Abstract
We are interested in (uniformly) parabolic PDEs with a nonlinear dependence of the leadingorder coefficients, driven by a rough right hand side. For simplicity, we consider a spacetime periodic setting with a single spatial variable:
∂2u−P(a(u)∂21u+σ(u)f)=0,
where P is the projection on meanzero functions, and f is a distribution which is only controlled in the low regularity norm of Cα−2 for α>23 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+∂1)−λ1 with λ1>13 is admissible. On the deterministic side we obtain a Cαestimate for u, assuming that we control products of the form v∂21v and vf with v solving the constantcoefficient equation ∂2v−a0∂21v=f. As a consequence, we obtain existence, uniqueness and stability with respect to (f,vf,v∂21v) of small spacetime 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 σ(u)f via a spacetime version of Gubinelli’s notion of controlled rough paths to the product a(u)∂21u, 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 ∂tu−P(a∂21u+σf)=0 with rough but given coefficient fields a and σ and then apply a fixed point argument. The PDE ingredient mimics the (kernelfree) Safonov approach to ordinary Schauder theory.
Original language  English 

Pages (fromto)  873950 
Number of pages  78 
Journal  Archive for Rational Mechanics and Analysis 
Volume  232 
Issue number  2 
Early online date  30 Nov 2018 
DOIs  
Publication status  Published  2 May 2019 
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Hendrik Weber
 Department of Mathematical Sciences  Professor of Probability
 Probability Laboratory at Bath
Person: Research & Teaching