arXiv subjects

Yichun Zhu

Publications and source records attributed to Yichun Zhu.

8 recordsLinked to original sources

arXiv2026

Well-Posedness of the Helmholtz Equation with Rough Coefficients

We establish the well-posedness of the Helmholtz equation with rough and compactly supported coefficients in Rd under sharp regularity assumptions. Using a paraproduct calculus in rescaled weighted Besov spaces, we rigorously define the product between the solution and the coefficient at the lowest regularity level without renormalization. A rescaled Lippmann-Schwinger formulation is shown to be equivalent to the Helmholtz equation with the Sommerfeld radiation condition. We prove existence, uniqueness, and explicit wavenumber dependent resolvent estimates in a general Lp setting, including an L2 theory relevant to scattering amplitudes. The results provide a sharp analytic foundation for wave propagation and scattering in highly irregular media.

math.AP↗

arXiv2025

Nonuniqueness in law of stochastic 3d navierstokes equations with general multiplicative noise

We are concerned with the three dimensional navier-stokes equations driven by a general multiplicative noise. For every divergence free and mean free initial condition in L2, we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, which implies non-uniqueness in law. Moreover, we prove the existence of infinitely many ergodic stationary solutions. Our results are based on a stochastic version of the convex integration and the Ito calculus.

math.PR↗

arXiv2025

An inverse potential problem for the stochastic heat equation with space-time noise

This paper investigates an inverse potential problem for the stochastic heat equation driven by space-time Gaussian noise, which is spatially colored and temporally white. The objective is to determine the covariance operator of the random potential. We establish that the covariance operator can be uniquely identified from the correlation of the mild solution to the stochastic heat equation at a final time, where the initial conditions are specified by a complete orthonormal basis. The analysis relies on characterizing a tensor product structure inherent in the problem and utilizing the monotonicity properties of the operators associated with the system.

math.PR↗

arXiv2024

On Green's function of the vorticity formulation for the 3D Navier-Stokes equations

We give a novel vorticity formulation for the 3D Navier-Stokes equations with Dirichlet boundary conditions. Via a resolvent argument, we obtain Green's function and establish an upper bound, which is the 3D analog of [24]. Moreover, we prove similar results for the corresponding Stokes problem with more general mixed boundary conditions.

math.AP↗

arXiv2022

Averaging principle for slow-fast systems of stochastic PDEs with rough coefficients

In this paper, we consider a class of slow-fast systems of stochastic partial differential equations where the nonlinearity in the slow equation is not continuous and unbounded. We first provide conditions that ensure the existence of a martingale solution. Then we prove that the laws of the slow motions are tight, and any of their limiting points is a martingale solution for a suitable averaged equation. Our results apply to systems of stochastic reaction-diffusion equations where the reaction term in the slow equation is only continuous and has polynomial growth.

math.PR↗

arXiv2022

The growth mechanism of boundary layers for the 2D Navier-Stokes equations

We give a detailed description of formation of the boundary layers in the inviscid limit problem. To be more specific, we prove that the magnitude of the vorticity near the boundary is growing to the size of $1/\sqrtν$ and the width of the layer is spreading out to be proportional the $\sqrtν$ in a finite time period. In fact, the growth time scaling is almost $ν$

math.AP↗

arXiv2020

An averaging approach to the Smoluchowski-Kramers approximation in the presence of a varying magnetic field

We study the small mass limit of the equation describing planar motion of a charged particle of a small mass $μ$ in a force field, containing a magnetic component, perturbed by a stochastic term. We regularize the problem by adding a small friction of intensity $\e>0$. We show that for all small but fixed frictions the small mass limit of $q_{μ, \e}$ gives the solution $q_\e$ to a stochastic first order equation, containing a noise-induced drift term. Then, by using a generalization of the classical averaging theorem for Hamiltonian systems by Freidlin and Wentzell, we take the limit of the slow component of the motion $q_\e$ and we prove that it converges weakly to a Markov process on the graph obtained by identifying all points in the same connected components of the level sets of the magnetic field intensity function.

math.PR↗

arXiv2020

A generalization of the Freidlin-Wentcell theorem on averaging of Hamiltonian systems

In this paper, we generalize the classical Freidlin-Wentzell's theorem for random perturbations of Hamiltonian systems. In stead of the two-dimensional standard Brownian motion, the coefficient for the noise term is no longer the identity matrix but a state-dependent matrix plus a state-dependent matrix that converges uniformly to 0 on any compact sets as $ε$ tends to 0. We also take the drift term into consideration where the drfit term also contains two parts, the state-dependent mapping and a state-dependent mapping that converges uniformly to 0 on any compact sets as $ε$ tends to 0. In the proof, we use the result of generalized differential operator. We also adapt a new way to prove the weak convergence inside the edge by constructing an auxiliary process and apply Girsanov's theorem in the proof of gluing condition.

math.PR↗