Existence and uniqueness results of a stochastic nonlinear heat equation with a constraint of codimension one

Abstract: In this work, we investigate the well-posedness of a stochastic heat equation with an arbitrary (but polynomial) nonlinearity in any dimension $d\geq 1$ perturbed by a multiplicative white noise in the Stratonovich form, subject to an $L^2-$norm constraint on the solution. In bounded smooth domains, we establish the existence of a martingale solution taking values in $H_0^1 \cap L^p$ for arbitrary $2 \le p < \infty$, using a modified Faedo-Galerkin scheme. By utilizing a sequence of self-adjoint operators which are bounded in $L^p$ for any $2 \le p < \infty$, we provide a novel proof of an Itô formula for the $L^p-$norm of the solution. Together with pathwise uniqueness of the martingale solution, the Yamada-Watanabe result then yields the existence of a strong solution and uniqueness in law.

• The paper establishes existence and uniqueness of both martingale and strong solutions for a stochastic heat equation under an L2-norm constraint.
• It introduces a novel Lp-Itô formula and utilizes Faedo-Galerkin approximations with energy estimates to address supercritical nonlinearities.
• The framework ensures solutions remain on the unit sphere, paving the way for applications in stochastic flows on manifolds and high-dimensional dynamics.

Existence and Uniqueness for a Stochastic Nonlinear Heat Equation with Codimension-One Constraint

Introduction and Problem Formulation

This work addresses the rigorous analysis of a class of stochastic nonlinear heat equations in bounded smooth domains, subject to a codimension-one constraint, specifically the $L^2$-norm sphere constraint. The SPDE considered is

$$du(t) = \left[ \Delta u(t) - |u(t)|^{p-2}u(t) + (\|\nabla u(t)\|_{L^2(O)}^2 + \|u(t)\|_{L^p(O)}^p)u(t) \right] dt + \sum_{i=1}^M N_i(u(t)) \circ dW_i(t)$$

on a domain $O \subset \mathbb{R}^d$, $d\geq 1$, with Dirichlet boundary conditions and initial data $u_0\in H_0^1 \cap L^p \cap M$, the $L^2$-unit sphere. The driving noise is a (potentially infinite-dimensional) multiplicative Gaussian process interpreted in the Stratonovich sense, with state-dependent coefficients $N_i$. The nonlinearity exponent $p\in [2,\infty)$ is arbitrary, allowing for high degrees of nonlinearity unconstrained by Sobolev-space limitations.

A critical structural property is that the drift and the noise act on the tangent space to the constraint manifold, ensuring preservation of the $L^2$-norm through time. The problem thus represents an SPDE on a manifold, formulated in both Stratonovich and Itô forms, the latter via explicit calculation of the Itô correction.

Analytical Framework and Main Contributions

The authors develop a systematic approach to prove well-posedness of the above SPDE under minimal restrictions:

1. Arbitrary Spatial Dimension and Nonlinearity: Existence and uniqueness are established for any $d\geq 1$ and $$du(t) = \left[ \Delta u(t) - |u(t)|^{p-2}u(t) + (\|\nabla u(t)\|_{L^2(O)}^2 + \|u(t)\|_{L^p(O)}^p)u(t) \right] dt + \sum_{i=1}^M N_i(u(t)) \circ dW_i(t)$$0, including supercritical regimes previously out of reach.
2. Martingale and Strong Solutions: Existence of martingale solutions in $$du(t) = \left[ \Delta u(t) - |u(t)|^{p-2}u(t) + (\|\nabla u(t)\|_{L^2(O)}^2 + \|u(t)\|_{L^p(O)}^p)u(t) \right] dt + \sum_{i=1}^M N_i(u(t)) \circ dW_i(t)$$1 is first proved via a tailored Faedo-Galerkin approximation coupled with delicate compactness arguments. Pathwise uniqueness is then established, yielding, by the generalized Yamada–Watanabe theorem, existence and uniqueness in law of strong solutions.
3. Invariant Manifold: The solutions remain on the $$du(t) = \left[ \Delta u(t) - |u(t)|^{p-2}u(t) + (\|\nabla u(t)\|_{L^2(O)}^2 + \|u(t)\|_{L^p(O)}^p)u(t) \right] dt + \sum_{i=1}^M N_i(u(t)) \circ dW_i(t)$$2-sphere almost surely for all time, for arbitrary initial data in $$du(t) = \left[ \Delta u(t) - |u(t)|^{p-2}u(t) + (\|\nabla u(t)\|_{L^2(O)}^2 + \|u(t)\|_{L^p(O)}^p)u(t) \right] dt + \sum_{i=1}^M N_i(u(t)) \circ dW_i(t)$$3.
4. Novel $$du(t) = \left[ \Delta u(t) - |u(t)|^{p-2}u(t) + (\|\nabla u(t)\|_{L^2(O)}^2 + \|u(t)\|_{L^p(O)}^p)u(t) \right] dt + \sum_{i=1}^M N_i(u(t)) \circ dW_i(t)$$4-Itô Formula: The authors devise a new proof of an $$du(t) = \left[ \Delta u(t) - |u(t)|^{p-2}u(t) + (\|\nabla u(t)\|_{L^2(O)}^2 + \|u(t)\|_{L^p(O)}^p)u(t) \right] dt + \sum_{i=1}^M N_i(u(t)) \circ dW_i(t)$$5-Itô formula for the solution, valid for all $$du(t) = \left[ \Delta u(t) - |u(t)|^{p-2}u(t) + (\|\nabla u(t)\|_{L^2(O)}^2 + \|u(t)\|_{L^p(O)}^p)u(t) \right] dt + \sum_{i=1}^M N_i(u(t)) \circ dW_i(t)$$6, bypassing the limitations of previous approaches (e.g., Krylov's lemma), which are restrictive in the context of high $$du(t) = \left[ \Delta u(t) - |u(t)|^{p-2}u(t) + (\|\nabla u(t)\|_{L^2(O)}^2 + \|u(t)\|_{L^p(O)}^p)u(t) \right] dt + \sum_{i=1}^M N_i(u(t)) \circ dW_i(t)$$7.
5. General Stochastic Forcing: The framework admits countably infinite driving Brownian motions, addressing both finite- and infinite-noise-dimensional cases.
6. Technical Innovations: Introduction of a specific family of self-adjoint operators ($$du(t) = \left[ \Delta u(t) - |u(t)|^{p-2}u(t) + (\|\nabla u(t)\|_{L^2(O)}^2 + \|u(t)\|_{L^p(O)}^p)u(t) \right] dt + \sum_{i=1}^M N_i(u(t)) \circ dW_i(t)$$8) ensures uniform boundedness in all relevant $$du(t) = \left[ \Delta u(t) - |u(t)|^{p-2}u(t) + (\|\nabla u(t)\|_{L^2(O)}^2 + \|u(t)\|_{L^p(O)}^p)u(t) \right] dt + \sum_{i=1}^M N_i(u(t)) \circ dW_i(t)$$9-spaces, facilitating convergence and energy estimates. Key Lipschitz estimates and monotonicity properties for noise coefficients and nonlinear drift are rigorously derived.

Technical Approach

• Approximation and Tightness: The solution is constructed as the limit of Galerkin approximations with careful projection onto finite-dimensional subspaces, enforcing the codimension-one constraint at every stage. High-order uniform moment bounds, Aldous tightness conditions, and Skorokhod representation theorems are leveraged.
• Itô Calculus in Manifold-Valued Setting: Precise calculation of the Itô-Stratonovich correction for the noise guarantees that the law of solutions is invariant on the unit sphere. The $O \subset \mathbb{R}^d$0-Itô formula is proved using spectral multiplier techniques, essential as classical approaches (e.g., those relying on regularization) do not extend beyond the Hilbertian ($O \subset \mathbb{R}^d$1) case when strong nonlinearities are present.
• Pathwise Uniqueness via Energy Inequalities: The Schmalfuss argument, employing suitably chosen Lyapunov functionals and Gronwall’s lemma, yields pathwise uniqueness through control of deviation between two solutions driven by the same noise realization.

Results

The principal results are as follows:

• Existence of Probabilistically Weak and Martingale Solutions: For any $O \subset \mathbb{R}^d$2, there exists a weak solution whose paths belong to $O \subset \mathbb{R}^d$3, satisfying the constraint at all times, and which solves the SPDE in the sense of Definition 1.2.
• $O \subset \mathbb{R}^d$4-Itô Formula: For any $O \subset \mathbb{R}^d$5, the solution satisfies an $O \subset \mathbb{R}^d$6-norm formula with drift, correction, and stochastic terms, valid for any $O \subset \mathbb{R}^d$7.
• Pathwise Uniqueness and Strong Solutions: Pathwise uniqueness of martingale solutions holds; thus, by the Yamada–Watanabe theory (Ondřejat's theorem), there exists a strong solution that is unique in law, adapted to the natural filtration of the noise.
• Energy Bounds: Uniform a-priori bounds on moments of $O \subset \mathbb{R}^d$8 and $O \subset \mathbb{R}^d$9 norms, higher (even) moment bounds, and bounds on higher-order Sobolev norms and nonlinearities are established. In particular,

$d\geq 1$0

uniformly in the approximation parameter.

Implications and Further Directions

The paper advances the theory of SPDEs on manifolds subject to infinite-dimensional noise and strong polynomial nonlinearities. The absence of constraints on $d\geq 1$1 or $d\geq 1$2 in existence and uniqueness is significant in the study of physically or geometrically motivated flows (e.g., norm-preserving heat flows, constrained Schrödinger or Navier-Stokes dynamics).

Practically, the methods developed provide tools for the construction of stochastic flows and invariant measures on manifolds, with potential applications in high-dimensional stochastic control, geometric statistics, and ergodic theory. The analytical framework is sufficiently flexible to be adapted to other classes of SPDEs with similar nonlinearities and constraints, including nonlocal nonlinearities, or higher-codimension constraints.

Future developments may include:

• Extension to non-polynomial or critical nonlinearities (e.g., for equations near the deterministic blow-up threshold).
• Numerical approximation schemes leveraging the invariant manifold property.
• Large deviation principles and statistical properties derived from ergodic solutions on manifolds.
• Analysis under weaker regularity assumptions on the domain or noise structure.

Conclusion

This work fully characterizes the existence, uniqueness, and regularity of solutions to a broad class of stochastic nonlinear heat equations under a codimension-one constraint. By overcoming critical technical barriers associated with highly nonlinear drift and general noise in infinite dimensions, and by introducing new techniques for the derivation of $d\geq 1$3-Itô formulas, the work establishes a robust foundation for future studies of stochastic evolution equations on manifolds.

Reference: "Existence and uniqueness results of a stochastic nonlinear heat equation with a constraint of codimension one" (2604.26549)