Journal of Functional Analysis最新文献

This paper concerns smooth solutions to Cauchy problem for isothermal Euler equations with damping depending on a relaxation time. We prove that the problem admits a unique solution when either the relaxation time or the initial datum is sufficiently small. In particular, this yields the global existence of a large smooth solution when the relaxation time is sufficiently small. We justify that, in an appropriate time scaling, the density of Euler equations with damping converges to the large solution of the heat equation as the relaxation time tends to zero. Moreover, we establish error estimates of such a convergence for the large solutions. A key step in proving these results is a uniform estimate of a quantity related to Darcy's law.

We make major progress on a folkloric conjecture in analysis by constructing a measure on the real line which is doubling on all n-adic intervals for any finite list of $n\in N$, yet not doubling overall. In particular, we extend previous results in the area, including those of Boylan-Mills-Ward and Anderson-Hu, by using a wide array of substantially new ideas. In addition, we provide several nontrivial applications to reverse Hölder weights, $A_p$ weights, Hardy spaces, BMO and VMO function classes, and connect our results with key principles and conjectures across number theory.

Lebesgue space bounds $L^{p_1}\left(R^1\right)\times L^{p_2}\left(R^1\right)\to L^q\left(R^1\right)$ are established for certain singular maximal bilinear operators. The proof combines a single scale trilinear smoothing inequality with Calderón-Zygmund theory.

We consider nonnegative solutions u of the Cauchy problem for a semilinear heat equation with space-dependent reaction: $u_t=\Delta u+\mu \left(x\right)u^p$, $u(x,0)=u_0\left(x\right)$, where $\mu \left(x\right)\ge 0$ satisfies some condition and the initial data $u_0\left(x\right)(\not\equiv 0)$ satisfies ${\Vert \tilde{\mu }{u}_0\Vert }_{L^{\infty }\left(R^N\right)}<\infty$ with $\tilde{\mu }={\mu }^{1/(p-1)}$. We study weighted solutions $\tilde{\mu }u$ which blow up at minimal blow-up time. Such a weighted solution blows up at space infinity in some direction (directional blow-up). We call this direction a blow-up direction of $\tilde{\mu }u$. We give a sufficient and necessary condition on $u_0$ for a weighted solution to blow up at minimal blow-up time. Moreover, we completely characterize blow-up directions of $\tilde{\mu }u$ by the profile of the initial data.

Considered herein is a particular nonlinear dispersive stochastic system consisting of Dirac and Klein-Gordon equations. They are coupled by nonlinear terms due to the Yukawa interaction. We consider a case of homogeneous multiplicative noise that seems to be very natural from the perspective of the least action formalism. We are able to show existence and uniqueness of a corresponding Cauchy problem in Bourgain spaces. Moreover, the regarded model implies charge conservation, known for the deterministic analogue of the system, and this is used to prove a global existence result for suitable initial data.

We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem in finite cones and establish optimal asymptotic expansions in terms of the corresponding solutions in infinite cones. The spherical domains over which cones are formed are allowed to have singularities. An elliptic operator on such spherical domains with coefficients singular on the boundary plays an important role. Due to the singularity of the spherical domains, extra care is needed for the study of the global regularity of the eigenfunctions and solutions of the associated singular Dirichlet problem.

In this paper, we systematically study the existence, asymptotic behaviors, uniqueness, and nonlinear orbital stability of traveling-wave solutions with small propagation speeds for the generalized surface quasi-geostrophic (gSQG) equation. Firstly we obtain the existence of a new family of global solutions via the variational method. Secondly we show the uniqueness of maximizers under our variational setting. Thirdly by using the variational framework, the uniqueness of maximizers and a concentration-compactness principle we establish some stability theorems. Moreover, after a suitable transformation, these solutions constitute the desingularization of traveling point vortex pairs.

This paper is devoted to stability results for the Gaussian logarithmic Sobolev inequality, with explicit stability constants.

We consider the class of weakly porous sets in Euclidean spaces. As our first main result we show that the distance weight $w\left(x\right)=\mathrm{dist}{(x,E)}^{-\alpha }$ belongs to the Muckenhoupt class $A_1$, for some $\alpha >0$, if and only if $E\subset R^n$ is weakly porous. We also give a precise quantitative version of this characterization in terms of the so-called Muckenhoupt exponent of E. When E is weakly porous, we obtain a similar quantitative characterization of $w\in A_p$, for $1<p<\infty$, as well. At the end of the paper, we give an example of a set $E\subset R$ which is not weakly porous but for which $w\in A_p\setminus A_1$ for every $0<\alpha <1$ and $1<p<\infty$.