arXiv subjects

Houqi Su

Publications and source records attributed to Houqi Su.

4 recordsLinked to original sources

arXiv2026

Strong convergence rates for numerical approximations of the stochastic Burgers equation with additive rough noise

We establish strong convergence rates for a fully discrete method of the one-dimensional stochastic Burgers equation driven by additive rough noise. Using the Cole-Hopf transformation, the equation is converted into a linear stochastic heat equation, which is discretized by a spectral Galerkin method in space and an accelerated exponential Euler method in time. A numerical approximation of the original solution is then reconstructed through a regularized inverse Cole-Hopf transformation. The analysis combines strong Sobolev error estimates with the positivity and inverse-moment bounds for the transformed solution. For additive space-time white noise, the numerical method achieves mean-square convergence rates arbitrarily close to 1/2 in space and 1/4 in time. The rates are slower when the noises are rougher.

math.NA↗

arXiv2026

On the stability of rarefaction for stochastic viscous conservation law

We study the asymptotic stability of rarefaction waves for one-dimensional stochastic viscous conservation laws driven by nonlinear conservative noise. In a critical scaling where stochastic energy injection and viscous dissipation compete at comparable magnitudes, standard kinetic and viscosity frameworks encounter obstructions due to regularity gaps and non-integrable profiles. To address this, we introduce a stochastic area inequality controlling accumulated energy fluctuations, a local $L^1$ contraction principle via stochastic Kružkov doubling-of-variables that yields pathwise uniqueness without global integrability, and a modified Galerkin scheme preserving the $H^2$ energy structure. Assuming local $H^2$ regularity, we prove almost sure algebraic convergence to the rarefaction wave. For sufficiently small initial perturbations, we establish global well-posedness and sharp decay estimates in expectation. The smallness condition identifies a regime where viscous dissipation dominates stochastic injection, reflecting a structural stability threshold rather than a technical artifact. Our approach extends the analytical framework for conservative SPDEs with rough fluxes.

math.PR↗

arXiv2025

Ergodicity of the viscous scalar conservation laws with a degenerate noise

This paper establishes the ergodicity in $H^\nn,\nn=\lfloor\frac{d}{2}+1\rfloor$ of the viscous scalar conservation laws on torus $\mT^d$ with general polynomial flux and a degenerate noise. The noise could appear in as few as several directions. We introduce a localized framework that restricts attention to trajectories with controlled energy growth, circumventing the limitations of traditional contraction-based approaches. This localized method allows for a demonstration of e-property and consequently proves the uniqueness of invariant measure under a H{ö}rmander-type condition. Furthermore, we characterize the absolute continuity of the invariant measure's projections onto any finite-dimensional subspaces under requirement on a new algebraically non-degenerate condition for the flux.

math.PR↗

arXiv2021

Large time behavior of strong solutions for stochastic Burgers equation

We consider the large time behavior of strong solutions to a kind of stochastic Burgers equation, where the position x is perturbed by a Brownian noise. It is well known that both the rarefaction wave and viscous shock wave are time-asymptotically stable for deterministic Burgers equation since the pioneer work of A. Ilin and O. Oleinik [21] in 1964. However, the stability of these wave patterns under stochastic perturbation is not known until now. In this paper, we give a definite answer to the stability problem of the rarefaction and viscous shock waves for the 1-d stochastic Burgers equation. That is, the rarefaction wave is still stable under white noise perturbation and the viscous shock is not stable yet. Moreover, a time-convergence rate toward the rarefaction wave is obtained. To get the desired decay rate, an important inequality (denoted by Area Inequality) is derived. This inequality plays essential role in the proof, and may have applications in the related problems for both the stochastic and deterministic PDEs.

math.PR↗