偏微分方程分析

We consider the initial value problem (IVP) for a two-parameter family of derivative nonlinear Schr\"odinger equations on the torus, known as the Majda-McLaughlin-Tabak (MMT) model arising in weak wave turbulence theory. For positive…

We consider the semilinear parabolic equation \[ \partial_t u = \Delta u + 2\mathfrak{q}\,\delta_{\mathbb{S}}\,\nabla u + |u|^{p-1}u \qquad \text{in } (0,\infty)\times\mathbb{R}^N, \] where $|\mathfrak{q}|\le 1$, $p>1$, and $\mathbb{S}$ is…

We consider solutions of uniformly elliptic equations with measurable coefficients. We assume that the lowest eigenvalue of the coefficient matrix is at least $K^{-1}$ and the largest eigenvalue is at most $K$. In three and higher…

This work presents the analysis and numerical simulation of a stationary drift-diffusion model for electrical discharge in micro-electro-mechanical systems (MEMS). The model couples Poisson's equation for the electrostatic potential with…

We investigate a variational problem for eigenvalues of the Laplace-Beltrami operator on smooth manifolds with respect to Radon measures belonging to a suitable class; we are motivated by conformal eigenvalues in dimension two. Our main…

For minimizers of a degenerate diffusion functional with a singular reaction term, we prove that the free boundary is $(n-1)$-rectifiable. The argument relies on a suitable integrability property, derived from a pointwise gradient estimate,…

We investigate the existence of segregated rotating waves, arising in the singular limit of competition-diffusion systems of the type \[ \partial_t u_i -\partial_{xx} u_i = f(u_i)-\beta u_i \sum_{j \neq i} a_{ij} u_j,\qquad…

We prove global-in-time strong pathwise well-posedness for a stochastic fluid-structure interaction problem coupling a two-dimensional incompressible Navier-Stokes fluid to a one-dimensional damped Kirchhoff plate. The coupling is imposed…

We study small--data solutions of a nonlinear scalar field equation on spatially flat $d$--dimensional FLRW spacetimes ($d\ge4$). In conformal time $\tau$ the field satisfies a damped semilinear wave/Klein--Gordon equation with…

We study wave-type equations on dynamical spacetimes that settle down to a subextremal Kerr black hole spacetime. We prove strong estimates for solutions of (tensorial) linear wave-type equations when the time-translation-invariant model…

In this paper, we introduce a suitable notion of flat solutions for the anisotropic surface diffusion equation with elasticity in three-dimensions, based on a minimizing movement scheme inspired by that introduced by Cahn and Taylor. Using…

We study small-amplitude solitary waves for two-dimensional capillary--gravity flows with arbitrary vorticity on the equatorial $f$-plane. The steady free-boundary problem is formulated as a reversible Hamiltonian spatial-dynamics system in…

Let $\mathbb{B}$ be the unit ball in $\mathbb{R}^2$, $W_0^{1,2} \left( \mathbb{B} \right)$ is a standard Sobolev space. Suppose a function $h_{\epsilon}(x)$ is radially symmetric, nonnegative, continuous on $\overline{\mathbb{B}}$ and…

We study the asymptotic behavior of global minimizers of a Ginzburg--Landau-type functional with general compact vacuum manifold $\mathcal{N}$ on bounded domains in $\mathbb{R}^3$, in the regime where the energy grows at a logarithmic rate.…

We study the dissipative Boussinesq problem, which extends the "good" Boussinesq equation by incorporating viscosity effects. It is well-known that this model supports monotone decreasing traveling kink solutions. We show that these kinks…

We study energy quantization for a class of Dirac systems on compact spin Einstein manifolds of dimension \(n\). For a sequence of solutions to a nonlinear Dirac system with uniformly bounded energy on a fixed spin Riemannian manifold, we…

In the context of hyperbolic formulations of Einstein's field equations obtained via gauge fixing, constraint damping is a desirable feature that ensures that violations of the gauge condition and thus of the constraint equations are…

We study the two-dimensional stochastic Navier-Stokes equations on the torus with horizontal dissipation and additive noise. First, we prove a uniform large deviation principle for the solution paths in the energy space $C([0,T];H).$ The…

We study a one-dimensional active-line equation arising as a thin-sheet reduced mechanism for high-Weissenberg Oldroyd-B dynamics. The unknown is a positive periodic line density rho=m+eta satisfying rho_t+c rho Lambda rho-c(H…

We prove a finite-scale estimate for vortex stretching in spatially filtered three-dimensional Navier--Stokes flow. The positive near-field part of the filtered stretching is bounded by a pairwise defect of filtered vorticity directions. A…