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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 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 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 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.

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 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.

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 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 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.

A. V. Balakrishnan, a co-founder and first, long-time Editor-in-Chief of Applied Mathematics and Optimization, passed away in Los Angeles on March 17, 2015. The scientific community, and in particular our journal’s community, is deeply saddened and mourns his departure. The present special issue of AMO intends to commemorate and cherish his memory while paying tribute to his lasting role and influence, by offering a selection of contributions of long-time friends, former advisees, collaborators and associates.

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.

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.

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 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 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 $\delta>0$.

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.

In this Note, the global existence of strong solutions of the primitive equations for the ocean in space dimension 3 with the Dirichlet boundary condition is obtained. The method of the proof can be easily adapted to treat full primitive equations in a domain with a varying bottom topography. To cite this article: I. Kukavica, M. Ziane, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

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.

Resume Dans cette Note, nous etudions la regularite globale des solutions des equations de Navier–Stokes dans un domaine de faible epaisseur Ω = [ 0 , L 1 ] × [ 0 , L 2 ] × [ 0 , ϵ ] avec des conditions aux limites periodiques. Nous montrons que si ‖ ∇ u 0 ‖ L 2 ( Ω ) ⩽ C ( L 1 , L 2 , δ ) / ϵ 1 / 2 − δ , ou u 0 est la donnee initiale et δ > 0 est arbitraire, alors il existe une solution reguliere globale avec la donnee initiale u 0 . Cette condition ameliore les resultats existants, en particulier la moyenne de la vitesse initiale dans la direction de l'epaisseur faible n'est pas supposee petite quand l'epaisseur est petite. Pour citer cet article : I. Kukavica, M. Ziane, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

We address the long time behavior of solutions of the stochastic Korteweg-de Vries equation $ du + (\partial^3_x u +u\partial_x u +\lambda u)dt = f dt+\Phi dW_t$ on ${\mathbb R}$ where $f$ is a deterministic force. We prove that the Feller property holds and establish the existence of an invariant measure. The tightness is established with the help of the asymptotic compactness, which is carried out using the Aldous criterion.

Resume Dans cette Note, nous donnons des conditions suffisantes de regularite des solutions des equations de Navier–Stokes. Nous montrons que si une solution de Leray–Hopf verifie l'une des trois conditions (i) ∂ u / ∂ x 3 ∈ L t s 0 L x r 0 ou 2 / s 0 + 3 / r 0 ⩽ 2 et 9 / 4 ⩽ r 0 ⩽ 3 , (ii) ∇ u 3 ∈ L t s 1 L x r 1 ou 2 / s 1 + 3 / r 1 ⩽ 11 / 6 et 54 / 23 ⩽ r 0 ⩽ 18 / 5 , ou (iii) u 3 ∈ L t s 0 L x r 0 ou 2 / s 0 + 3 / r 0 ⩽ 5 / 8 et 24 / 5 ⩽ r 0 ⩽ ∞ , alors elle est reguliere. Ces conditions ameliorent les resultats existants sur la regularite conditionnelle des equations de Navier–Stokes. Pour citer cet article : I. Kukavica, M. Ziane, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

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 $u_0$.

We prove that any solution of the Kuramoto–Sivashinsky equation either belongs to the global attractor or it cannot be continued to a solution defined for all negative times. This extends a previous result of the first author who proved that solutions which do not belong to the global attractor have superexponential backward growth. A particular consequence of the result is that the global attractor can be characterized as the maximal invariant set.