L -decay rates of planar viscous rarefaction wave for scalar conservation law with degenerate viscosity in n -dimensions

We investigate the Cauchy problem and the initial-boundary value problem for multi-dimensional conservation laws with degenerate viscosity in the whole space and in the half-space respectively. We give the optimal decay estimates in the W^1,p(1≤p≤∞)$W^{1,p}(1\le p\le \infty )$ norm for the perturbation from the planar viscous rarefaction wave. The analysis based on the new L^p$L^p$-energy method and L¹$L^1$-estimates.     

Weighted dispersive estimates for two-dimensional Schrödinger operators with Aharonov–Bohm magnetic field

We consider two-dimensional Schrödinger operators H   with an Aharonov–Bohm magnetic field and an additional electric potential. We obtain an explicit leading term of the asymptotic expansion of the unitary group e^−itH$e^{-itH}$ for t→∞$t\to \infty$ in weighted L²$L^2$-spaces. In particular, we show that the magnetic field improves the decay of e^−itH$e^{-itH}$ with respect to the unitary group of non-magnetic Schrödinger operators, and that the decay rate in time is determined by the magnetic flux.     

Quantitative Grothendieck property

A Banach space X is Grothendieck if the weak and the weak^? convergence of sequences in the dual space X^?$X^{?}$ coincide. The space ?^∞${?}^{\infty }$ is a classical example of a Grothendieck space due to Grothendieck. We introduce a quantitative version of the Grothendieck property, we prove a quantitative version of the above-mentioned Grothendieck?s result and we construct a Grothendieck space which is not quantitatively Grothendieck. We also establish the quantitative Grothendieck property of L^∞(μ)$L^{\infty }(\mu )$ for a σ-finite measure μ.     

Preserving closedness of operators under summation

We give a sufficient condition for the sum of two closed operators to be closed. In particular, we study the sum of two sectorial operators with the sum of their sectoriality angles greater than π  . We show that if one of the operators admits bounded H^∞$H^{\infty }$-calculus and the resolvent of the other operator satisfies a boundedness condition stronger than the standard sectoriality, but weaker than the bounded imaginary powers property in the case of UMD spaces, then the sum is closed. We apply the result to the abstract parabolic problem and give a sufficient condition for L^p$L^p$-maximal regularity.     

Exponential growth of the vorticity gradient for the Euler equation on the torus

We prove that there are solutions to the Euler equation on the torus with C^1,α$C^{1,\alpha }$ vorticity and smooth except at one point such that the vorticity gradient grows in L^∞$L^{\infty }$ at least exponentially as t→∞$t\to \infty$. The same result is shown to hold for the vorticity Hessian and smooth solutions. Our proofs use a version of a recent result by Kiselev and Šverák [6].     

From homogenization to averaging in cellular flows

We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude A  , in a two-dimensional domain with L²$L^2$ cells. For fixed A  , and L→∞$L\to \infty$, the problem homogenizes, and has been well studied. Also well studied is the limit when L   is fixed, and A→∞$A\to \infty$. In this case the solution equilibrates along stream lines.     

Riesz transforms for Dunkl Hermite expansions

In the present paper, we establish that Riesz transforms for Dunkl Hermite expansions introduced by Nowak and Stempak are singular integral operators with Hörmander's type condition. We prove that they are bounded on L^p(R^d,d_μκ)$L^p({\mathbb{R}}^d,d{\mu }_{\kappa })$ for 1<p<∞$1<p<\infty$ and from L¹(R^d,d_μκ)$L^1({\mathbb{R}}^d,d{\mu }_{\kappa })$ into L^1,∞(R^d,d_μκ)$L^{1,\infty }({\mathbb{R}}^d,d{\mu }_{\kappa })$.     

The cutoff phenomenon for Ehrenfest chains

We consider families of Ehrenfest chains and provide a simple criterion on the L^p$L^p$-cutoff and the L^p$L^p$-precutoff with specified initial states for 1≤p<∞$1\le p<\infty$. For the family with an L^p$L^p$-cutoff, a cutoff time is described and a possible window is given. For the family without an L^p$L^p$-precutoff, the exact order of the L^p$L^p$-mixing time is determined. The result is consistent with the well-known conjecture on cutoffs of Markov chains proposed by Peres in 2004, which says that a cutoff exists if and only if the multiplication of the spectral gap and the mixing time tends to infinity.     

Global solutions to a chemotaxis system with non-diffusive memory

In this article, an existence theorem of global solutions with small initial data belonging to L¹∩L^p$L^1\cap L^p$, (n<p?∞)$(n<p?\infty )$ for a chemotaxis system is given on the whole space Rⁿ${\mathbb{R}}^n$, n?3$n?3$. In the case p=∞$p=\infty$, our global solution is integrable with respect to the space variable on some time interval, and then conserves the mass for a short time, at least. The system consists of a chemotaxis equation with a logarithmic term and an ordinary equation without diffusion term.     

Some remarks on the asymptotic profiles of solutions for strongly damped wave equations on the 1-D half space

We consider a mixed problem with the Dirichlet null boundary conditions on the 1-dimensional half space (0,+∞)$(0,+\infty )$. We will derive an asymptotic profiles of the corresponding solutions in the case when the initial data belong to a weighted L^1,2(0,∞)$L^{1,2}(0,\infty )$ space by employing a method recently introduced in [7] or [8].     

Existence and nonlinear stability of stationary solutions to the full compressible Navier–Stokes–Korteweg system

This paper is concerned with the existence, uniqueness, and nonlinear stability of stationary solutions to the Cauchy problem of the full compressible Navier–Stokes–Korteweg system effected by the given mass source, the external force of general form, and the energy source in R³${\mathbb{R}}^3$. Based on the weighted L²$L^2$-method and some delicate L^∞$L^{\infty }$ estimates on solutions to the linearized problem, the existence and uniqueness of stationary solution are obtained by the contraction mapping principle. The proof of the stability result is given by an elementary energy method and relies on some intrinsic properties of the full compressible Navier–Stokes–Korteweg system.     

Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations

We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical (α<1/2$\alpha <1/2$) dissipation α(−Δ)${(-\mathrm{\Delta })}^{\alpha }$. 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$\alpha =1/2$) QG equation [L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, arXiv: math.AP/0608447, 2006]. Their approach successively increases the regularity levels of Leray–Hopf weak solutions: from L²$L^2$ to L^∞$L^{\infty }$, from L^∞$L^{\infty }$ to Hölder (Cδ$C^{\delta }$, δ>0$\delta >0$), and from Hölder to classical solutions. In the supercritical case, Leray–Hopf weak solutions can still be shown to be L^∞$L^{\infty }$, but it does not appear that their approach can be easily extended to establish the Hölder continuity of L^∞$L^{\infty }$ solutions. In order for their approach to work, we require the velocity to be in the Hölder space C1−2α$C^{1-2\alpha }$. Higher regularity starting from Cδ$C^{\delta }$ with δ>1−2α$\delta >1-2\alpha$ can be established through Besov space techniques and will be presented elsewhere [P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press].     

Korn type inequalities in Orlicz spaces

We exhibit balance conditions between a Young function A and a Young function B   for a Korn type inequality to hold between the L^B$L^B$ norm of the gradient of vector-valued functions and the L^A$L^A$ norm of its symmetric part. In particular, we extend a standard form of the Korn inequality in L^p$L^p$, with 1<p<∞$1<p<\infty$, and an Orlicz version involving a Young function A   satisfying both the _Δ2${\mathrm{\Delta }}_2$ and the _∇2${\mathrm{\nabla }}_2$ condition.     

Type III factors with unique Cartan decomposition

We prove that for any free ergodic nonsingular nonamenable action Γ?(X,μ)$\Gamma ?(X,\mu )$ of all Γ   in a large class of groups including all hyperbolic groups, the associated group measure space von Neumann algebra L^∞(X)?Γ${\mathrm{L}}^{\infty }(X)?\Gamma$ has L^∞(X)${\mathrm{L}}^{\infty }(X)$ as its unique Cartan subalgebra, up to unitary conjugacy. This generalizes the probability measure preserving case that was established in Popa and Vaes (in press) [38]. We also prove primeness and indecomposability results for such crossed products, for the corresponding orbit equivalence relations and for arbitrary amalgamated free products _M1?BM2$M_1{?}_B{M}_2$ over a subalgebra B of type I.     

Hypergroups and invariant complemented subspaces

Let K   be a hypergroup with a Haar measure. The purpose of the present paper is to initiate a systematic approach to the study of the class of invariant complemented subspaces of _L∞(K)$L_{\infty }(K)$ and _C0(K)$C_0(K)$, the class of left translation invariant w^?$w^{?}$-subalgebras of _L∞(K)$L_{\infty }(K)$ and finally the class of non-zero left translation invariant C^?$C^{?}$-subalgebras of _C0(K)$C_0(K)$ in the hypergroup context with the goal of finding some relations between these function spaces. Among other results, we construct two correspondences: one, between closed Weil subhypergroups and certain left translation invariant w^?$w^{?}$-subalgebras of _L∞(K)$L_{\infty }(K)$, and another, between compact subhypergroups and a specific subclass of the class of left translation invariant C^?$C^{?}$-subalgebras of _C0(K)$C_0(K)$. By the help of these two characterizations, we extract some results about invariant complemented subspaces of _L∞(K)$L_{\infty }(K)$ and _C0(K)$C_0(K)$.     

The wave equation for the Bessel Laplacian

We study radial solutions of the Cauchy problem for the wave equation in the multidimensional unit ball B^d$B^d$, d≥1$d\ge 1$. In this case, the operator that appears is the Bessel Laplacian and the solution u(t,x)$u(t,x)$ is given in terms of a Fourier–Bessel expansion. We prove that, for initial L^p$L^p$ data, the series converges in the L²$L^2$ norm. The analysis of a particular operator, the adjoint of the Riesz transform for Fourier–Bessel series, is needed for our purposes, and may be of independent interest. As applications, certain L^p−L²$L^p-L^2$ estimates for the solution of the heat equation and the extension problem for the fractional Bessel Laplacian are obtained.     

An alternate relaxation approximation to conservation laws

In this paper we introduce a slight modification to the relaxation system of Jin and Xin which approximates a conservation law. The proposed alternate system satisfies an integral constraint that is more consistent than the standard one while retaining the semilinear structure. We establish L^∞$L^{\infty }$ estimates under the usual subcharacteristic condition and also construct a convex entropy.