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In this paper, we consider a Navier-Stokes-Lame system modeling a fluid-structure interaction. For a general domain, we establish local well-posedness for strong solutions in which initial velocity u0 belongs to H1 while the initial data (w0, w1) for the elasticity equation belongs to (H3/2+k, H1/2+k) for any k [?] (0, k0) where k0 is an explicit positive constant. [ABSTRACT FROM AUTHOR]

Abstract  We address the problem of local uniqueness of weak solutions to the Navier–Stokes system, with the initial datum in a subspace of $${BMO^{-1}(\mathbb{R}^{n})}$$. The existence and uniqueness of local mild solutions has been proven by Koch and Tataru (Adv Math 157:22–35, 2001). We present a necessary and sufficient condition for two weak solutions to evolve from the same initial datum, and for weak solutions to be mild. [ABSTRACT FROM AUTHOR]

We prove the existence of global strong solutions of the primitive equations of the ocean in the case of the Dirichlet boundary conditions on the side and the bottom boundaries including the varying bottom topography. Previously, the existence of global strong solutions was known in the case of the Neumann boundary conditions in a cylindrical domain (Cao and Titi 2007 Ann. Math. 166 245-67). [ABSTRACT FROM AUTHOR]

We prove that a weak solution of the Navier–Stokes system satisfies the energy equality if the associated pressure is locally square integrable. A similar statement is shown to hold for the Euler system. [ABSTRACT FROM AUTHOR]

We address the backward uniqueness property for the equation $u_t-\Delta u = w_j\partial_{j}u+v u$ in ${\mathbb R}^n\times(T_0,0]$. We show that under rather general conditions on $v$ and $w$, $u|_{t=0}=0$ implies that $u$ vanishes to infinite order for all points $(x,0)$. It follows that the backward uniqueness holds if $w=0$ and $v\in L^{\infty}([0,T_0],L^p({\mathbb R}^n))$ when $p>n/2$. The borderline case $p=n/2$ is also covered with an additional continuity and smallness assumption. [ABSTRACT FROM AUTHOR]

For dissipative equations of the form [ABSTRACT FROM AUTHOR]

We show that the normal derivative of a harmonic function which vanishes on an open subset of the boundary of a Dini domain cannot vanish on a subset of positive surface measure. [ABSTRACT FROM AUTHOR]

We derive a priori estimates for the incompressible free-boundary Euler equations with surface tension in three spatial dimensions. Working in Lagrangian coordinates, we provide a priori estimates for the local existence when the initial velocity, which is rotational, belongs to H3, with some additional regularity on the normal component of the initial velocity. This lowers the requirement on the regularity of initial data in the Lagrangian setting. Our methods are direct and involve three key elements: estimates for the pressure, the boundary regularity provided by the mean curvature, and the Cauchy invariance. [ABSTRACT FROM AUTHOR]

We address the interior regularity of suitable weak solutions of the 3D incompressible Navier–Stokes equations. We prove a partial regularity criterion based on only one component of the velocity. Furthermore, the uniform estimate obtained in this criterion allows us to prove a generalization of the Leray regularity condition. Finally, we extend the regularity due to Escauriaza–Serëgin–Šverák. Namely, we prove that if is a suitable weak solution and one of the velocity components belongs to , while the remaining two are in the larger Morrey space , and if the pressure satisfies , then the solution is regular. [ABSTRACT FROM AUTHOR]

We address the system of partial differential equations modeling motion of an elastic body inside an incompressible fluid. The fluid is modeled by the incompressible Navier–Stokes equations while the structure is represented by the damped wave equation with interior damping. The additional boundary stabilization γ, considered in our previous paper, is no longer necessary. We prove the global existence and exponential decay of solutions for small initial data in a suitable Sobolev space. [ABSTRACT FROM AUTHOR]

A modification of the classical primitive equations of the atmosphere is considered in order to take into account important phase transition phenomena due to air saturation and condensation. We provide a mathematical formulation of the problem that appears to be new in this setting, by making use of differential inclusions and variational inequalities, and which allows to develop a rather complete theory for the solutions to what turns out to be a nonlinearly coupled system of non-smooth partial differential equations. Specifically we prove the global existence of quasi-strong and strong solutions, along with uniqueness results and maximum principles of physical interest. [ABSTRACT FROM AUTHOR]