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We derive a priori estimates for the compressible free-boundary Euler equations with surface tension in three spatial dimensions in the case of a liquid. These are estimates for local existence in Lagrangian coordinates when the initial velocity and initial density belong to $H^3$, with an extra regularity condition on the moving boundary, thus lowering the regularity of the initial data. Our methods are direct and involve two key elements: the boundary regularity provided by the mean curvature, and a new compressible Cauchy invariance., Comment: arXiv admin note: text overlap with arXiv:1708.00086

Our main result is the existence of solutions to the free boundary fluid-structure interaction system. The system consists of a Navier-Stokes equation and a wave equation defined in two different but adjacent domains. The interaction is captured by stress and velocity matching conditions on the free moving boundary lying in between the two domains. We prove the local existence of a solution when the initial velocity of the fluid belongs to $H^{3}$ while the velocity of the elastic body is in $H^{2}$.

We provide a new simple proof of local-in-time existence of regular solutions to the Euler equation on a domain with a free moving boundary and without surface tension in 2 space dimensions. We prove the existence under the condition that the initial velocity belongs to the Sobolev space $H^{2.5+δ}$ where $\delta>0$ is arbitrary.

Let $u$ be a local weak solution of the Navier-Stokes system in a space-time domain $D\subseteq\mathbb R^{n}\times\mathbb R$. We prove that for every $q>3$ there exists $\epsilon>0$ with the following property: If $(x_0,t_0)\in D$ and if there exists $r_0>0$ such that sup $|x-x_0|+\sqrt{t-t_0} < r_0$ sup $r\in(0,r_0)$ $ \frac{1}{r^{n+2-q}} \int_{t-r^2}^{t+r^2}\ \ \ \int_{|y-x|\le r} |u(y,s)|^{q}\,dy\,ds \le \epsilon $ then the solution $u$ is regular in a neighborhood of $(x_0,t_0)$. There is no assumption on the integrability of the pressure or the vorticity.

We consider the partial regularity of suitable weak solutions of the Navier-Stokes equations in a domain $D$. We prove that the parabolic Hausdorff dimension of space-time singularities in $D$ is less than or equal to 1 provided the force $f$ satisfies $f\in L^{2}(D)$. Our argument simplifies the proof of a classical result of Caffarelli, Kohn, and Nirenberg, who proved the partial regularity under the assumption $f\in L^{5/2+\delta}$ where $\delta>0$.

Let $\Omega=[0,L_1]\times[0,L_2]\times[0,\epsilon]$ where $L_1,L_2>0$ and $\epsilon\in(0,1)$. We consider the Navier-Stokes equations with periodic boundary conditions and prove that if $ \|\| \nabla u_0\|\|_{L^2(\Omega)} \le \frac{1}{C(L_1,L_2)\epsilon^{1/6}} $ then there exists a unique global smooth solution with the initial datum $u_0$.

For the dissipative equations of the form $ u_{t}-u_{x x}+f(x,u,u_x)=0$ we prove that the global attractor can be parametrized by a finite number of Fourier modes and that the number of modes is algebraic in parameters. This improves our earlier result [15], where the number of required modes is exponential. The method extends to equations of order higher than two.

Friz and Robinson showed that analytic global attractors consisting of periodic functions can be parametrised using the values of the solution at a finite number of points throughout the domain, a result applicable to the $2$d Navier-Stokes equations with periodic boundary conditions. In this paper we extend the argument to cover any attractor consisting of analytic functions; in particular we are now able to treat the $2$d Navier-Stokes equations with Dirichlet boundary conditions.

We consider $L^2$ bounds on the gradient of solutions of the Navier-Stokes equations on a general bounded 2D domain with Dirichlet boundary conditions. We obtain an upper bound for this norm on any compact subset of a given domain. We show that the bound is uniform on the global attractor and depends polynomially on the Grashof number.

Three approaches for the rigorous study of the 2D Navier-Stokes equations (NSE) are applied to the Lorenz system. Analysis of time averaged solutions leads to a description of invariant probability measures on the Lorenz attractor which is much more complete than what is known for the NSE. As is the case for the NSE, solutions on the Lorenz attractor are analytic in a strip about the real time axis. Rigorous estimates are combined with numerical computation of Taylor coefficients to estimate the width of this strip. Approximate inertial forms originally developed for the NSE are analyzed for the Lorenz system, and the dynamics for the latter are completely described.