arXiv subjects
Publications and source records attributed to Peter Constantin.
We prove nonlinear lower bounds and commutator estimates for the Dirichlet fractional Laplacian in bounded domains. The applications include bounds for linear drift-diffusion equations with nonlocal dissipation and global existence of weak solutions of critical surface quasi-geostrophic equations.
We consider the incompressible Euler equations on ${\mathbb R}^d$, where $d \in \{ 2,3 \}$. We prove that: (a) In Lagrangian coordinates the equations are locally well-posed in spaces with fixed real-analyticity radius (more generally, a fixed Gevrey-class radius). (b) In Lagrangian coordinates the equations are well-posed in highly anisotropic spaces, e.g.~Gevrey-class regularity in the label $a_1$ and Sobolev regularity in the labels $a_2,...,a_d$. (c) In Eulerian coordinates both results (a) and (b) above are false.
We consider the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy's law. We establish that as long as the slope of the interface between the two fluids remains bounded and uniformly continuous, the solution remains regular. The proofs exploit the nonlocal nonlinear parabolic nature of the equations through a series of nonlinear lower bounds for nonlocal operators. These are used to deduce that as long as the slope of the interface remains uniformly bounded, the curvature remains bounded. The nonlinear bounds then allow us to obtain local existence for arbitrarily large initial data in the class $W^{2,p}$, $1<p\leq \infty$. We provide furthermore a global regularity result for small initial data: if the initial slope of the interface is sufficiently small, there exists a unique solution for all time.
We consider the global attractor of the critical SQG semigroup $S(t)$ on the scale-invariant space $H^1(\mathbb{T}^2)$. It was shown in~\cite{CTV13} that this attractor is finite dimensional, and that it attracts uniformly bounded sets in $H^{1+δ}(\mathbb{T}^2)$ for any $δ>0$, leaving open the question of uniform attraction in $H^1(\mathbb{T}^2)$. In this paper we prove the uniform attraction in $H^1(\mathbb{T}^2)$, by combining ideas from DeGiorgi iteration and nonlinear maximum principles.
We present a Lagrangian-Eulerian strategy for proving uniqueness and local existence of solutions in path spaces of limited smoothness for a class of incompressible hydrodynamic models including Oldroyd-B type complex fluid models and zero magnetic resistivity magneto-hydrodynamics equations.
We present a simple, short and elementary proof that if $v$ is a Beltrami flow with a finite energy in $\mathbb R^3$ then $v=0$. In the case of the Beltrami flows satisfying $v\in L^\infty _{loc} (\Bbb R^3) \cap L^q(\Bbb R^3)$ with $q\in [2, 3)$, or $|v(x)|=O(1/|x|^{1+\varepsilon})$ for some $\varepsilon >0$, we provide a different, simple proof that $v=0$.
We discuss general incompressible inviscid models, including the Euler equations, the surface quasi-geostrophic equation, incompressible porous medium equation, and Boussinesq equations. All these models have classical unique solutions, at least for short time. We show that they have real analytic Lagrangian paths. More precisely, we show that as long as a solution of any of these equations is in a class of regularity that assures Hölder continuous gradients of velocity, the corresponding Lagrangian paths are real analytic functions of time. The method of proof is conceptually straightforward and general, and we address the combinatorial issues head-on.
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.
This paper considers the three dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an $L^2$ maximum principle for the fluid interface. We also show global in time existence for strong and weak solutions with initial data controlled by explicit constants. Furthermore we refine the estimates from our paper \cite{PDPB} to obtain global existence and uniqueness for strong solutions with larger initial data than we previously had in 2D. Finally we provide global in time results in critical spaces, giving solutions with bounded slope and time integrable bounded curvature.
We give an example of a well posed, finite energy, 2D incompressible active scalar equation with the same scaling as the surface quasi-geostrophic equation and prove that it can produce finite time singularities. In spite of its simplicity, this seems to be the first such example. Further, we construct explicit solutions of the 2D Boussinesq equations whose gradients grow exponentially in time for all time. In addition, we introduce a variant of the 2D Boussinesq equations which is perhaps a more faithful companion of the 3D axisymmetric Euler equations than the usual 2D Boussinesq equations.
We provide local formulas for the pressure of incompressible fluids. The pressure can be expressed in terms of its average and averages of squares of velocity increments in arbitrary small neighborhoods. As application, we give a brief proof of the fact that $C^α$ velocities have $C^{2α}$ (or Lipschitz) pressures. We also give some regularity criteria for 3D incompressible Navier-Stokes equations.
We prove the existence of a compact global attractor for the dynamics of the forced critical surface quasi-geostrophic equation (SQG) and prove that it has finite fractal (box-counting) dimension. In order to do so we give a new proof of global regularity for critical SQG. The main ingredient is the nonlinear maximum principle in the form of a nonlinear lower bound on the fractional Laplacian, which is used to bootstrap the regularity directly from $L^\infty$ to $C^α$, without the use of De Giorgi techniques. We prove that for large time, the norm of the solution measured in a sufficiently strong topology becomes bounded with bounds that depend solely on norms of the force, which is assumed to belong merely to $L^\infty \cap H^1$. Using the fact that the solution is bounded independently of the initial data after a transient time, in spaces conferring enough regularity, we prove the existence of a compact absorbing set for the dynamics in $H^1$, obtain the compactness of the linearization and the continuous differentiability of the solution map. We then prove exponential decay of high yet finite dimensional volume elements in $H^1$ along solution trajectories, and use this property to bound the dimension of the global attractor.
We prove absence of anomalous dissipation of energy for forced critical SQG in two space dimensions.
We establish the existence and uniqueness of an ergodic invariant measure for 2D fractionally dissipated stochastic Euler equations on the periodic box, for any power of the dissipation term.
We prove global regularity of solutions of Oldroyd-B equations in 2 spatial dimensions with spatial diffusion of the polymeric stresses.
We obtain a family of nonlinear maximum principles for linear dissipative nonlocal operators, that are general, robust, and versatile. We use these nonlinear bounds to provide transparent proofs of global regularity for critical SQG and critical d-dimensional Burgers equations. In addition we give applications of the nonlinear maximum principle to the global regularity of a slightly dissipative anti-symmetric perturbation of 2d incompressible Euler equations and generalized fractional dissipative 2d Boussinesq equations.
In this paper we derive a probabilistic representation of the deterministic 3-dimensional Navier--Stokes equations in the presence of spatial boundaries. The formulation in the absence of spatial boundaries was done by the authors in [Comm. Pure Appl. Math. 61 (2008) 330--345]. While the formulation in the presence of boundaries is similar in spirit, the proof is somewhat different. One aspect highlighted by the formulation in the presence of boundaries is the nonlocal, implicit influence of the boundary vorticity on the interior fluid velocity.
This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasi-geostrophic (SQG) equation with the velocity field $u$ related to the scalar $θ$ by $u=\nabla^\perpΛ^{β-2}θ$, where $1<β\le 2$ and $Λ=(-Δ)^{1/2}$ is the Zygmund operator. The borderline case $β=1$ corresponds to the SQG equation and the situation is more singular for $β>1$. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions and the local existence of patch type solutions. The second family is a dissipative active scalar equation with $u=\nabla^\perp (\log(I-Δ))^μθ$ for $μ>0$, which is at least logarithmically more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. This result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani \cite{Oh}.