[Paper Review] A flow approach to the generalized KPZ equation

Ajay Chandra, Léonard Ferdinand|arXiv (Cornell University)|Feb 5, 2024

Advanced Mathematical Physics Problems4 citations

TL;DR

This paper extends the flow approach of Duch (2021) to establish local well-posedness for the generalized Kardar-Parisi-Zhang (gKPZ) equation with non-polynomial, smooth nonlinearities. By introducing flow coordinates based on elementary differentials and adapting probabilistic and deterministic analysis, the authors prove existence and uniqueness of solutions in a subcritical regime, even for rough noise with H"older-regularity $\alpha > 1/4$ in one dimension.

ABSTRACT

We show that the flow approach of Duch [Duc21] can be adapted to prove local well-posedness for the generalized Kardar-Parisi-Zhang equation. The key step is to extend the flow approach so that it can accommodate semi-linear equations involving smooth, non-polynomial, functions of the solution - this is accomplished by introducing coordinates for the flow built out of elementary differentials.

Motivation & Objective

• To establish local well-posedness for the generalized KPZ equation with smooth, non-polynomial nonlinearities in the drift and noise terms.
• To extend the flow approach—previously used for polynomial or regularized equations—to handle semi-linear SPDEs with general smooth functions of the solution.
• To adapt probabilistic and deterministic analysis tools to control stochastic objects and remainders in the presence of rough noise with H"older-regularity $\alpha > 1/4$.
• To construct a non-stationary effective force via a flow-based renormalization procedure, enabling control of divergent terms in the solution expansion.
• To provide a robust framework for singular SPDEs that can be generalized to other subcritical equations, including those with fractional Laplacians.

Proposed method

• Adapt the flow approach of Duch (2021) by introducing new coordinates based on elementary differentials to represent the solution flow in a non-polynomial setting.
• Define an effective scale and remainder flow to decompose the solution into a regular part and a remainder term, enabling iterative control.
• Introduce a weighted norm $|||\cdot|||_{1,N}$ to control the flow evolution, combining parabolic regularity and stochastic cancellations.
• Use a Kolmogorov-type argument to control the growth of stochastic objects, relying on cumulant estimates and moment bounds.
• Apply a deterministic post-processing step to absorb residual singularities and ensure integrability of the flow at small scales.
• Leverage regularizing kernels $K_{N,\mu}$ and effective Green's functions to control the propagation of regularity and singularities in the solution.

Experimental results

Research questions

• RQ1 Can the flow approach be extended to handle non-polynomial, smooth nonlinearities in the generalized KPZ equation?
• RQ2 How can the flow framework be adapted to control stochastic objects with low regularity, particularly when $\alpha \leq 1$?
• RQ3 What modifications are required to maintain integrability and convergence in the flow iteration when the nonlinearity is not polynomial?
• RQ4 Can the effective force be constructed non-stationarily to absorb divergent terms arising from rough noise?
• RQ5 Does the method yield local well-posedness for the gKPZ equation in the full subcritical regime, including $n=1$, $\alpha > 1/4$?

Key findings

• The authors prove local well-posedness for the generalized KPZ equation with smooth, non-polynomial nonlinearities in the drift and noise terms.
• The flow approach is successfully extended to non-polynomial settings by introducing coordinates based on elementary differentials, enabling control of the solution flow.
• The remainder term in the flow expansion is shown to satisfy $|||\hat{w}_\mu \zeta^a_{\varepsilon,\mu}|||_{N^{2\Gamma + o(a)+1}_1} \lesssim \mu^{|a| - 2\eta}$, ensuring integrability at $\mu = 0$.
• The stochastic objects are controlled via cumulant analysis and a Kolmogorov argument, yielding bounds of order $\nu^{|a| - \eta + 1}$ in the flow evolution.
• The method applies to the full subcritical regime, including the case $n=1$, $\alpha > 1/4$, which was previously excluded in some probabilistic approaches due to divergent variances.
• The framework is robust and can be extended to equations with fractional Laplacians, as demonstrated by the possibility of proving well-posedness for $s > 1$ in the modified equation $\partial_{x_0} + (-\Delta_x)^s$.