Abstract
We are interested in (uniformly) parabolic PDEs with a nonlinear dependence of the leading-order coefficients, driven by a rough right hand side. For simplicity, we consider a space-time periodic setting with a single spatial variable:
where P is the projection on mean-zero functions, and f is a distribution which is only controlled in the low regularity norm of \({ C^{\alpha-2}}\) for \({\alpha > \frac{2}{3}}\) on the parabolic Hölder scale. The example we have in mind is a random forcing f and our assumptions allow, for example, for an f which is white in the time variable x2 and only mildly coloured in the space variable x1; any spatial covariance operator \({(1 + |\partial_1|)^{-\lambda_1 }}\) with \({\lambda_1 > \frac13}\) is admissible. On the deterministic side we obtain a \({C^\alpha}\)-estimate for u, assuming that we control products of the form \({v\partial_1^2v}\) and vf with v solving the constant-coefficient equation \({\partial_2 v-a_0\partial_1^2v=f}\). As a consequence, we obtain existence, uniqueness and stability with respect to \({(f, vf, v \partial_1^2v)}\) of small space-time periodic solutions for small data. We then demonstrate how the required products can be bounded in the case of a random forcing f using stochastic arguments. For this we extend the treatment of the singular product \({\sigma(u)f}\) via a space-time version of Gubinelli’s notion of controlled rough paths to the product \({a(u)\partial_1^2u}\), which has the same degree of singularity but is more nonlinear since the solution u appears in both factors. In fact, we develop a theory for the linear equation \({\partial_t u - P(a\partial_1^2 u +\sigma f)=0}\) with rough but given coefficient fields a and \({\sigma}\) and then apply a fixed point argument. The PDE ingredient mimics the (kernel-free) Safonov approach to ordinary Schauder theory.
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Open access funding provided by Max Planck Society. HW is supported by the Royal Society through the University Research Fellowship UF140187.
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Otto, F., Weber, H. Quasilinear SPDEs via Rough Paths. Arch Rational Mech Anal 232, 873–950 (2019). https://doi.org/10.1007/s00205-018-01335-8
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DOI: https://doi.org/10.1007/s00205-018-01335-8