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We address the local well-posedness for the stochastic Navier–Stokes system with multiplicative cylindrical noise in the whole space. More specifically, we prove that there exists a unique local strong solution to the system in$$L^p(\mathbb {R}^3)$$$L^p\left(R^3\right)$for $$p>3$$$p>3$.

We address a system of equations modeling an incompressible fluid interacting with an elastic body. We prove the local existence when the initial velocity belongs to the space$$H^{1.5+\epsilon }$$$H^{1.5+ϵ}$and the initial structure velocity is in$$H^{1+\epsilon }$$$H^{1+ϵ}$, where$$\epsilon \in (0, 1/20)$$$ϵ\in (0,1/20)$.

We examine a micro-scale model of superfluidity derived by Pitaevskii (Sov. Phys. JETP 8:282-287, 1959) which describes the interacting dynamics between superfluid He-4 and its normal fluid phase. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of global weak solutions in$${\mathbb {T}}^3$$$T^3$for a power-type nonlinearity, beginning from small initial data. The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining time-independent a priori estimates.

We investigate a micro-scale model of superfluidity derived by Pitaevskii (1959Sov. Phys. JETP8282–7) to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model involves the nonlinear Schrödinger equation (NLS) and the Navier–Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. Depending on the nature of the nonlinearity in the NLS, we prove global/almost global existence of solutions to this system in${\mathbb{T}}^2$—strong in wavefunction and velocity, and weak in density.

We address a system of equations modeling a compressible fluid interacting with an elastic body in dimension three. We prove the local existence and uniqueness of a strong solution when the initial velocity belongs to the space$$H^{2+\epsilon }$$$H^{2+ϵ}$and the initial structure velocity is in$$H^{1.5+\epsilon }$$$H^{1.5+ϵ}$, where$$\epsilon \in (0,1/2)$$$ϵ\in (0,1/2)$.

We consider the local existence and uniqueness of solutions for a system consisting of an inviscid fluid with a free boundary, modeled by the Euler equations, in a domain enclosed by an elastic boundary, which evolves according to the wave equation. We derive a priori estimates for the local existence of solutions and also conclude the uniqueness. Both, existence and uniqueness are obtained under the assumption that the Euler data belongs to$$H^{r}$$$H^r$, where$$r>2.5$$$r>2.5$, which is known to be the borderline exponent for the Euler equations. Unlike the setting of the Euler equations with vacuum, the membrane is shown to stabilize the system in the sense that the Rayleigh–Taylor condition does not need to be assumed.