L-Kuramoto–Sivashinsky SPDEs in one-to-three dimensions: L-KS kernel, sharp Hölder regularity, and Swift–Hohenberg law equivalence

Abstract

Abstract. Generalizing the L-Kuramoto-Sivashinsky (L-KS) kernel from our earlier work, we give a novel explicit-kernel formulation useful for a large class of fourth order deterministic, stochastic, linear, and nonlinear PDEs in multispatial dimensions. These include pattern formation equations like the SwiftHohenberg and many other prominent and new PDEs. We first establish existence, uniqueness, and sharp dimension-dependent spatio-temporal Hölder regularity for the canonical (zero drift) L-KS SPDE, driven by white noise on. The spatio-temporal Hölder exponents are exactly the same as the striking ones we proved for our recently introduced Browniantime Brownian motion (BTBM) stochastic integral equation, associated with time-fractional PDEs. The challenge here is that, unlike the positive BTBM density, the L-KS kernel is the Gaussian average of a modified, highly oscillatory, and complex Schrödinger propagator. We use a combination of harmonic and delicate analysis to get the necessary estimates. Second, attaching order parameters ε 1 to the L-KS spatial operator and ε 2 to the noise term, we show that the dimension-dependent critical ratio ε 2 /ε d/8 1 controls the limiting behavior of the L-KS SPDE, as ε 1 , ε 2 ց 0; and we compare this behavior to that of the less regular second order heat SPDEs. Finally, we give a change-ofmeasure equivalence between the canonical L-KS SPDE and nonlinear L-KS SPDEs. In particular, we prove uniqueness in law for the Swift-Hohenberg and the law equivalence-and hence the same Hölder regularity-of the SwiftHohenberg SPDE and the canonical L-KS SPDE on compacts in one-to-three dimensions.2010 Mathematics Subject Classification. 35R60, 60H15, 35R11, 35G99, 60H20, 60H30, 42A38, 45H05, 45R05, 60J45, 60J35, 60J60, 60J65.Key words and phrases. Fourth order (S)PDEs, L-Kuramoto-Sivashinsky (S)PDEs, SwiftHohenberg (S)PDE, linear and nonlinear fourth order (S)PDEs, L-KS kernel, imaginary-Browniantime-Brownian-angle process, Gaussian average of angle modified Shrödinger propagator, kernel stochastic integral equations.

L-Kuramoto–Sivashinsky SPDEs vs. time-fractional SPIDEs: Exact continuity and gradient moduli, 1/2-derivative criticality, and laws

“…As was done in [1] for L-KS SPDEs, we can extend the results in [9,8] to our β-time-fractional SPIDEs (1.2). The almost sure L 2 condition in [9,8,1], which is much weaker than the usual Novikov condition typically found in change-of-measure results, allows us to state an equivalence in law-and thus in all almost sure regularity results-between both (1.1) and (1.2) and their nonlinear versions the Swift-Hohenberg SPDEs in the mild kernel formulations (1.47) and (1.55) (with a ≡ 1) to account for the boundary conditions 23 , and we replace R d with S. The linear-nonlinear equivalence result is now stated. For completeness, we restate the Swift-Hohenberg conclusions from [1].…”

“…In [5,1], motivated by [6], he introduced and gave the explicit kernel stochastic integral equation formulation for a large class of stochastic equations he called L-KS SPDEs. This class includes stochasric versions of prominent nonlinear equations like the Swift-Hohenberg PDE, variants of the Kuramoto-Sivashinsky PDE, as well as many new ones (see [1]). He established in [1], among other things, the existence of a pathwise unique solution U to the nonlinear L-KS SPDE (1.1) with Lipschitz diffusion coefficient a:…”

“…Is the spatial modulus of continuity a more discriminating measure of regularity that depends on β ∈ {1/2 k ; k ∈ N}, and is β = 1/2 critical on (0, 1/2]? (Q3) It was established in [3,2,1] that the L-KS SPDE (1.9) and the β = 1/2 time-fractional SIEs (time fractional SPIDEs in (1.6)) have identical spatio-temporal Hölder regularity. Does the continuity modulus capture the rougher regularity for the case β = 1/2 time-fractional SPIDEs (1.6) (since, by footnote 6, (1.6) is also associated with the rougher positive bi-Laplacian PDE (1.10))?…”

“…The spatial gradient spatio-temporal Hölder regularity in Theorems 1.2 and 1.3 tell us that the gradient of L-KS SPDEs, ∂ x U , is rougher (γ-Hölder with γ ∈ (0, 1/2)) in space than the continuously differentiable (and hence Lipschitz) solution U . More surprisingly, ∂ x U is also rougher in time than U (γ-Hölder with γ ∈ (0, 1/8) vs. γ ∈ (0, 3/8) as in Allouba [1]). Compared to the second order heat SPDE, the L-KS gradient ∂ x U has the same spatial Hölder regularity as that of the solution to the heat SPDE;…”

“…Rigorous kernel stochastic integral equations formulations. For the L-KS SPDE (1.9), as done in [1,5], we use the linearized Kuramoto-Sivashinsky kernel introduced in [6, 5, 1] to define their rigorous mild SIE formulation. This L-KS kernel is the fundamental solution to the deterministic version of (1.9) (a ≡ 0), as shown in [6,5,1], and is given by:…”