An energy method for rough partial differential equations

Existence, uniqueness and stability of semi-linear rough partial differential equations

“…equipped with its natural Banach structure. (This is in contrast to [9,Thm. 2] where compactness argument lead to some sub-optimal spaces in the continuity statement.…”

“…The following lemma is the main technical tool of the paper. It replaces the tensorization argument in [1], [3] and [9].…”

“…We then again use the product formula, Lemma 6, and a solution to a backward problem defined on W 3,∞ (R d ) to transform the equation for u 2 into an expression without a rough path integral term. This method replaces the so-called "Rough Gronwall" in [3], [9].…”

“…This setting already contains the model problem du t = ∆u t dt + V · ∇u(t)dW t , with u 0 = u 0 (x) ∈ L 2 (R d ) and a vector field V = V (x), studied in [3], subsequently analyzed in detail in [9] with general second (resp. first) linear differential operators.…”

“…L = ∆) and Γ of pure transport type and use an intricate doubling of the variables argument and a rough version of the Gronwall lemma, first introduced in [3], to obtain energy estimates. Using these techniques, the work [9] study the forward equation in (1.3) in divergence form under optimal conditions on the coefficients in the drift term.…”