Related papers: A sharp stability criterion for Euler equations vi…
Sobolev-type embeddings on metric measure spaces encode a subtle interaction between the analytic regularity of functions and the geometry of the underlying domain space. In this paper we develop an embedding theory for variable…
Euler equations are the basic system in fluid dynamics describing the motion of incompressible and inviscid ideal fluids. For a bounded smooth domain $\Omega$ in $\mathbb{R}^n$. The well-posedness of Euler equations is well-known in Sobolev…
Sobolev embeddings, of arbitrary order, are considered into function spaces on domains of $\mathbb R^n$ endowed with measures whose decay on balls is dominated by a power $d$ of their radius. Norms in arbitrary rearrangement-invariant…
We construct and investigate the properties of tempered ultradistribution spaces in Sobolev spaces. A new Sobolev space preserving the original properties and condition whose derivatives are linear continuous operators embedding in $L^p$…
In this paper, we study the long-time stability behavior of a class of linear stochastic evolution equations in a Hilbert space with multiplicative noise. Explicit sufficient conditions for $p$-th moment and almost sure exponential…
We present a novel method for establishing large data local well-posedness in low regularity Sobolev spaces for general quasilinear Schr\"odinger equations with non-degenerate and nontrapping metrics. Our result represents a definitive…
For the $d$-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space $H^s(\mathbb R^d)$, $s>s_c:=d/2+1$. The borderline case $s=s_c$ was a folklore open problem.…
Entropy stabilization of the compressible Euler system is achieved by adapting the averages that are applied to the density and internal energy variables. The approach achieves non-linear robustness despite the use of simplified symmetric…
We provide a complete local well-posedness theory in $H^s$ based Sobolev spaces for the free boundary incompressible Euler equations with zero surface tension on a connected fluid domain. Our well-posedness theory includes: (i) Local…
This article is concerned with the well-posedness of the incompressible Euler equations describing a stably stratified ocean, reformulated in isopycnal coordinates. Our motivation for using this reformulation is twofold: first, its quasi-2D…
In this paper, we establish the stability for the Hardy-Littlewood-Sobolev (HLS) inequalities with explicit lower bounds. By establishing the relation between the stability of HLS inequalities and the stability of fractional Sobolev…
We prove a characterization of Hardy's inequality in Sobolev-Slobodecki\u{\i} spaces in terms of positive local weak supersolutions of the relevant Euler-Lagrange equation. This extends previous results by Ancona and Kinnunen & Korte for…
The paper is concerned with the sparse approximation of functions having hybrid regularity borrowed from the theory of solutions to electronic Schr\"odinger equations due to Yserentant [43]. We use hyperbolic wavelets to introduce…
We study existence, uniqueness and stability of radial solutions of the Lane-Emden-Fowler equation $-\Delta_g u=|u|^{p-1}u$ in a class of Riemannian models $(M,g)$ of dimension $n\ge 3$ which includes the classical hyperbolic space $\mathbb…
This paper is devoted to improvements of functional inequalities based on scalings and written in terms of relative entropies. When scales are taken into account and second moments fixed accordingly, deficit functionals provide explicit…
Accurate simulations of ice sheet dynamics, mantle convection, lava flow, and other highly viscous free-surface flows involve solving the coupled Stokes/free-surface equations. In this paper, we theoretically analyze the stability and…
We consider a one-parameter family of beam equations with Hamiltonian non-linearity in one space dimension under periodic boundary conditions. In a unified functional framework we study the long time evolution of initial data in two…
For the $2D$ Euler equation in vorticity formulation, we construct localized smooth solutions whose critical Sobolev norms become large in a short period of time, and solutions which initially belong to $L^\infty \cap H^1$ but escapes $H^1$…
We develop arbitrarily high-order, stationarity-preserving stabilized finite element methods for multidimensional nonlinear hyperbolic balance laws on Cartesian grids. We aim at approximating all the steady states of the problem at hand,…
We propose a functional framework of fractional Sobolev spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak H\"ormander condition. We characterize these spaces as real interpolation of natural order intrinic…