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Well-posedness, Regularity, and Strong Approximations of Superlinear Stochastic Reaction-Diffusion Equation

Zhihui Liu

arXiv:2609.16979v1math.NAcs.NAmath.PR

TL;DR

Whether tamed-FEM stability and strong convergence extend to reaction–diffusion equations with superlinear diffusion remained open, with well-posedness and moment estimates among the main difficulties. The paper extends variational well-posedness and moment estimates to superlinear diffusion, then applies these results and regularity analysis to construct and study tamed-FEMs. The tamed-FEMs are unconditionally stable, and achieve optimal strong convergence rates in one dimension that are uniform in time under certain dissipativity conditions.

  • Problem

    Whether tamed-FEM stability and strong convergence extend to reaction–diffusion equations with superlinear diffusion remained open, with well-posedness and moment estimates among the main difficulties.

    Source: §1.1 · p. 2

  • Method

    The paper extends variational well-posedness and moment estimates to superlinear diffusion, then applies these results and regularity analysis to construct and study tamed-FEMs.

    Source: §1.3 · p. 4, §2.3 · p. 11

  • Results

    The tamed-FEMs are unconditionally stable, and achieve optimal strong convergence rates in one dimension that are uniform in time under certain dissipativity conditions.

    Source: §1.3 · p. 4

  • Takeaways & Limitations

    To the authors’ knowledge, this is the first strong approximation result for SPDEs with superlinear diffusion coefficients.

    Source: §1.3 · p. 4

    Theorem 8’s strong error estimate cannot be established for d = 2, 3, despite the finite-α₁ estimate being available.

    Source: §4.3 · p. 29

Abstract

from arXiv · show

This paper develops a general framework for the well-posedness, regularity, and strong approximation of the stochastic reaction--diffusion equation (SRDE) with superlinear drift and diffusion coefficients. We first extend the well-posedness results in \emph{W. Liu and M. R\"ockner, J. Funct. Anal., 2902--2922, 2010} and \emph{W. Liu, J. Differential Equations, 572--592, 2013} to the case of superlinear diffusion in the Gelfand triple \(V \hookrightarrow H \hookrightarrow V^*\), with \(V\) equipped with the norm \(\|\cdot\|_V\), and derive a moment estimate by establishing a new It\^o formula for \(\|X\|_V^p\) with general \(p \ge 2\). We then apply this abstract result to the SRDE, establish higher spatial regularity \(\dot H^{1+\gamma}\) for any \(\gamma \in [0,1]\) whenever the initial datum lies in the same Sobolev space, and obtain temporal H\"older regularity. Finally, we construct a family of tamed finite element methods (tamed-FEMs) for the SRDE under general assumptions on the tamed functions, derive their long-time unconditional stability, and establish optimal strong convergence rates. To our knowledge, this is the first strong approximation result for SPDEs with superlinear diffusion coefficients.

1. Introduction

The paper studies the stochastic reaction–diffusion equation with polynomial-growth coefficients and addresses open well-posedness and approximation questions for superlinear diffusion. It develops moment and regularity results and uses them to analyze tamed finite element methods.

1.1. Numerical Motivations.

  • The EM scheme can blow up in the p-th moment for superlinear drift with additive white noise, while strong convergence and Lyapunov estimates for tamed-FEMs with superlinear diffusion remained open.Earlier work on tamed-FEMs addressed superlinear drift with Lipschitz diffusion, rather than superlinear diffusion.Source: §1.1 · p. 2

1.2. Mathematical Motivations.

  • The proposed coefficient conditions allow polynomial-growth diffusion without the earlier restriction on q, and the resulting estimates generalize prior results and answer ’s continuity conjecture.Earlier conditions restricted the polynomial order to q < 2 in dimensions 2 and 3, and the variational framework had not studied SPDEs with superlinear diffusion.Source: §1.2 · p. 3, §1.3 · p. 3

1.3. Main Results.

  • For tamed-FEMs, the paper establishes unconditional stability and optimal strong convergence rates in one dimension, with rates uniform in time under certain dissipativity conditions.The analysis uses Sobolev and Hölder regularity of the exact solution and a variational error decomposition; the authors describe this as the first strong approximation result for SPDEs with superlinear diffusion.Source: §1.3 · p. 4

1.4. Notations.

  • The paper sets conditional-expectation notation and mixed temporal, sample-path, and spatial norms for later statements.It defines E_n[·] as conditional expectation given F_tn.Source: §1.4 · p. 4

2. Well-posedness and Moment Estimates of Monotone SDEs with Superlinear Drift and Diffusion Operators

This section establishes well-posedness and V-norm moment estimates for monotone SDEs with superlinear drift and diffusion, using a general-p Itô formula. It also extends these results, including long-time results, under related assumptions.

2.1. Preliminaries.

  • The framework imposes (H0)–(H4)—orthogonality, hemicontinuity, monotonicity, coercivity, and growth—and allows α > 1; coefficients are taken independent of (t, ω) for simplicity.The SRDE corresponds to α = 2.Source: §2.1 · p. 5

2.2. Existence of Solution in V .

  • Under p ≥ 2, X0∈Lp(Ω;V), and (H0)–(H4), Eq. (1.2) has a solution in L∞_{t,ω}V0 that is a Markov process.The existence proof uses uniform a priori estimates for finite-dimensional approximations.Source: §2.2 · p. 8

2.3. Itˆo formula and Moment Estimates.

  • For p ≥ 2, an Itô formula for ||X||_V^p supports moment estimates, and under (H0)–(H4) with X0∈Lp(Ω;V), Eq. (1.2) has a unique solution.The formula is developed for general p and extended to V; Theorem 2 states uniqueness and moment estimates.Source: §2.3 · pp. 8, 11, 12
  • Stability estimate (2.19) gives uniqueness for solutions with the same initial condition, and pathwise uniqueness follows from continuity in H.Source: §2.3 · p. 12
  • For general p ≥ 2(α0 − 1), estimates for the first terms in (2.26) and (2.27) follow; the remaining two terms follow from uniform estimates for corresponding Galerkin approximations and weak lower semicontinuity of norms.Source: §2.3 · p. 13
  • The framework also establishes well-posedness in V for locally monotone, polynomial-growth diffusion and in H under coupled coercivity and polynomial growth.Source: §2.3 · p. 13
  • The moment estimates answer the continuity conjecture for ∥X∥_V^p and generalize [30, Theorem 1.1] and [29, Theorem 1.1] to polynomial-growth diffusion.Source: §2.3 · p. 14
  • In H, the transition semigroup is Feller; compact embedding gives an invariant measure, and c0 < 0 ensures uniqueness, finite second moment, and exponential convergence.Source: §2.3 · p. 14

3. Well-posedness and Regularity of SRDE

For the SRDE, the section establishes well-posedness under polynomial-growth assumptions and develops higher spatial and temporal regularity, with some estimates uniform in time under an additional condition.

3. Well-posedness and Regularity of SRDE

  • The framework also applies to general SPDEs, including stochastic p-Laplace, porous media, and fast diffusion equations.The section focuses primarily on the SRDE.Source: §3 · p. 15

3.1. Well-posedness of SRDE.

  • For X0 ∈ Lp(Ω; ˙H1) with p ≥ 2(q + 1), the SRDE has a unique solution when Assumption 1 and the stated structural conditions hold.The result uses (H0)–(H2), (H3’), and (H4’).Source: §3.1 · p. 16
  • The well-posedness arguments extend to gradient noise of the form G(v)gm = γ0∇v gm + g gm, with g satisfying Assumption 1.Source: §3.1 · p. 18

3.2. Regularity of SRDE.

  • The section establishes higher spatial regularity in ˙H1+γ and temporal Hölder regularity; estimates (3.21) and (3.22) are uniform in time under condition (3.15).The stated spatial regularity result covers the 3-D case with an additional moment-exponent condition.Source: §3.2 · pp. 18–19

4. Strong convergence rates of tamed-FEM

This section develops tamed finite element methods for stochastic reaction–diffusion equations with superlinear coefficients, establishing long-time unconditional stability and analyzing strong errors through auxiliary-process estimates. It derives strong convergence in one dimension, identifies conditions for uniform-in-time estimates, and describes limitations on higher-order spatial convergence.

4.1. Tamed-FEM and Long-time Stability.

  • The tamed-FEM admits a general-p Lyapunov estimate and, under the stated assumptions, has long-time unconditional stability for sufficiently small time steps.The Lyapunov estimate is stronger than the result in [36] for p = 2 with Lipschitz diffusion.Source: §4 · p. 21, §4.1 · pp. 22–23

4.2. Assumptions on Tamed-FEM.

  • The tamed functions constructed from f and g satisfy the tamed coercivity condition and the conditions imposed for strong-error estimates, but coupled monotonicity does not follow in general without further assumptions.The paper also gives a counterexample to deriving coupled monotonicity under the stated hypotheses.Source: §4.1 · p. 23, §4.2 · pp. 25–26

4.3. Auxiliary Process and Moment Estimates.

  • Under (3.15), estimates (4.26) and (4.27) are uniform in time.Source: §4.3 · p. 28
  • The auxiliary process's H^(1+γ) regularity is unavailable in dimensions 2 and 3 under the stated bounds, preventing the strong-error estimate in Theorem 8 for those dimensions.The obstruction is the unavailable L∞ bound for Gτ(X), for which only finite-exponent L^(2α1) boundedness is available.Source: §4.3 · p. 29

4.4. Strong Error Estimate for Auxiliary Process.

  • Under Assumptions 1 and 2, the auxiliary process admits error estimates for any ε∈(0,1); under (3.15), estimates (4.29) and (4.30) are uniform in time.The proof absorbs the |ln τ| factor into a τ^-ε/2 term.Source: §4.4 · pp. 29, 30, 32

4.5. Strong Error Estimate for Tamed-FEM.

  • In one dimension, the tamed drift and diffusion operator differences also satisfy strong error estimates for initial data in H˙^(1+γ), γ ∈ [0, 1).The result assumes p ≥ 2, p0 = max{2p(q + 1), 2pq(q + 1)}, and Assumptions 1 and 2; its estimate is uniform in time under (3.15).Source: §4.5 · p. 32
  • Higher-order spatial convergence for superlinear diffusion requires L∞ bounds on both the exact solution and auxiliary process, which are unavailable for the auxiliary process in dimensions two and three.The paper notes that this blocks establishing its H˙^(1+γ) regularity there, despite proving that regularity for the exact solution.Source: §4.5 · p. 33
  • In one dimension, the tamed-FEM converges strongly under Assumptions 1–3 for initial data in H˙^(1+γ), with γ ∈ [0, 1), and sufficiently small time steps.The theorem requires p* > p ≥ 2 and τ ∈ (0, τmax); under (3.15) and M0 < 2λ1, its estimate is uniform in time.Source: §4.5 · p. 33
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