Mathematics

We consider the Hausdorff dimension of the divergence set on which the pointwise convergence lim t?? e it ?��???f(x)=f(x) fails when f??H s ( R d ) . We especially prove the conjecture raised by Barceló, Bennett, Carbery and Rogers \cite{BBCR} for d=3 , and improve the previous results in higher dimensions d?? . We also show that a Strichartz type estimate for f??e it ?��???f with the measure dtdμ(x) is essentially equivalent to the estimate for the spherical average of μ ? which has been extensively studied for the Falconer distance set problem. The equivalence provides shortcuts to the recent results due to B. Liu and K. Rogers.

For β< 1 3 , we consider C β ( T 3 ?[0,T]) weak solutions of the incompressible Euler equations that do not conserve the kinetic energy. We prove that for such solutions the closed and non-empty set of singular times B satisfies dim H (B)??2β 1?��?. This lower bound on the Hausdorff dimension of the singular set in time is intrinsically linked to the Hölder regularity of the kinetic energy and we conjecture it to be sharp. As a first step in this direction, for every β< β ??< 1 3 we are able to construct, via a convex integration scheme, non-conservative C β ( T 3 ?[0,T]) weak solutions of the incompressible Euler system such that dim H (B)??1 2 + 1 2 2 β ??1??β ??. The structure of the wild solutions that we build allows moreover to deduce non-uniqueness of C β ( T 3 ?[0,T]) weak solutions of the Cauchy problem for Euler from every smooth initial datum.

We introduce a dispersive regularization of the compressible Euler equations in Lagrangian coordinates, in the one-dimensional torus. We assume a Van der Waals pressure law, which presents both hyperbolic and elliptic zones. The dispersive regularization is of Schroedinger type. In particular, the regularized system is complex-valued. It has a conservation law, which, for real unknowns, is identical to the energy of the unregularized physical system. The regularized system supports high-frequency solutions, with an existence time or an amplitude which depend strongly on the pressure law.

The existence of dissipative solutions to the compressible isentropic Navier-Stokes equations was established in this paper. This notion was inspired by the concept of dissipative solutions to the incompressible Euler equations of Lions (\cite{Lions-1996}, Section 4.4). Our method is to recover such solutions by passing to the limits from approximated solutions, thanks to compactness argument.

We prove explicit doubling inequalities and obtain uniform upper bounds (under (d??) -dimensional Hausdorff measure) of nodal sets of weak solutions for a family of linear elliptic equations with rapidly oscillating periodic coefficients. The doubling inequalities, explicitly depending on the doubling index, are proved at different scales by a combination of convergence rates, a three-ball inequality from certain "analyticity", and a monotonicity formula of a frequency function. The upper bounds of nodal sets are shown by using the doubling inequalities, approximations by harmonic functions and an iteration argument.

Consider a tensor product of simple dyadic shifts defined below. We prove here that for dyadic bi-parameter repeated commutator its norm can be estimated from below by Chang-Fefferman BMO norm pertinent to its symbol. See Theorems in Section 8 at the end of this article. But this is done below under an extra assumption on the Haar--Fourier side of the symbol. In Section 7 we carefully analyze what goes wrong in the absence of this extra assumption. At the end of this note we also list a counterexample to the existing proof of characterization of bi-parameter repeated commutator with the Hilbert transforms. This is a counterexample to the proof, and it is not a counterexample to the statement of factorization result in bi-disc, or to Nehari's theorem in bi-disc. To the best of our knowledge Nehari's theorem on bi-disc is still open. Moreover its dyadic bi-parameter version considered in the present paper is also still open for general symbol without any extra restrictions.

In this paper we study a dynamical optimal transport problem on a network that allows for transport of mass between different edges if a penalty κ is paid. We show existence of minimisers using duality and discuss the relationships of the distance-functional to other metrics such as the Fisher-Rao and the classical Wasserstein metric and analyse the resulting distance functional in the limiting case κ?��? .

In this paper, we study the initial boundary value problem of the important hyperbolic Kirchhoff equation u_{tt}-\left(a \int_\Omega |\nabla u|^2 \dif x +b\right)\Delta u = \lambda u+ |u|^{p-1}u , where a , b>0 , p>1 , λ?�R and the initial energy is arbitrarily large. We prove several new theorems on the dynamics such as the boundedness or finite time blow-up of solution under the different range of a , b , λ and the initial data for the following cases: (i) 1<p<3 , (ii) p=3 and a>1/? , (iii) p=3 , a??/? and $\lam <b\lam_1$, (iv) p=3 , a<1/? and $\lam >b\lam_1$, (v) p>3 and $\lam\leq b\lam_1$, (vi) p>3 and $\lam> b\lam_1$, where $\lam_1 = \inf\left\{\|\nabla u\|^2_2 :~ u\in H^1_0(\Omega)\ {\rm and}\ \|u\|_2 =1\right\}$, and ?=inf{?��?u ??4 2 : u??H 1 0 (Ω) and ?�u ??4 =1} . Moreover, we prove the invariance of some stable and unstable sets of the solution for suitable a , b and $\lam$, and give the sufficient conditions of initial data to generate a vacuum region of the solution. Due to the nonlocal effect caused by the nonlocal integro-differential term, we show many interesting differences between the blow-up phenomenon of the problem for a>0 and a=0 .

We consider the time-harmonic elastic wave scattering from a general (possibly anisotropic) inhomogeneous medium with an embedded impenetrable obstacle. We show that the impenetrable obstacle can be effectively approximated by an isotropic elastic medium with a particular choice of material parameters. We derive sharp estimates to rigorously verify such an effective approximation. Our study is strongly motivated by the related studies of two challenging inverse elastic problems including the inverse boundary problem with partial data and the inverse scattering problem of recovering mediums with buried obstacles. The proposed effective medium theory readily yields some interesting applications of practical significance to these inverse problems.

The effects of internal adaptation dynamics on the self-organized aggregation of chemotactic bacteria are investigated by Monte Carlo (MC) simulations based on a two-stream kinetic transport equation coupled with a reaction-diffusion equation of the chemoattractant that bacteria produce. A remarkable finding is a nonmonotonic behavior of the peak aggregation density with respect to the adaptation time; more specifically, aggregation is the most enhanced when the adaptation time is comparable to or moderately larger than the mean run time of bacteria. Another curious observation is the formation of a trapezoidal aggregation profile occurring at a very large adaptation time, where the biased motion of individual cells is rather hindered at the plateau regimes due to the boundedness of the tumbling frequency modulation. Asymptotic analysis of the kinetic transport system is also carried out, and a novel asymptotic equation is obtained at the large adaptation-time regime while the Keller-Segel type equations are obtained when the adaptation time is moderate. Numerical comparison of the asymptotic equations with MC results clarifies that trapezoidal aggregation is well described by the novel asymptotic equation, and the nonmonotonic behavior of the peak aggregation density is interpreted as the transient of the asymptotic solutions between different adaptation time regimes.