Global well-posedness and long-time dynamics for a higher order quasi-geostrophic type equation

In this paper we study a higher order viscous quasi-geostrophic type equation. This equation was derived in [11] as the limit dynamics of a singularly perturbed Navier–Stokes–Korteweg system with Coriolis force, when the Mach, Rossby and Weber numbers go to zero at the same rate.The scope of the present paper is twofold. First of all, we investigate well-posedness of such a model on the whole space ${\mathbb{R}}^2$: we prove that it is well-posed in $H^s$ for any $s\ge 3$, globally in time. Interestingly enough, we show that this equation owns two levels of energy estimates, for which one gets existence and uniqueness of weak solutions with different regularities (namely, $H^3$ and $H^4$ regularities); this fact can be viewed as a remainder of the so called BD-entropy structure of the original system.In the second part of the paper we investigate the long-time behavior of these solutions. We show that they converge to the solution of the corresponding linear parabolic type equation, with same initial datum and external force. Our proof is based on dispersive estimates both for the solutions to the linear and non-linear problems.     

Fractional partial differential equations with boundary conditions

We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posedness of the associated Cauchy problems in $C_0(\mathrm{\Omega })$ and $L_1(\mathrm{\Omega })$. In order to do so we develop a new method of embedding finite state Markov processes into Feller processes on bounded domains and then show convergence of the respective Feller processes. This also gives a numerical approximation of the solution. The proof of well-posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the Lax–Richtmyer Equivalence Theorem to prove convergence without checking well-posedness.     

A note on the performance of the gamma kernel estimators at the boundary

The gamma kernel estimator is proposed in Chen [Chen, S.X., 2000. Probability density function estimation using gamma kernels. Annals of the Institute of Statistical Mathematics 52, 471–480] to estimate densities with support $[0,\infty )$. It is shown in his paper that the gamma kernel estimator is non-negative, free of boundary bias, and achieves the optimal rate of convergence for the mean integrated squared error. Numerical results reported in Chen’s paper show that, in the boundary region, the gamma kernel estimator even outperforms some widely used boundary corrected density estimators such as the boundary kernel estimator. However, our study finds that the gamma kernel estimator at $x=0$ is actually the reflection estimator when the double exponential kernel is used and is only boundary problem free when the estimated density has a shoulder at $x=0$ (i.e., the first derivative of the density at $x=0$ is zero). For densities not satisfying the shoulder condition, we show that the gamma kernel estimator has a severe boundary problem and its performance is inferior to that of the boundary kernel estimator.     

Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow

In this paper, we prove the well-posedness of the linearized Prandtl equation around a non-monotonic shear flow in Gevrey class $2?\theta$ for any $\theta >0$. This result is almost optimal by the ill-posedness result proved by Gérard-Varet and Dormy, who construct a class of solution with the growth like $e^{\sqrt{k}t}$ for the linearized Prandtl equation around a non-monotonic shear flow.     

On the local existence for the Euler equations with free boundary for compressible and incompressible fluids

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.     

Partial regularity for solutions to subelliptic eikonal equations

On a bounded domain Ω in the Euclidean space ${\mathbb{R}}^n$, we study the homogeneous Dirichlet problem for the eikonal equation associated with a system of smooth vector fields, which satisfies Hörmander's bracket generating condition. We prove that the solution is smooth in the complement of a closed set of Lebesgue measure zero.     

Self-adjoint and skew-symmetric extensions of the Laplacian with singular Robin boundary condition

We study the Laplacian in a bounded domain, with a varying Robin boundary condition singular at one point. The associated quadratic form is not semi-bounded from below, and the corresponding Laplacian is not self-adjoint, it has a residual spectrum covering the whole complex plane. We describe its self-adjoint extensions and exhibit a physically relevant skew-symmetric one. We approximate the boundary condition, giving rise to a family of self-adjoint operators, and we describe its spectrum by the method of matched asymptotic expansions. A part of the spectrum acquires a strange behavior when the small perturbation parameter $\epsilon >0$ tends to zero, namely it becomes almost periodic in the logarithmic scale $|\ln ?\epsilon |$, and in this way “wanders” along the real axis at a speed $O({\epsilon }^{?1})$.     

Global well-posedness for a drug transport model in tumor multicell spheroids

In this paper we study global existence of solutions of a mathematical model for drug transport in tumor multicell spheroids. The model is a free boundary problem of a system of partial differential equations. It contains one nonlinear first-order equation describing the distribution of live tumor cells, and two nonlinear reaction diffusion equations describing the evolution of nutrient concentration and drug concentration, respectively. By using the method of characteristics for first-order equations, the $L^p$-theory for parabolic equations, the Banach fixed point theorem and the extension method, we prove that this problem has a unique global solution.     

Sharp Moser–Trudinger inequalities for the Laplacian without boundary conditions

We derive a sharp Moser–Trudinger inequality for the borderline Sobolev imbedding of $W^{2,n/2}(B_n)$ into the exponential class, where $B_n$ is the unit ball of ${\mathbb{R}}^n$. The corresponding sharp results for the spaces $W_0^{d,n/d}(\Omega )$ are well known, for general domains Ω, and are due to Moser and Adams. When the zero boundary condition is removed the only known results are for $d=1$ and are due to Chang–Yang, Cianchi and Leckband. The proof of our result is based on a new integral representation formula for the “canonical” solution of the Poisson equation on the ball, that is, the unique solution of the equation $\mathrm{\Delta }u=f$ which is orthogonal to the harmonic functions on the ball. The main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation, expressed in terms of its distribution function.     

A large deviations principle for stochastic flows of viscous fluids

We study the well-posedness of a stochastic differential equation on the two dimensional torus ${\mathbb{T}}^2$, driven by an infinite dimensional Wiener process with drift in the Sobolev space $L^2(0,T;H^1\left({\mathbb{T}}^2\right))$. The solution corresponds to a stochastic Lagrangian flow in the sense of DiPerna Lions. By taking into account that the motion of a viscous incompressible fluid on the torus can be described through a suitable stochastic differential equation of the previous type, we study the inviscid limit. By establishing a large deviations principle, we show that, as the viscosity goes to zero, the Lagrangian stochastic Navier–Stokes flow approaches the Euler deterministic Lagrangian flow with an exponential rate function.     

Quasistatic crack growth in 2d-linearized elasticity

In this paper we prove a two-dimensional existence result for a variational model of crack growth for brittle materials in the realm of linearized elasticity. Starting with a time-discretized version of the evolution driven by a prescribed boundary load, we derive a time-continuous quasistatic crack growth in the framework of generalized special functions of bounded deformation ($GSBD$). As the time-discretization step tends to zero, the major difficulty lies in showing the stability of the static equilibrium condition, which is achieved by means of a Jump Transfer Lemma generalizing the result of [19] to the $GSBD$ setting. Moreover, we present a general compactness theorem for this framework and prove existence of the evolution without imposing a-priori bounds on the displacements or applied body forces.     

On a free boundary problem for an American put option under the CEV process

We consider an American put option under the CEV process. This corresponds to a free boundary problem for a PDE. We show that this free boundary satisfies a nonlinear integral equation, and analyze it in the limit of small $\rho =2r/{\sigma }^2$, where $r$ is the interest rate and $\sigma$ is the volatility. We use perturbation methods to find that the free boundary behaves differently for five ranges of time to expiry.