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A solution theory for quasilinear singular SPDEs. (English) Zbl 1458.60077

The authors deal with quasilinear singular stochastic partial differential equations (SPDEs) of the form \[ (\partial_t -a(u) \partial_x^2 )u = F(u,\xi), \] where \(a\) is a smooth function, \(F\) is a non-linearity that is sub-critical (in the sense of the theory of regularity structures) and \(\xi\) is a noise term. For instance, the considered class of equations includes systems with KPZ-type non-linearities driven by space-time white noise. Within the theory of regularity structures the authors provide an approach allowing to build local renormalized solutions to the above class of quasilinear SPDEs. The key observation is that it is possible to rewrite the quasilinear SPDEs in terms of a family of Green’s functions. As an application of the generally developed approach, the authors apply their construction to a quasilinear version of the KPZ equation and show, in this case, that the counter terms induced by the renormalisation procedure are local functions of the solution, which is a priori not clear in general by the authors’ construction.

MSC:

60H17 Singular stochastic partial differential equations
35K59 Quasilinear parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
60L30 Regularity structures
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