Comparing the stochastic nonlinear wave and heat equations: a case study

Abstract

We study the two-dimensional stochastic nonlinear wave equation (SNLW) and stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional derivative (of order α > 0) of a space-time white noise. In particular, we show that the well-posedness theory breaks at α = 1 2 for SNLW and at α = 1 for SNLH. This provides a first example showing that SNLW behaves less favorably than SNLH. (i) As for SNLW, Deya (2020) essentially proved its local well-posedness for 0 < α < 1 2 . We first revisit this argument and establish multilinear smoothing of order 1 4 on the second order stochastic term in the spirit of a recent work by Gubinelli, Koch, and Oh (2018). This allows us to simplify the local well-posedness argument for some range of α. On the other hand, when α ≥ 1 2 , we show that SNLW is ill-posed in the sense that the second order stochastic term is not a continuous function of time with values in spatial distributions. This shows that a standard method such as the Da Prato-Debussche trick or its variant, based on a higher order expansion, breaks down for α ≥ 1 2 . (ii) As for SNLH, we establish analogous results with a threshold given byThese examples show that in the case of rough noises, the existing well-posedness theory for singular stochastic PDEs breaks down before reaching the critical values (α = 3 4 in the wave case and α = 2 in the heat case) predicted by the scaling analysis (due to Deng, Nahmod, and Yue (2019) in the wave case and due to Hairer (2014) in the heat case).

On ill-posedness of nonlinear stochastic wave equations driven by rough noise

“…This phenomenon can be compared with the situation of a general SDE driven by a two-dimensional fractional noise of index H ď 1 4 . Our results clarify and extend previous similar properties exhibited in [3] or in [14].…”

“…Indeed, while 9 B can only be regarded as a general space-time distribution, it is possible to construct as a function of time (we have seen it in Proposition 1.2). Many similar behaviours have been exhibited for the nonlinear heat equation on the torus: see for instance the contrast between Lemma 16 and Proposition 24 in [4], or between items (i) and (ii) in [14,Proposition 1.9].…”

“…The transition from these uniform bounds to the assertions piq-paq and piq-pbq of Theorem 2.4 is then a matter of standard arguments, that have been detailled many times in the literature. Such a procedure can be found for instance in [2, Section 2.1] or [3, Section 3] (see also [13,Proposition 3.6] or [14,Lemma 2.6] for results in the same spirit). We leave it to the reader to check that our setting in this regard is not different from the one in the latter references.…”

On the Parabolic and Hyperbolic Liouville Equations

“…Furthermore, in this paper, we study both SNLH (1.1) and SdNLW (1.2), which allows us to point out similarity and difference between the analysis of the stochastic heat and wave equations. See also [57] for a comparison of the stochastic heat and wave equations on T 2 with a quadratic nonlinearity driven by fractional derivatives of a space-time white noise.…”

Invariant Gibbs dynamics for the dynamical sine-Gordon model

“…is now well understood [15,17,29,30]. See also [33,39] for related results on twodimensional compact Riemannian manifolds [33] and on R 2 [39].…”