Analysis of PDEs (math.AP)

Abstract: We establish the existence of at least two solutions of the {\it Prandtl-Batchelor} like elliptic problem driven by a power nonlinearity and a singular term. The associated energy functional is nondifferentiable and hence the usual variational techniques do not work. We shall use a novel approach in tackling the associated energy functional by a sequence of $C^1$ functionals and a {\it cutoff function}. Our main tools are fundamental elliptic regularity theory and the mountain pass theorem.

Abstract: We prove some Liouville-type theorems for stable solutions (and solutions stable outside a compact set) of quasilinear anisotropic elliptic equations. Our results cover the particular case of the pure Finsler p-Laplacian.

Abstract: In this paper, we prove $L^2 \to L^p$ estimates, where $p>2$, for spectral projectors on a wide class of hyperbolic surfaces. More precisely, we consider projections in small spectral windows $[\lambda-\eta,\lambda+\eta]$ on geometrically finite hyperbolic surfaces of infinite volume. In the convex cocompact case, we obtain optimal bounds with respect to $\lambda$ and $\eta$, up to subpolynomial losses. The proof combines the resolvent bound of Bourgain-Dyatlov and improved estimates for the Schr\"odinger group (Strichartz and smoothing estimates) on hyperbolic surfaces.

Abstract: This paper concerns with the global dynamics of classical solutions to an important alarm-taxis ecosystem, which demonstrates the behaviors of prey that attract secondary predator when threatened by primary predator. And the secondary predator pursues the signal generated by the interaction of the prey and primary predator. However, it seems that the necessary gradient estimates for global existence cannot be obtained in critical case due to strong coupled structure. Thereby, we develop a new approach to estimate the gradient of prey and primary predator which takes advantage of slightly higher damping power. Then the boundedness of classical solutions in two dimension with Neumann boundary conditions can be established by energy estimates and semigroup theory. Moreover, by constructing Lyapunov functional, it is proved that the coexistence homogeneous steady states is asymptotically stability and the convergence rate is exponential under certain assumptions on the system coefficients.

Abstract: Several extensions of the classical optimal transport distances to the quantum setting have been proposed. In this paper, we investigate the pseudometrics introduced by Golse, Mouhot and Paul in [Commun Math Phys 343:165-205, 2016] and by Golse and Paul in [Arch Ration Mech Anal 223:57-94, 2017]. These pseudometrics serve as a quantum analogue of the Monge--Kantorovich--Wasserstein distances of order $2$ on the phase space. We prove that they are comparable to negative Sobolev norms up to a small term in the semiclassical approximation, which can be expressed using the Wigner--Yanase Skew information. This enables us to improve the known results in the context of the mean-field and semiclassical limits by requiring less regularity on the initial data.

Abstract: We study the electron magnetohydrodynamics (MHD) in two dimensional geometry, which has a rich family of steady states. In an anisotropic resistivity context, we show global in time existence of small smooth solution near a shear type steady state. Convergence rate of the solution to the steady state is also obtained.