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We consider the stochastic Navier–Stokes equations in $${\mathbb T}^{3}$$ with multiplicative white noise. We construct a unique local strong solution with initial data in $$L^p$$ , where $$p>5$$ . We also address the global existence of the solution when the initial data is small in $$L^p$$ , with the same range of p.

We address the inviscid limit for the Navier–Stokes equations in a half space, with initial datum that is analytic only close to the boundary of the domain, and that has Sobolev regularity in the complement. We prove that for such data the solution of the Navier–Stokes equations converges in the vanishing viscosity limit to the solution of the Euler equation, on a constant time interval.

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

We consider the three-dimensional incompressible free-boundary Euler equations in a bounded domain and with surface tension. Using Lagrangian coordinates, we establish a priori estimates for solutions with minimal regularity assumptions on the initial data.

We consider the Navier–Stokes equations posed on the half space, with Dirichlet boundary conditions. We give a direct energy-based proof for the instantaneous space–time analyticity and Gevrey-class regularity of the solution, uniformly up to the boundary of the half space.

We address preservation of the Lagrangian analyticity radius of solutions to the Euler equations belonging to natural analytic space based on the size of Taylor (or Gevrey) coefficients. We prove that if the solution belongs to such space, then the solution also belongs to it for a positive amount of time. We also prove the local analog of this result for a sufficiently large Gevrey parameter; however, we show that the preservation holds independently of the size of the radius. Finally, we construct a solution which shows that the first result does not hold in the Eulerian setting.

We address the question of quantitative uniqueness for the equation $$\Delta u=V u$$ with either periodic or Dirichlet boundary conditions in a disk. We construct solutions u corresponding to potential functions V such that u vanishes of order $$\mathrm{const} \Vert V\Vert _{L^\infty }^{2/3}$$ . The example also shows sharpness of recently obtained bounds in the case of a parabolic equation $$u_t-\Delta u= V u$$ .

We consider the free boundary compressible and incompressible Euler equations with surface tension. In both cases, we provide a priori estimates for the local existence with the initial velocity in H 3 , with the H 3 condition on the density in the compressible case. An additional condition is required on the free boundary. Compared to the existing literature, both results lower the regularity of initial data for the Lagrangian Euler equation with surface tension.

We establish a sufficient regularity condition for local solutions of the Navier–Stokes equations. For a suitable weak solution (u, p) on a domain D we prove that if $$\partial _3 u$$ belongs to the space $$L_t^{s_0}L_x^{r_0}(D)$$ where $$2/s_0 + 3/r_0 \le 2 $$ and $$9/4 \le r_0\le 5/2$$ , then the solution is Holder continuous in D.

We address long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity in the cases of the torus, ${\mathbb R}^2$, and on a bounded domain with Lions or Dirichlet boundary conditions. In all the cases, we obtain bounds on the long time behavior for the norms of the velocity and the vorticity. In particular, we obtain that the norm $\Vert (u,\rho)\Vert_{H^2\times H^{1}}$ is bounded by a single exponential, improving earlier bounds., Comment: Journal of Dynamics and Differential Equations, to appear

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.

We address the local existence and uniqueness of solutions for the 3D Euler equations with a free interface. We prove the local well-posedness in the rotational case when the initial datum $$u_0$$ satisfies $$u_0\in H^{2.5+\delta }$$ and , where $$\delta >0$$ is arbitrarily small, under the Taylor condition on the pressure.

We prove a local-in-time existence of solutions result for the two dimensional incompressible Euler equations with a moving boundary, with no surface tension, under the Rayleigh---Taylor stability condition. The main feature of the result is a local regularity assumption on the initial vorticity, namely $$H^{1.5+\delta }$$H1.5+� Sobolev regularity in the vicinity of the moving interface in addition to the global regularity assumption on the initial fluid velocity in the $$H^{2+\delta }$$H2+� space. We use a special change of variables and derive a priori estimates, establishing the local-in-time existence in $$H^{2+\delta }$$H2+�. The assumptions on the initial data constitute the minimal set of assumptions for the existence of solutions to the rotational flow problem to be established in 2D.

We introduce a new method for establishing local analyticity and estimating the local analyticity radius of a solutions to the 3D Navier–Stokes equations at interior points. The approach is based on rephrasing the problem in terms of second order parabolic systems which are then estimated using the mild solution approach. The estimates agree with the global analyticity radius from [16] up to a logarithm.

We address the global regularity for the 2D Boussinesq equations with positive viscosity and zero diffusivity. We prove that for data (u0, ρ0 )i nH s × H s−1 ,w here 1 1.

We address the long-time behavior of solutions to damped dispersive stochastic partial differential equations, namely the KdV equation and the nonlinear Schrodinger equation on the whole space. We prove that the transition semigroup is Feller and establish the existence of an invariant measure using the asymptotic compactness property of the transition semigroup and the Aldous criterion.

We address the well-posedness of the primitive equations of the ocean with continuous initial data. We show that the splitting of the initial data into a regular finite energy part and a small bounded part is preserved by the equations, thus leading to existence and uniqueness of solutions.

We address a fluid?structure system which consists of the incompressible Navier?Stokes equations and a damped linear wave equation defined on two dynamic domains. The equations are coupled through transmission boundary conditions and additional boundary stabilization effects imposed on the free moving interface separating the two domains. Given sufficiently small initial data, we prove the global-in-time existence of solutions by establishing a key energy inequality which in addition provides exponential decay of solutions.

We address the global regularity of solutions of the Navier–Stokes equations in a domain \({Q = \mathbb{R}^2 \times [-1/2, 1/2]}\) with periodic boundary conditions. We prove that the solutions are smooth under some structural conditions and some natural conditions on the horizontal derivatives of the horizontal components of the velocity and the derivative in the vertical direction. The obtained conditions admit data whose vertical average is large in BMO−1. We emphasize that unlike previously known results, our data do not need to oscillate in any direction. Also, the results allow non-zero force and lead to solutions which do not decay in time.

We address a question posed by Glatt-Holtz and Ziane in [GHZ09, Remark 2.1 (ii)], regarding moments of strong pathwise solutions to the Navier-Stokes equations in a two-dimensional bounded domain O. We prove that Eφ(‖u(t)‖2H1(O)) 0, where φ(x) = log(1 + log(1 + x)). Such moment bounds may be used to study statistical properties of the long time behavior of the equation. In addition, we obtain algebraic moment bounds on compact subdomains O0 of the form Eφe(‖u(t)‖2H1(O0)) 0 and any e > 0.

We consider an elliptic equation with a divergence-free drift b. We prove that an inequality of Harnack type holds under the assumption b ∈ L ∩L where δ > 0. As an application we provide a one sided Liouville’s theorem provided that b ∈ L(R) ∩ L(R).

We find a new class of data for which the Prandtl boundary layer equations and the hydrostatic Euler equations are locally in time well-posed. In the case of the Prandtl equations, if the initial datum $u_0$ is monotone on a number of intervals (on some strictly increasing, on some strictly decreasing) and analytic on the complement of these intervals, we show that the local existence and uniqueness hold. The same result is true for the hydrostatic Euler equations if we assume these conditions for the initial vorticity $\omega_0=\partial_y u_0$.

We address the local existence of solutions of the 2D and 3D water wave problems. For the space dimension three, we consider the irrotational datum u 0 and prove that the local in time existence holds for initial velocities belonging to H 2.5+? , where ?>0 is arbitrary. For the space dimension two, the data does not need to be irrotational. We prove the local in time existence when u 0 belongs to H 2+? and $\operatorname{curl}u_{0}$ to H 1.5+? , where ?>0 is arbitrary.

We are concerned with global regularity of solutions in a periodic domain Q=[0,1]3. We prove that, in the class of solutions oscillating in the vertical direction, the solutions are smooth under natural conditions on the horizontal derivatives of the horizontal components of the velocity, the derivative in the vertical direction and the vertical average of the initial data. The obtained conditions admit data whose BMO−1-norm has algebraic dependence on 1/h where h is the period of oscillation and generate global solutions. Also, the results allow non-zero force and large data which do not decay in time.

We establish a sharp regularity result for the normal trace of the solution to the anisotropic linear elasticity system with Dirichlet boundary condition on a Lipschitz domain. Using this result we obtain a new existence result for a fluid-structure interaction model in the case when the structure is an anisotropic elastic body.

We consider solutions of the Stokes system in a neighborhood of a point in which the velocity $u$ vanishes of order $d$. We prove that there exists a divergence-free polynomial $P$ in $x$ with $t$-dependent coefficients of degree $d$ which approximates the solution $u$ of order $d+\alpha$ for certain $\alpha>0$. The polynomial $P$ satisfies a Stokes equation with a forcing term which is a sum of two polynomials in $x$ of degrees $d-1$ and $d$. The results extend to Oseen systems and to the Navier-Stokes equation.

We establish the strong unique continuation property for differences of solutions to the Navier–Stokes system with Gevrey forcing. For this purpose, we provide Carleman-type inequalities with the same singular weight for the Laplacian and the heat operator.

We address the system of fluid–structure interaction consisting of a compressible Navier–Stokes equation coupled with an elasticity equation, with the velocity and stress continuity requirements across the free moving interface. We prove the a priori estimates for existence of solutions when the initial velocity belongs to H 3 , the initial density is bounded from below and belongs to H 3/2+r , where r> 0, and the initial velocity of the displacement is in H 3/2+r .

We estimate the parabolic fractal (or parabolic box-counting) dimension of the singular set for suitable weak solutions of the Navier?Stokes equations in a bounded domain D. We prove that the parabolic fractal dimension is bounded by 45/29 improving an earlier result from (Kukavica 2009 Nonlinearity 22 2889?900). Also, we introduce the new (parabolic) ?-fractal dimension, where ? is a parameter, which for ??=?1 agrees with the parabolic fractal and for ??=?? with the parabolic Hausdorff dimension. We prove that for a certain range of ?, the dimension of the singular set is bounded by 3/2.

Article history: We address the strong unique continuation problem for higher order elliptic partial differential equations in 2D with Gevrey co- efficients. We provide a quantitative estimate of unique continu- ation (observability estimate) and prove that the solutions satisfy the strong unique continuation property for ranges of the Gevrey exponent strictly including non-analytic Gevrey classes. As an ap- plication, we obtain a new upper bound on the Hausdorff length of the nodal sets of solutions with a polynomial dependence on the coefficients. Published by Elsevier Inc.

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.

We address the question of well-posedness in spaces of analytic functions for the Cauchy problem for the hydrostatic incompressible Euler equations (inviscid primitive equations) on domains with boundary. By a suitable extension of the Cauchy–Kowalewski theorem we construct a locally in time, unique, real-analytic solution and give an explicit rate of decay of the radius of real-analyticity.

In the paper, we address the partial regularity of solutions of the Navier–Stokes system. Earlier, we have proved that the one-dimensional parabolic Hausdorff measure of the singular set is zero under the assumption that the force f belongs locally to L 5 / 3 . Here we prove the same statement under a more general assumption that the Morrey norm in L 5 / 7 10 / 7 of the force is sufficiently small. We do so by establishing a fractional integration theorem using the Morrey spaces and by a suitable iteration using a localized version of the Morrey norm.

In this paper, we consider a Navier?Stokes?Lam? 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.

We address the question of well-posedness in spaces of analytic functions for the hydrostatic incompressible Euler equations (inviscid primitive equations) on domains with boundary, with a novel side-boundary condition.

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 question of to what extent the temporal decay ‖ u ( ⋅ , t ) ‖ L 2 = O ( t − γ 0 ) for solutions u of the Navier–Stokes equation influences the decay of weighted norms. We prove that ‖ | x | a u ( ⋅ , t ) ‖ L 2 = O ( t − γ 0 + a / 2 ) holds under the condition 0 ≤ a n / 2 + 1 . This extends the range of weights obtained in [I. Kukavica, J.J. Torres, Weighted L p decay for solutions of the Navier–Stokes equations, Commun. Partial Differential Equations 32 (2007) 819–831]. A wider range of exponents is also obtained for the decay of L p norms.

We address the problem of analyticity of smooth solutions u of the incompressible Euler equations. If the initial datum is real-analytic, the solution remains real-analytic as long as f t 0 ∥∇u(·, s)∥ L∞ ds < ∞. Using a Gevrey-class approach we obtain lower bounds on the radius of space analyticity which depend algebraically on exp ∫ t 0 ∥∇u(·,s)∥ L∞ ds. In particular, we answer in the positive a question posed by Levermore and Oliver.

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.

The aim of this paper is to prove that the solutions of the primitive equations converge, in the zero viscosity limit, to the solutions of the hydrostatic Euler equations. We construct the solution of the primitive equations through a matched asymptotic expansion involving the solution of the hydrostatic Euler equation and boundary layer correctors as the first order term, and an error that we show to be \({O(\sqrt{\nu})}\). The main assumption is spatial analyticity of the initial datum.