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Well-posedness of semilinear SPDEs with singular drift: a variational approach

Luca Scarpa, University College London

Sala conferenze IMATI-CNR, Pavia - Martedì 7 Novembre 2017 h.15:00

Abstract. Well-posedness is proved for singular semilinear SPDEs of the form
 B(t, Xt) dWt ∈ dXt + AXt dt + β(Xt) dt  in (0, T) × D,
X(0) = X0 in D, 
where D ⊆ Ris a smooth bounded domain, T > 0, A is a linear coercive maximal monotone operator in L2(D) and β is a maximal monotone graph everywhere defined on R, on which no growth nor smoothness conditions are required. Moreover, W is a cylindrical Wiener process on a suitable Hilbert space U, B takes values in the Hilbert-Schmidt operators from U to L2(D) and satisfies classical Lipschitz continuity hypotheses in the second variable. The proof consists in approximating the equation, finding uniform estimates both pathwise and in expectation on the approximated solutions, and then passing to the limit using compactness and lower semicontinuity results. Finally, possible generalizations are discussed.
This study is based on a joint work with Carlo Marinelli (University College London). 

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Dipartimento di Matematica ''F. Casorati''

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