In this paper, we establish a weighted Trudinger-Moser type inequality with the full Sobolev norm constraint on the whole Euclidean space. Main tool is the singular Trudinger-Moser inequality on the whole space recently established by Adimurthi and Yang, and a transformation of functions. We also discuss the existence and non-existence of maximizers for the associated variational problem.

In this paper, we consider minimizers for nonlocal energy functionals generalizing elastic energies that are connected with the theory of peridynamics [19] or nonlocal diffusion models [1]. We derive nonlocal versions of the Euler-Lagrange equations under two sets of growth assumptions for the integrand. Existence of minimizers is shown for integrands with joint convexity (in the function and nonlocal gradient components). By using the convolution structure, we show regularity of solutions for certain Euler-Lagrange equations. No growth assumptions are needed for the existence and regularity of minimizers results, in contrast with the classical theory.

We demonstrate norm inflation for nonlinear nonlocal equations, which extend the Korteweg-de Vries equation to permit fractional dispersion, in the periodic and non-periodic settings. That is, an initial datum is smooth and arbitrarily small in a Sobolev space but the solution becomes arbitrarily large in the Sobolev space after an arbitrarily short time.

We study the filtered Euler equations that are the regularized Euler equations derived by filtering the velocity field. The filtered Euler equations are a generalization of two well-known regularizations of incompressible inviscid flows, the Euler-$\alpha$ equations and the vortex blob method. We show the global existence of a unique weak solution for the two-dimensional (2D) filtered Euler equations with initial vorticity in the space of Radon measure that includes point vortices and vortex sheets. Moreover, a sufficient condition for the global well-posedness is described in terms of the filter and thus our result is applicable to various filtered models. We also show that weak solutions of the 2D filtered Euler equations converge to those of the 2D Euler equations in the limit of the regularization parameter provided that initial vorticity belongs to the space of bounded functions.

We revisit an example of a semi-Riemannian geodesic that was discussed by Musso, Pejsachowicz and Portaluri in 2007 to show that not every conjugate point is a bifurcation point. We point out a mistake in their argument, showing that on this geodesic actually every conjugate point is a bifurcation point. Finally, we provide an improved example which shows that the claim in our title is nevertheless true.

We study the uniqueness of singular radial (forward and backward) self-similar positive solutions of the equation $ u_t-\Delta u = u^p, $ $ x\in\mathbb R^n,\ t>0, $ where $p\geq(n+2)/(n-2)_+$.

The aim of this paper is to prove the existence of a weak-renormalized solution to a simplified model of turbulence of the $k-\varepsilon$ kind in spatial dimension $N=2$. The unknowns are the average velocity field and pressure, the mean turbulent kinetic energy and an appropriate time dependent variable. The motion equation and the additional PDE are respectively solved in the weak and renormalized senses.