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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 Mach limit problem for the Euler equations in the analytic spaces. We prove that, given analytic data, the solutions to the compressible Euler equations are uniformly bounded in a suitable analytic norm and then show that the convergence toward the incompressible Euler solution holds in the analytic norm. We also show that the same results hold more generally for Gevrey data with the convergence in the Gevrey norms., 37 pages

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 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 prove existence of global-in-time weak solutions of the incompressible Navier-Stokes equations in the half-space $\mathbb{R}^3_+$ with initial data in a weighted space that allow non-uniformly locally square integrable functions that grow at spatial infinity in an intermittent sense. The space for initial data is built on cubes whose sides $R$ are proportional to the distance to the origin and the square integral of the data is allowed to grow as a power of $R$. The existence is obtained via a new a priori estimate and stability result in the weighted space, as well as new pressure estimates. Also, we prove eventual regularity of such weak solutions, up to the boundary, for $(x,t)$ satisfying $t>c_1|x|^2 + c_2$, where $c_1,c_2>0$, for a large class of initial data $u_0$, with $c_1$ arbitrarily small. As an application of the existence theorem, we construct global discretely self-similar solutions, thus extending the theory on the half-space to the same generality as the whole space., 32 pages, 1 figure

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 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)

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 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 obtain a global existence result for the three-dimensional Navier-Stokes equations with a large class of data allowing growth at spatial infinity. Namely, we show the global existence of suitable weak solutions when the initial data belongs to the weighted space $\mathring M^{2,2}_{\mathcal C}$ introduced in [Z. Bradshaw and I. Kukavica, Existence of suitable weak solutions to the Navier-Stokes equations for intermittent data, J. Math. Fluid Mech. to appear]. This class is strictly larger than currently available spaces of initial data for global existence and includes all locally square integrable discretely self-similar data. We also identify a sub-class of data for which solutions exhibit eventual regularity on a parabolic set in space-time.

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 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.