Quasilinear Parabolic Stochastic Evolution Equations

• The topic is defined as a class of SPDEs where the leading diffusion term depends nonlinearly on the solution and is perturbed by stochastic processes.
• Methodologies include Sobolev space frameworks, energy estimates, and controlled rough path techniques to ensure global existence and regularity of solutions.
• Applications span stochastic convection–diffusion, porous media, and nonlinear transport, with analysis of large deviations and blow-up criteria offering critical insights.

Quasilinear parabolic stochastic evolution equations are a class of stochastic partial differential equations (SPDEs) in which the leading-order (diffusive) and possibly lower-order coefficients depend nonlinearly on the solution itself, and the system is perturbed by a stochastic process, typically a cylindrical Wiener process. These equations generalize semilinear models by allowing for quasilinear (solution-dependent) structure in the principal part and arise in modeling a wide range of physical phenomena involving stochastic transport, porous media, nonlinear diffusions, and convection in random environments. Rigorous analysis centers on existence, regularity, and large deviation principles in appropriate function spaces under precise structural hypotheses on the coefficients and the noise.

1. Structural Formulation and Analytical Framework

A prototypical quasilinear parabolic SPDE on the $d$-dimensional periodic torus $\Td$ takes the form

$du(t) + \Div(B(u(t)))\,dt = \Div(A(u(t))\nabla u(t))\,dt + \sigma(u(t))\,dW(t),$

where $B:\R\to\R^d$ (flux), $A:\R\to\R^{d\times d}$ (diffusion matrix), $\sigma(u):U\to H$ (multiplicative noise mapping from a separable Hilbert space $U$ to $H=L^2(\Td)$), and $W$ is a $U$-cylindrical Wiener process (Dong et al., 2017).

The associated function space framework typically involves Sobolev spaces $H^a(\Td)$, their duals $H^{-a}(\Td)$, and pathwise solution spaces $\mathcal{E} = C([0,T];H)\cap L^2([0,T];H^1)$. Compactness and regularity analysis relies on Rellich-Kondrachov and Aubin-Lions embeddings, ensuring the appropriate compactness for passage to limits and for the construction of skeleton equations in LDP analysis (Dong et al., 2017, Zhang, 2019).

Structural hypotheses (uniform ellipticity, local/global Lipschitz regularity for $A$ and $B$, Hilbert–Schmidt noise with polynomial or quadratic growth and Lipschitz continuity for $\sigma$) guarantee global existence and uniqueness of strong or weak solutions via energy estimates: $\E\!\left[\sup_{t\le T}\|u(t)\|_H^2+\int_0^T \|\nabla u(t)\|_{L^2}^2\,dt\right]<\infty.$ (Dong et al., 2017)

2. Existence, Uniqueness, and Regularity Theory

Global pathwise existence and uniqueness is established for nondegenerate quasilinear systems under $C^1_{\mathrm{Lip}}$ regularity of $A$ and $B$ and suitable growth and Lipschitz conditions on $\sigma$, both in $L^2$ and higher Sobolev spaces. For degenerate cases (where the diffusion $A(u)$ may vanish or be singular), existence, uniqueness, and $L^1$ contraction are advanced via kinetic formulation: solutions are measures solving a velocity-dependent equation, and uniqueness is established by doubling variable techniques and generalized Itô formulas for weak solutions (Debussche et al., 2013).

Regularity theory proceeds via deterministic PDE bootstrapping and stochastic convolution estimates. The decomposition $u = y + z$ separates the solution into a deterministic PDE part $y$ and a stochastic convolution $z$, allowing for iterative upgrades of spatial and temporal regularity via Schauder theory, Krylov–Safonov techniques, and functional analytic interpolation. Under $C^k$ regularity of $A,B,F$ and compatible initial data, regularity up to $C^{k+\vartheta}$ (space) and $C^{(k+\vartheta)/2}$ (time) is achieved for all $\vartheta<1/2$ (Debussche et al., 2014).

3. Well-Posedness in Banach and Hilbert Scales; Blow-up Criterion

Abstract quasilinear stochastic evolution equations in Banach or Hilbert spaces are formulated as

$$du(t) + A(u(t))u(t)\,dt = F(t,u(t))\,dt + B(t,u(t))\,dW(t)$$

with sectorial operator families $A(u)$, locally Lipschitz nonlinearities $F, B$, and initial data in suitable trace or interpolation spaces. The maximal $L^p$ regularity theory supplies strong solution existence and uniqueness up to a maximal stopping time, characterized by a blow-up alternative: the solution either exists globally or blows up in the high regularity norm, with explosion prevented under uniform continuity and $L^p$-control (Hornung, 2016).

Paradifferential operator techniques reduce fully nonlinear stochastic parabolic PDEs to quasilinear form amenable to analytic and probabilistic treatment via the bounded $H^\infty$-calculus and maximal regularity (Agresti, 2018). Fixed-point arguments using controlled semigroups and stopping-time localizations produce pathwise mild solutions in quasilinear Cauchy problems, e.g., for cross-diffusion and Landau–Lifshitz–Gilbert systems (Kuehn et al., 2018, Hocquet et al., 2022).

4. Large Deviations and Probabilistic Asymptotics

A central theme is the Freidlin–Wentzell large deviation principle (LDP) for solution trajectories as noise amplitude $\varepsilon\to0$. For quasilinear SPDEs with multiplicative noise, the family $\{u^\varepsilon\}$ satisfies an LDP in $\mathcal{E}$, quantified by the good rate function

$$I(\varphi) = \inf_{\substack{h\in L^2\ \varphi = G^0\left(\int_0^\cdot h(s)ds\right)}} \frac12 \int_0^T \|h(s)\|_U^2 ds,$$

where $G^0$ maps controls $h$ to solutions of the deterministic skeleton equation

$du^h + \Div(B(u^h))dt = \Div(A(u^h)\nabla u^h)dt + \sigma(u^h)h(t)dt, \quad u^h(0) = u_0$

(Dong et al., 2017, Zhang, 2019, Cerrai et al., 2022).

The proof utilizes the Budhiraja–Dupuis weak convergence framework, requiring compactness (Aubin–Lions lemma), exponential equivalence between the nonlinear and reference Gaussian systems, and tightness of controlled approximations. This provides asymptotic estimates for probabilities of rare deviations from the deterministic limit and is crucial for quantifying stochastic effects in nonlinear models.

Extensions include small-time asymptotics, central limit theorems, moderate deviations, and metastability analysis, with current methods covering non-degenerate sectors and periodic domains. Open directions involve degenerate diffusion, exit-time asymptotics, importance sampling for rare event simulation, and extensions to manifold or network settings (Zhang, 2019).

5. Deterministic-Stochastic Interplay and Rough Path Extensions

Quasilinear SPDE analysis integrates deterministic PDE techniques (Schauder theory, Krylov–Safonov interior regularity, paradifferential calculus) with probabilistic methods (martingale inequalities, stochastic factorization, controlled rough paths).

Recent advances employ controlled path and rough path expansions: $$du_t - L_t(u_t)u_t\,dt = N_t(u_t)\,dt + F(u_t)\,d\mathbf{X}_t$$ where $\mathbf{X}$ is a $\gamma$-Hölder rough path ($\gamma\in(1/3,1/2)$), and the mild solution is constructed via sewing maps and fixed-point arguments in Banach scales. This framework admits stochastic drivers beyond classical Wiener processes, allowing for fractional noise or general rough signals (Hocquet et al., 2022, Otto et al., 2016).

6. Applications and Representative Models

The theory covers stochastic quasilinear convection–diffusion equations,

$du + \Div(u^m)\,dt = \Delta(u^n)\,dt + \sqrt{\varepsilon}\sigma(u)dW, \quad m,n \geq 1,$

stochastic porous media, sedimentation–consolidation, two-phase flow, nonlinear reaction-diffusion, generalized non-Newtonian Navier–Stokes, cross-diffusion systems (Shigesada–Kawasaki–Teramoto), and Landau–Lifshitz–Gilbert magnetization evolution (Dong et al., 2017, Hornung, 2016, Kuehn et al., 2018, Hocquet et al., 2022).

The Freidlin–Wentzell LDP provides asymptotic probabilities for substantial stochastic fluctuations, critical for understanding metastability and stochastic bifurcations. Regularity results ensure spatial-temporal Hölder continuity up to the limits implied by the noise and coefficient smoothness, supporting advanced numerical and analytical approaches.

---

Table 1: Comparison of Regularity Methods in Quasilinear Parabolic SPDEs

Method	Applicability	Regularity Achieved
Schauder/Krylov–Safonov	Smooth initial data, uniform ellipticity	$C^{k+\vartheta}$ in space, $C^{(k+\vartheta)/2}$ in time (Debussche et al., 2014)
Kinetic Formulation	Possibly degenerate diffusion, weak solutions	$L^1$ contraction, existence/uniqueness, measurability (Debussche et al., 2013)
Controlled Rough Paths	Rough noise (fractional Brownian), Banach scales	$C^{\gamma-\varepsilon}$ temporal, $B^{a-\sigma}$ spatial continuity (Hocquet et al., 2022)

These analytic strategies enable treatment of both regular and irregular data, handling additive and multiplicative noise, and accommodating the full range of quasilinear phenomena encountered in stochastic evolution models.