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1. Local-in-time existence of a free-surface 3D Euler flow with H 2+δ initial vorticity in a neighborhood of the free boundary

   https://doi.org/10.1088/1361-6544/aca5e3  

   Kukavica, I ; Ożański, W S ( December 2022 , Nonlinearity)

   We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. We assume that$u_0\in H^{2.5+\delta }$is such that$\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{u}_0\in H^{2+\delta }$in an arbitrarily small neighborhood of the free boundary, and we use the Lagrangian approach to derive an a priori estimate that can be used to prove local-in-time existence and uniqueness of solutions under the Rayleigh–Taylor stability condition.

    

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2. The relativistic Euler equations with a physical vacuum boundary: Hadamard local well-posedness, rough solutions, and continuation criterion

   https://doi.org/10.1007/s00205-022-01783-3  

   Disconzi, Marcelo M. ; Ifrim, Mihaela ; Tataru, Daniel ( May 2022 , Archive for Rational Mechanics and Analysis)

   In this paper we provide a complete local well-posedness theory for the free boundary relativistic Euler equations with a physical vacuum boundary on a Minkowski background. Specifically, we establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and continuous dependence on the data; (ii) low regularity solutions: our uniqueness result holds at the level of Lipschitz velocity and density, while our rough solutions, obtained as unique limits of smooth solutions, have regularity only a half derivative above scaling; (iii) stability: our uniqueness in fact follows from a more general result, namely, we show that a certain nonlinear functional that tracks the distance between two solutions (in part by measuring the distance between their respective boundaries) is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions; (v) a sharp continuation criterion, at the level of scaling, showing that solutions can be continued as long as the velocity is in$$L^1_t Lip$$$L_t^{1}Lip$and a suitable weighted version of the density is at the same regularity level. Our entire approach is in Eulerian coordinates and relies on the functional framework developed in the companion work of the second and third authors on corresponding non relativistic problem. All our results are valid for a general equation of state$$p(\varrho )= \varrho ^\gamma $$$p\left(\varrho \right)={\varrho }^{\gamma }$,$$\gamma > 1$$$\gamma >1$.

    

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3. On the Local Existence of Solutions to the compressible Navier–Stokes-Wave System with a Free Interface

   https://doi.org/10.1007/s00021-024-00861-8  

   Kukavica, Igor ; Li, Linfeng ; Tuffaha, Amjad ( March 2024 , Journal of Mathematical Fluid Mechanics)

   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)$.

    

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4. Local martingale solutions and pathwise uniqueness for the three-dimensional stochastic inviscid primitive equations

   https://doi.org/10.1007/s40072-022-00266-6  

   Hu, Ruimeng ; Lin, Quyuan ( July 2022 , Stochastics and Partial Differential Equations: Analysis and Computations)

   We study the stochastic effect on the three-dimensional inviscid primitive equations (PEs, also called the hydrostatic Euler equations). Specifically, we consider a larger class of noises than multiplicative noises, and work in the analytic function space due to the ill-posedness in Sobolev spaces of PEs without horizontal viscosity. Under proper conditions, we prove the local existence of martingale solutions and pathwise uniqueness. By adding vertical viscosity,i.e., considering the hydrostatic Navier-Stokes equations, we can relax the restriction on initial conditions to be only analytic in the horizontal variables with Sobolev regularity in the vertical variable, and allow the transport noise in the vertical direction. We establish the local existence of martingale solutions and pathwise uniqueness, and show that the solutions become analytic in the vertical variable instantaneously as$$t>0$$$t>0$and the vertical analytic radius increases as long as the solutions exist.

    

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5. Well-Posedness for Stochastic Fractional Navier–Stokes Equation in the Critical Fourier–Besov Space

   https://doi.org/10.1007/s10959-021-01152-y  

   Yin, Xiuwei ; Wu, Jiang-Lun ; Shen, Guangjun ( January 2022 , Journal of Theoretical Probability)

   The well-posedness of stochastic Navier–Stokes equations with various noises is a hot topic in the area of stochastic partial differential equations. Recently, the consideration of stochastic Navier–Stokes equations involving fractional Laplacian has received more and more attention. Due to the scaling-invariant property of the fractional stochastic equations concerned, it is natural and also very important to study the well-posedness of stochastic fractional Navier–Stokes equations in the associated critical Fourier–Besov spaces. In this paper, we are concerned with the three-dimensional stochastic fractional Navier–Stokes equation driven by multiplicative noise. We aim to establish the well-posedness of solutions of the concerned equation. To this end, by utilising the Fourier localisation technique, we first establish the local existence and uniqueness of the solutions in the critical Fourier–Besov space$$\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}$$${\dot{B}}_{p,r}^{4-2\alpha -\frac{3}{p}}$. Then, under the condition that the initial date is sufficiently small, we show the global existence of the solutions in the probabilistic sense.

    

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