Analysis of PDEs Papers of the Week

We study the long-time behavior of the kinetic Cucker-Smale model under initial data with unbounded support in both position and velocity.

We solve the equation by defining three synthetic cases - linear (\(\eta=0.001\), \(\gamma=10^{-6}\)), shock-dominated (\(\eta=0.02\), \(\gamma=10^{-6}\)), and dissipation-dominated (\(\eta=0.01\), \(\gamma=5\times10^{-4}\)) - and using Physics-Informed Neural Networks (PINNs) to showcase the solution.

This paper examines the stability threshold at high Reynolds number $\mathbf{Re}$ for the 3D Boussinesq equations with rotation near Couette flow and a vertical stratified temperature.

In particular, $\textbf{S}$ acts on vectors defined over the shrinking boundary layer but behaves asymptotically like a known operator. In this paper, we use a variational argument to determine conditions for which the eigenvalues and eigenvectors of $\textbf{S}$ converge to the eigenvalues and eigenfunctions of a Dirichlet-to-Neumann map, and we provide evidence of this spectral convergence through a numerical example.

We consider a family of nonlinear Schrödinger equations with odd-power nonlinearities and fractional dispersion in an energy-critical/supercritical regime.

Our aim in this paper is to derive an effective bound for the scattering norm $\|u\|_{L_t^3([0,\infty);L_x^6(\mathbb{R}^3))}$, for scattering energy class solutions $u$ with data sufficiently near $Q$ and close to the threshold manifold.

Using an averaging method over descendant levels, we prove the density of finitely supported functions in $W_a^{1,p}(T,m)$ for coefficients satisfying a weighted moment condition.

We study the interior regularity of convex solutions to $$\det D^2u = f(u)$$ in planar domains, with no regularity or monotonicity assumptions on $f$.

We study Fisher-KPP equations on hyperbolic space with reactions periodic in a Busemann coordinate or in the distance from a point or a geodesic.

We establish Bramson corrections for Busemann step data at positive, zero and negative speeds and, above the periodic extinction threshold, for the largest Busemann coordinate and radial level positions of finitely supported solutions. When the periodic spectral minimum is nonpositive but the geodesic spectral threshold is positive, we construct a smallest positive stationary solution with exponential transverse decay, prove uniqueness under an exponential bound, and determine longitudinal spreading with its logarithmic correction.

In this paper, we study Egorov's theorem for Laplacian eigenfunctions on compact hyperbolic surfaces in Zelditch's quantization.

We study optimal potentials for the finite-rank fractional Lieb--Thirring problem at the eigenvalue-counting endpoint, for \(0<s<1\) and \(d>2s\).

We study attainment of the best constant in \begin{equation*} \int_{\mathbb R^{m+n}} \bigl(|\nabla_xu|^2+|x|^2|\nabla_yu|^2+a|u|^2\bigr) \geq S_a\left(\int_{\mathbb R^{m+n}} (1+|x|^2)^{n/(Q-2)}|u|^q\right)^{2/q}, \end{equation*} where \(m,n\geq1\), \(Q=m+2n\), \(q=2Q/(Q-2)\), and \(a>0\).

The main mathematical challenges are the nonlinear coupling between the free shock boundary and the moving piston, alongside potential gradient catastrophes induced by spherical geometric source terms. To overcome these difficulties, we reformulate the problem using Riemann invariants and introduce a tailored directional derivative along the unperturbed piston trajectory.

The proof integrates Vitali's covering lemma, Hardy-Littlewood maximal theory, self-improving estimates, and compactness arguments.

We study the one-dimensional nonlinear half-wave equation with a logarithmically renormalized point interaction.

We establish the existence of solutions to the Neumann problem for the Hessian equation on a smooth, bounded strictly static convex domain in hyperbolic space.

We consider the so-called Weighted Energy-Dissipation approximation of gradient flows in metric spaces.

We study positive solutions \(Q\in H^1(\mathbb R^N)\) of \[-\Delta Q+Q=(I_\alpha*Q^p)Q^{p-1},\] where \(I_\alpha\) is the Riesz potential of order \(\alpha\), \(N\ge1\), \(0<\alpha<N\), \(p>1+\alpha/N\), and \((N-2)p<N+\alpha\).