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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 show that a suitable weak solution to the incompressible Navier-Stokes equations on ${\mathbb{R}^3\times(-1,1)}$ is regular on $\mathbb{R}^3\times(0,1]$ if $\partial_3 u $ belongs to $M^{2p/(2p-3),\alpha } ((-1,0);L^p (\mathbb{R}^3 ))$ for any $\alpha >1$ and $p\in (3/2,\infty)$, which is a logarithmic-type variation of a Morrey space in time. For each $\alpha >1$ this space is, up to a logarithm, critical with respect to the scaling of the equations, and contains all spaces $L^q ((-1,0);L^p (\mathbb{R}^3 ))$ that are subcritical, that is for which $2/q+3/p, Comment: 10 pages

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 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.

Local in time weak solutions to the 3D Navier–Stokes are constructed for a class of initial data in $$L^2_\mathrm {loc}$$. In contrast to other constructions (e.g. Lemarie-Rieusset in Recent developments in the Navier–Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, vol 431. Chapman & Hall/CRC, Boca Raton, 2002; Kikuchi and Seregin in Weak solutions to the Cauchy problem for the Navier–Stokes equations satisfying the local energy inequality. Nonlinear equations and spectral theory, American Mathematical Society translations: series 2, vol 220. American Mathematical Society, Providence, pp 141–164, 2007; Kwon and Tsai in Global Navier–Stokes flows for non-decaying initial data with slowly decaying oscillation. arXiv:1811.03249 ), the initial data is not required to be uniformly locally square integrable and, in particular, can exhibit growth in a local $$L^2$$ sense. This class of initial data includes vector fields in the critical Morrey space and discretely self-similar vector fields in $$L^2_\mathrm {loc}$$.

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 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 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 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 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.

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 $H^{2}$.

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.

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 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.

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 $\delta>0$.

We consider bounded strong solutions of the incompressible Navier-Stokes equations in n dimensions, where n ≥ 2, with potential forcing. Assuming that the initial datum is well localized, and assuming that a solution u satisfies for some γ0 ≥ 0, we prove that for all p ∊ [2, ∞) and for all a ∊ [0, n/p′) where p′ = p/(p − 1).

We study backward uniqueness properties for equations of the form u' + Au = f. Under mild regularity assumptions on A and f, it is shown that u(0) = 0 implies u(t) = 0 for t < 0. The argument is based on α-log and log-log convexity. The results apply to mildly nonlinear parabolic equations and systems with rough coefficients and the 2D Navier-Stokes system.

We prove that a weak solution $$u \in L_{\rm loc}^{\infty} ([0,T),L^{2}({\mathbb R}^{3})) \cap L_{\rm loc}^{2} ([0,T),H^{1}({\mathbb R}^{3}))$$ 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.

We establish sufficient conditions for the regularity of solutions of the Navier–Stokes system based on conditions on one component of the velocity. The first result states that if , where and 54/23 ≤ r ≤ 18/5, then the solution is regular. The second result is that if , where and 24/5 ≤ r ≤ ∞, then the solution is regular. These statements improve earlier results on one component regularity.

We study decay in space and time for derivatives of solutions of the Navier–Stokes equations in . The main results concern the ranges of exponents b, c and d for validity of the bounds and where u is a solution of the Navier–Stokes equation and ω is its vorticity.

We address local uniqueness of weak solutions of the Navier–Stokes system with initial data in L ∞ ( R n ) . We show that uniqueness of solutions does not hold. We classify all solutions as all those which can be obtained from the solution of the integral equation (mild solution) by means of a certain transformation.

We provide an upper bound on the size of the nodal set of the vorticity ω for solutions of the 2D periodic Navier–Stokes equation. The upper bound depends polynomially on the initial condition, viscosity, the size of Ω, and 1/t. The result is based on a new bound on the order of vanishing for solutions of .

In this paper, we address the partial regularity of suitable weak solutions of the incompressible Navier--Stokes equations. We prove an interior regularity criterion involving only one component of the velocity. Namely, if $(u,p)$ is a suitable weak solution and a certain scale-invariant quantity involving only $u_3$ is small on a space-time cylinder $Q_r(x_0,t_0)$, then $u$ is regular at $(x_0,t_0)$., Comment: 15 pages

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.

We consider the convergence in the $L^2$ norm, uniformly in time, of the Navier-Stokes equations with Dirichlet boundary conditions to the Euler equations with slip boundary conditions. We prove that if the Oleinik conditions of no back-flow in the trace of the Euler flow, and of a lower bound for the Navier-Stokes vorticity is assumed in a Kato-like boundary layer, then the inviscid limit holds., Comment: Improved the main result and fixed a number of typos

We show that for the periodic 2D Navier-Stokes equations (NSE) the set of initial data for which the solution exists for all negative times and has exponential growth is rather rich. We study this set and show that it is dense in the phase space of the NSE. This yields to a positive answer to a question in [BT]. After an appropriate rescaling the large Reynolds limit dynamics on this set is Eulerian.

In the case of the 3D Navier-Stokes equations, it is proved that there exists a constantɛ>0 with the following property: Every time-periodic solution with a period smaller thanɛ is necessarily a stationary solution. An explicit formula for ɛ is also provided.

In the case of solutions of the two-diensional Navier-Stokes equations, the following analyticity property is established. If the initial datum lies on the global attractor and is close enough to a stationary solution, then the analyticity radius att = 0 of the solution can be made arbitrarily large.

We address a fluid-structure interaction model describing the motion of an elastic body immersed in an incompressible fluid. We establish a priori estimates for the local existence of solutions for a class of initial data which also guarantees uniqueness.

We consider sufficient conditions for the regularity of Leray-Hopf solutions of the Navier-Stokes equations. We prove that if the third derivative of the velocity ∂u∕∂x3 belongs to the space Lts0Lxr0, where 2∕s0+3∕r0⩽2 and 9∕4⩽r0⩽3, then the solution is regular. This extends a result of Beirao da Veiga [Chin. Ann. Math., Ser. B 16, 407–412 (1995); C. R. Acad. Sci, Ser. I: Math. 321, 405–408 (1995)] by making a requirement only on one direction of the velocity instead of on the full gradient. The derivative ∂u∕∂x3 can be substituted with any directional derivative of u.

In this paper, we prove the existence of local-in-time smooth solutions to the nonlinear fluid structure interaction model first introduced in [J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, 1969] and considered in [V. Barbu, Z. Grujic, I. Lasiecka, A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid–structure interaction model, in: Fluids and Waves, in: Contemp. Math., vol. 440, Amer. Math. Soc., Providence, RI, 2007, pp. 55–82; V. Barbu, Z. Grujic, I. Lasiecka, A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid–structure interaction model, Indiana Univ. Math. J. 57 (3) (2008) 1173–1207]. In particular, the strong solutions here are obtained given initial datum for the Navier–Stokes equation in the space H 1 , and initial data for the wave equation w 0 and w 1 in the spaces H 2 ( Ω e ) and H 1 ( Ω e ) respectively.

We address uniqueness of mild solutions of the Navier–Stokes system in L t ∞ L x 3 . We prove that solutions for which the pressure is locally square integrable are unique.

We address the global regularity of solutions of the Navier–Stokes equations in a thin domain Ω = [ 0 , L 1 ] × [ 0 , L 2 ] × [ 0 , ϵ ] with periodic boundary conditions, where L 1 , L 2 > 0 and ϵ ∈ ( 0 , 1 / 2 ) . We prove that if ‖ ∇ u 0 ‖ L 2 ( Ω ) ⩽ ν C ( L 1 , L 2 ) ϵ 1 / 2 | log ϵ | 3 / 2 where u 0 is the initial datum, then there exists a unique global smooth solution with the initial datum u 0 . Also, if ‖ u 0 ‖ H ˙ 1 / 2 ( Ω ) ⩽ ν C ( L 1 , L 2 ) | log ϵ | 1 / 4 the global regularity of the corresponding solution holds.

We derive an asymptotic expansion for smooth solutions of the Navier–Stokes equations in weighted spaces. This result removes previous restrictions on the number of terms of the asymptotics, as well as on the range of the polynomial weights. We also write the expansion in terms of expressions involving non-linear quantities.

By a result of Foias and Saut, the quotient of the enstrophy and the energy of a solution of the Navier–Stokes equations with potential forces converges to the square root of an eigenvalue Λ k of the Stokes operator. We study the Hausdorff length ( H 1 ) of the level sets of the vorticity and the stream function of solutions. We provide upper bounds for these quantities, and for large t we express them in terms of the corresponding eigenvalue Λ k .