### Abstract

Original language | English |
---|---|

Pages (from-to) | 873-950 |

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 - 1 May 2019 |

### Cite this

*Archive for Rational Mechanics and Analysis*,

*232*(2), 873-950. https://doi.org/10.1007/s00205-018-01335-8

**Quasilinear SPDEs via Rough Paths.** / Otto, Felix; Weber, Hendrik.

Research output: Contribution to journal › Article

*Archive for Rational Mechanics and Analysis*, vol. 232, no. 2, pp. 873-950. https://doi.org/10.1007/s00205-018-01335-8

}

TY - JOUR

T1 - Quasilinear SPDEs via Rough Paths

AU - Otto, Felix

AU - Weber, Hendrik

PY - 2019/5/1

Y1 - 2019/5/1

N2 - 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: ∂2u−P(a(u)∂21u+σ(u)f)=0, where P is the projection on mean-zero 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 Cα-estimate for u, assuming that we control products of the form v∂21v and vf with v solving the constant-coefficient equation ∂2v−a0∂21v=f. As a consequence, we obtain existence, uniqueness and stability with respect to (f,vf,v∂21v) 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 σ(u)f via a space-time 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 (kernel-free) Safonov approach to ordinary Schauder theory.

AB - 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: ∂2u−P(a(u)∂21u+σ(u)f)=0, where P is the projection on mean-zero 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 Cα-estimate for u, assuming that we control products of the form v∂21v and vf with v solving the constant-coefficient equation ∂2v−a0∂21v=f. As a consequence, we obtain existence, uniqueness and stability with respect to (f,vf,v∂21v) 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 σ(u)f via a space-time 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 (kernel-free) Safonov approach to ordinary Schauder theory.

U2 - 10.1007/s00205-018-01335-8

DO - 10.1007/s00205-018-01335-8

M3 - Article

VL - 232

SP - 873

EP - 950

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 2

ER -