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We consider the local existence and uniqueness of solutions for a system consisting of an inviscid fluid with a free boundary, modeled by the Euler equations, in a domain enclosed by an elastic boundary, which evolves according to the wave equation. We derive a priori estimates for the local existence of solutions and also conclude the uniqueness. Both, existence and uniqueness are obtained under the assumption that the Euler data belongs to $$H^{r}$$ H r , where $$r>2.5$$ r > 2.5 , which is known to be the borderline exponent for the Euler equations. Unlike the setting of the Euler equations with vacuum, the membrane is shown to stabilize the system in the sense that the Rayleigh–Taylor condition does not need to be assumed.

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 global existence of solutions for the 2D Kuramoto-Sivashinsky equations in a periodic domain $$[0,L_1]\times [0,L_2]$$ with initial data satisfying $$\Vert u_0\Vert _{L^2}\le C^{-1}L_2^{-2}$$ , where C is a constant. We prove that the global solution exists under the condition $$L_2\le 1/C L_1^{3/5}$$ , improving earlier results. The solutions are smooth and decrease energy until they are dominated by $$C L_1^{3/2}L_2^{1/2}$$ , implying the existence of an absorbing ball in $$L^2$$ .

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 address the problem of controllability of the MHD system in a rectangular domain with a control prescribed on the side boundary. We identify a necessary and sufficient condition on the data to be null controllable, i.e., can be driven to the zero state. We also show that the validity of this condition allows the states to be stirred to each other. If the condition is not satisfied, one can move from one state to another with the help of a simple shear external magnetic force., Comment: 14 pages

We establish the validity of the Euler$+$Prandtl approximation for solutions of the Navier-Stokes equations in the half plane with Dirichlet boundary conditions, in the vanishing viscosity limit, for initial data which are analytic only near the boundary, and Sobolev smooth away from the boundary. Our proof does not require higher order correctors, and works directly by estimating an $L^{1}$-type norm for the vorticity of the error term in the expansion Navier-Stokes$-($Euler$+$Prandtl$)$. An important ingredient in the proof is the propagation of local analyticity for the Euler equation, a result of independent interest.

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 address the quantitative uniqueness properties of the solutions of the parabolic equation $ \partial_t u - \Delta u = w_j (x,t) \partial_j u + v(x,t) u $ where $v$ and $w$ are bounded. We prove that for solutions $u$, the order of vanishing is bounded by $C(\Vert v\Vert_{L^\infty}^{2/3}+\Vert w\Vert_{L^\infty}^2)$ matching the upper bound previously established in the elliptic case. in the elliptic case., Comment: Indiana Univ. Math. J. (to appear)

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 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.

In this paper, we address the long time behavior of solutions of the stochastic Schrodinger equation $$du+(\lambda u+i\Delta u +i\alpha |u|^{2\sigma }u)dt=\Phi dW_t$$ in $${{\mathbb {R}}}^{d}$$ . We prove the existence of an invariant measure in $$H^{1}$$ for $$\sigma

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.

Resume Dans cette Note, nous etudions l'existence locale des solutions pour l'equation du scalaire actif ∂ t θ + u ⋅ ∇ θ = 0 ou u = − ∇ ⊥ ( − Δ ) − 1 + β / 2 θ . Cette equation correspond a l'equation d'Euler quand β = 0 et a l'equation quasi-geostrophique quand β = 1 . Nous montrons l' existence locale dans l'espace H 1 + β + ϵ , ou ϵ > 0 , pour 1 β 2 . Un resultat recent de Chae, Constantin, Cordoba, Gancedo et Wu montre l'existence locale dans H 4 . L'amelioration est due a des nouvelles estimations plus precises du commutateur, et l'exposant fractionnaire est obtenu par un traitement different de la non-linearite en utilisant une inegalite double du commutateur.

In this chapter, we introduce a system modeling an interaction of an incompressible fluid and an elastic body. We provide the local existence theory for this model suitable for applications to the global existence of solutions with small initial data. We also provide the crucial energy and equipartition estimates for the system.

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 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 $H^{2}$.