL}(5,\mathbb{R})$. This action preserves a quadratic form with signature $(2,3)$. Thus, it acts conformally on the 3-dimensional Einstein universe $\mathbb{E}{\mathrm{in}}^{1,2}$. We describe the orbits induced in $\mathbb{E}{\mathrm{in}}^{1,2}$ and its complement in $\mathbb{R}{\mathbb{P}}^4$. This work completes the study in [2], and is one element of the classification of cohomogeneity one actions on $\mathbb{E}{\mathrm{in}}^{1,2}$[5].     

On the stability of the solitary waves to the (generalized) Kawahara equation

In this paper we investigate the orbital stability of solitary waves to the (generalized) Kawahara equation (gKW) which is a fifth order dispersive equation. For some values of the power of the nonlinearity, we prove the orbital stability in the energy space $H^2(\mathbb{R})$ of two branches of even solitary waves of gKW by combining the well-known spectral method introduced by Benjamin [4] with continuity arguments. We construct the first family of even solitons by applying the implicit function theorem in the neighborhood of the explicit solitons of gKW found by Dey et al. [9]. The second family consists of even traveling waves with low speeds. They are solutions of a constraint minimization problem on the line and rescaling of perturbations of the soliton of gKdV with speed 1.     

Space-time asymptotics of the two dimensional Navier–Stokes flow in the whole plane

We consider the space-time behavior of the two dimensional Navier–Stokes flow. Introducing some qualitative structure of initial data, we succeed to derive the first order asymptotic expansion of the Navier–Stokes flow without moment condition on initial data in $L^1({\mathbb{R}}^2)\cap L_{\sigma }^2({\mathbb{R}}^2)$. Moreover, we characterize the necessary and sufficient condition for the rapid energy decay ${6u(t)6}_2=o(t^{?1})$ as $t\to \infty$ motivated by Miyakawa–Schonbek [21]. By weighted estimated in Hardy spaces, we discuss the possibility of the second order asymptotic expansion of the Navier–Stokes flow assuming the first order moment condition on initial data. Moreover, observing that the Navier–Stokes flow $u(t)$ lies in the Hardy space $H^1({\mathbb{R}}^2)$ for $t>0$, we consider the asymptotic expansions in terms of Hardy-norm. Finally we consider the rapid time decay ${6u(t)6}_2=o(t^{?\frac{3}{2}})$ as $t\to \infty$ with cyclic symmetry introduced by Brandolese [2].     

The topological structure of complete noncompact submanifolds in spheres

In this paper, we first show that δ-super stable complete noncompact minimal submanifolds in $S^{m+n}$ or $R^{m+n}$ with $\delta >{(\frac{m?1}{m})}^2$ admit no nontrivial $L^2$ harmonic 1-forms and have only one nonparabolic end, which generalizes Cao–Shen–Zhu's result in [2] on stable minimal hypersurface in $R^{m+1}$ and Lin's result in [13] on $\frac{m?1}{m}$-super stable minimal submanifolds in $R^{m+n}$. Second, we prove that the dimension of the space of $L^2$ harmonic p-forms on $M^m$ is zero or finite and there is only one nonparabolic end or finitely many nonparabolic ends of M under the assumptions on the Schrödinger operators involving the squared norm of the traceless second fundamental form.     

A sharp bound on the expected number of upcrossings of an L2-bounded Martingale

For a martingale $M$ starting at $x$ with final variance ${\sigma }^2$, and an interval $(a,b)$, let $\Delta =\frac{b?a}{\sigma }$ be the normalized length of the interval and let $\delta =\frac{|x?a|}{\sigma }$ be the normalized distance from the initial point to the lower endpoint of the interval. The expected number of upcrossings of $(a,b)$ by $M$ is at most $\frac{\sqrt{1+{\delta }^2}?\delta }{2\Delta }$ if ${\Delta }^2\le 1+{\delta }^2$ and at most $\frac{1}{1+{(\Delta +\delta )}^2}$ otherwise. Both bounds are sharp, attained by Standard Brownian Motion stopped at appropriate stopping times. Both bounds also attain the Doob upper bound on the expected number of upcrossings of $(a,b)$ for submartingales with the corresponding final distribution. Each of these two bounds is at most $\frac{\sigma }{2(b?a)}$, with equality in the first bound for $\delta =0$. The upper bound $\frac{\sigma }{2}$ on the length covered by $M$ during upcrossings of an interval restricts the possible variability of a martingale in terms of its final variance. This is in the same spirit as the Dubins & Schwarz sharp upper bound $\sigma$ on the expected maximum of $M$ above $x$, the Dubins & Schwarz sharp upper bound $\sigma \sqrt{2}$ on the expected maximal distance of $M$ from $x$, and the Dubins, Gilat & Meilijson sharp upper bound $\sigma \sqrt{3}$ on the expected diameter of $M$.