Topic overview

math.AP

9009 works10210 researchers

Open map Browse papers

Map preview

Start with the graph, then narrow the list

9009works

10210researchers

Next steps

Use the topic as a working map

Open the full map for clusters, then return here to scan ranked papers and people.

Inspect nearby papers, researchers, institutions and communities without opening a separate graph page.

Building this map preview

BZPEER is loading the nearby papers, people, topics and institutions for this page.

preprint2017arXiv

The Monge problem with vanishing gradient penalization: Vortices and asymptotic profile

We investigate the approximation of the Monge problem (minimizing \int\_$Ω$ |T (x) -- x| d$μ$(x) among the vector-valued maps T with prescribed image measure T \# $μ$) by adding a vanishing Dirichlet energy, namely $ε$ \int\_$Ω$ |DT |^2. We study the $Γ$-convergence as $ε$ $\rightarrow$ 0, proving a density result for Sobolev (or Lipschitz) transport maps in the class of transport plans. In a certain two-dimensional framework that we analyze in details, when no optimal plan is induced by an H ^1 map, we study the selected limit map, which is a new "special" Monge transport, possibly different from the monotone one, and we find the precise asymptotics of the optimal cost depending on $ε$, where the leading term is of order $ε$| log $ε$|.

preprint2015arXiv

Existence of positive solutions for an approximation of stationary mean-field games

Here, we consider a regularized mean-field game model that features a low-order regularization. We prove the existence of solutions with positive density. To do so, we combine a priori estimates with the continuation method. In contrast with high-order regularizations, the low-order regularizations are easier to implement numerically. Moreover, our methods give a theoretical foundation for this approach.

preprint2017arXiv

Inverse Strichartz estimates for 1d Schrödinger operators with potentials of quadratic growth

We prove inverse Strichartz theorems at $L^2$ regularity for a family of Schrödinger evolutions in one space dimension. Prior results rely on spacetime Fourier analysis and are limited to the translation-invariant equation $i\partial_t u = -\tfrac{1}{2} Δu$. Motivated by applications to the mass-critical Schrödinger equation with external potentials (such as the harmonic oscillator) we use a physical space approach.

preprint2017arXiv

Three-dimensional Navier-Stokes equations driven by space-time white noise

In this paper we study 3D Navier-Stokes (NS) equation driven by space-time white noise by using regularity structure theory introduced in [Hai14] and paracontrolled distribution proposed in [GIP13]. We obtain local existence and uniqueness of solutions to the 3D Navier-Stokes equation driven by space-time white noise.

preprint2017arXiv

Degeneracy in finite time of 1D quasilinear wave equations II

We consider the large time behavior of solutions to the following nonlinear wave equation: $\partial_{t}^2 u = c(u)^{2}\partial^2_x u + λc(u)c'(u)(\partial_x u)^2$ with the parameter $λ\in [0,2]$. If $c(u(0,x))$ is bounded away from a positive constant, we can construct a local solution for smooth initial data. However, if $c(\cdot )$ has a zero point, then $c(u(t,x))$ can be going to zero in finite time. When $c(u(t,x))$ is going to 0, the equation degenerates. We give a sufficient condition that the equation with $0\leq λ< 2$ degenerates in finite time.

preprint2017arXiv

Notes on solution maps of abstract FDEs

It is shown that the solution maps of an abstract functional differential equations (FDEs) are $α$-contractions in the phase space equipped with an equivalent norm under appropriate assumptions. This result can be applied to time-delayed reaction-diffusion equations and other evolution systems with time delay.

preprint2017arXiv

On weighted mixed-norm Sobolev estimates for some basic parabolic equations

Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation, the harmonic oscillator evolution equation $$\partial_tu=Δu-|x|^2u+f,$$ and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-Δ)^su=f$ and $(\partial_t-Δ+|x|^2)^su=f$, for $0<s<1$.

preprint2016arXiv

The global nonlinear stability of Minkowski spacetime for self-gravitating massive fields. A brief overview

This is a short review of a series of papers which, in collaboration with Yue Ma, establish several novel existence results for systems of coupled wave-Klein-Gordon equation. Our method, the Hyperbolic Hyperboloidal Method, has allowed us to address the global evolution problem for the Einstein equations of general relativity and investigate the global geometry of matter spacetimes that are initially close to Minkowski spacetime.

preprint2014arXiv

Negative Orlicz-Sobolev norms and strongly nonlinear systems in fluid mechanics

We prove a version of the negative norm theorem in Orlicz-Sobolev spaces. A study of continuity properties of the Bogovskii -operator between Orlicz spaces is a crucial step, of independent interest, in our approach. Applications to the problem of pressure reconstruction for Non-Newtonian fluids governed by constitutive laws, which are not necessarily power type, are presented. A key inequality for a numerical analysis of the underlying elliptic system is also derived.

preprint2016arXiv

Nonlinear Flows for Displacement Correction and Applications in Tomography

In this paper we derive nonlinear evolution equations associated with a class of non-convex energy functionals which can be used for correcting displacement errors in imaging data. We study properties of these filtering flows and provide experiments for correcting angular perturbations in tomographical data.

preprint2017arXiv

Existence of non-trivial solutions for nonlinear fractional Schrödinger-Poisson equations

We prove the existence of non-trivial solutions for a fractional Schr$\ddot{o}$dinger-Poisson equation in $\mathbb{R}^{3}$. The proof is based on the perturbation method and the mountain pass theorem.

preprint2016arXiv

Local Petrovskii lacunas at parabolic singular points of wavefronts of strictly hyperbolic PDE's

We enumerate the local Petrovskii lacunas (that is, the domains of local regularity of the principal fundamental solutions) of strictly hyperbolic PDE's with constant coefficients in $R^N$ at the parabolic singular points of their wavefronts. These points form the next difficult family of classes of the natural classification of singular points after the so-called simple singularities, studied previously. Also we promote a computer program counting for topologically different morsifications of critical points of smooth functions, and hence also for local components of the complement of a generic wavefront at its singular points.

preprint2016arXiv

The A-Stokes approximation for non-stationary problems

Let $\mathcal A$ be an elliptic tensor. A function $v\in L^1(I;LD_{div}(B))$ is a solution to the non-stationary $\mathcal A $-Stokes problem iff \begin{align}\label{abs} \int_Q v\cdot\partial_tϕ\,dx\,dt-\int_Q \mathcal A(\varepsilon(v),\varepsilon(ϕ))\,dx\,dt=0\quad\forallϕ\in C^{\infty}_{0,div}(Q), \end{align} where $Q:=I\times B$, $B\subset\mathbb R^d$ bounded. If the l.h.s. is not zero but small we talk about almost solutions. We present an approximation result in the fashion of the $\mathcal A$-caloric approximation for the non-stationary $\mathcal A $-Stokes problem. Precisely, we show that every almost solution $v\in L^p(I;W^{1,p}_{div}(B))$, $1<p<\infty$, can be approximated by a solution in the $L^s(I;W^{1,s}(B))$-sense for all $s<p$. So, we extend the stationary $\mathcal A$-Stokes approximation by Breit-Diening-Fuchs to parabolic problems.

preprint2017arXiv

A note on local $W^{1,p}$-regularity estimates for weak solutions of parabolic equations with singular divergence-free drifts

We investigate weighted Sobolev regularity of weak solutions of non-homogeneous parabolic equations with singular divergence-free drifts. Assuming that the drifts satisfy some mild regularity conditions, we establish local weighted $L^p$-estimates for the gradients of weak solutions. Our results improve the classical one to the borderline case by replacing the $L^\infty$-assumption on solutions by solutions in the John-Nirenberg \textup{BMO} space. The results are also generalized to parabolic equations in divergence form with small oscillation elliptic symmetric coefficients and therefore improve many known results.

preprint2017arXiv

Extrema of the dynamic pressure in an irrotational regular wave train with underlying currents and infinite depth

In this paper, we investigate the maximum and minimum of the dynamic pressure in a regular wave train with underlying currents and infinite depth respectively. The result is obtained using maximum principles in combination with exploiting some of the physical structures of the problem itself.

preprint2017arXiv

A decomposition result for a singular elliptic equation on compact Riemannian manifolds

On compact Riemannian manifolds, we prove a decomposition theorem for arbitrarily bounded energy sequence of solutions of a singular elliptic equation.

preprint2016arXiv

Existence results for a coupled viscoplastic-damage model in thermoviscoelasticity

In this paper we address a model coupling viscoplasticity with damage in thermoviscoelasticity. The associated PDE system consists of the momentum balance with viscosity and inertia for the displacement variable, at small strains, of the plastic and damage flow rules, and of the heat equation. It has a strongly nonlinear character and in particular features quadratic terms on the right-hand side of the heat equation and of the damage flow rule, which have to be handled carefully. We propose two weak solution concepts for the related initial-boundary value problem, namely `entropic' and `weak energy' solutions. Accordingly, we prove two existence results by passing to the limit in a carefully devised time discretization scheme. Finally, in the case of a prescribed temperature profile, and under a strongly simplifying condition, we provide a continuous dependence result, yielding uniqueness of weak energy solutions.

preprint2017arXiv

Trust-Region Methods for Nonlinear Elliptic Equations with Radial Basis Functions

We consider the numerical solution of nonlinear elliptic boundary value problems with Kansa's method. We derive analytic formulas for the Jacobian and Hessian of the resulting nonlinear collocation system and exploit them within the framework of the trust-region algorithm. This ansatz is tested on semilinear, quasilinear and fully nonlinear elliptic PDEs (including Plateau's problem, Hele-Shaw flow and the Monge-Ampère equation) with excellent results. The new approach distinctly outperforms previous ones based on linearization or finite-difference Jacobians.

preprint2016arXiv

Transonic Flows with Shocks Past Curved Wedges for the Full Euler Equations

We establish the existence, stability, and asymptotic behavior of transonic flows with a transonic shock past a curved wedge for the steady full Euler equations in an important physical regime, which form a nonlinear system of mixed-composite hyperbolic-elliptic type. To achieve this, we first employ the coordinate transformation of Euler-Lagrange type and then exploit one of the new equations to identify a potential function in Lagrangian coordinates. By capturing the conservation properties of the Euler system, we derive a single second-order nonlinear elliptic equation for the potential function in the subsonic region so that the transonic shock problem is reformulated as a one-phase free boundary problem for a second-order nonlinear elliptic equation with the shock-front as a free boundary. One of the advantages of this approach is that, given the shock location or quivalently the entropy function along the shock-front downstream, all the physical variables can expressed as functions of the gradient of the potential function, and the downstream asymptotic behavior of the potential function at the infinite exit can be uniquely determined with uniform decay rate. To solve the fre

preprint2015arXiv

Lieb-Thirring inequalities on the torus

We consider the Lieb-Thirring inequalities on the d-dimensional torus with arbitrary periods. In the space of functions with zero average with respect to the shortest coordinate we prove the Lieb-Thirring inequalities for the $γ$-moments of the negative eigenvalues with constants independent of ratio of the periods. Applications to the attractors of the damped Navier-Stokes system are given.

preprint2015arXiv

Stochastic Navier-Stokes equations for compressible fluids

We study the Navier-Stokes equations governing the motion of isentropic compressible fluid in three dimensions driven by a multiplicative stochastic forcing. In particular, we consider a stochastic perturbation of the system as a function of momentum and density, which is affine linear in momentum and satisfies suitable growth assumptions with respect to density, and establish existence of the so-called finite energy weak martingale solution under the condition that the adiabatic constant satisfies $γ>3/2$. The proof is based on a four layer approximation scheme together with a refined stochastic compactness method and a careful identification of the limit procedure.

preprint2016arXiv

Palindromic discontinuous Galerkin method for kinetic equations with stiff relaxation

We present a high order scheme for approximating kinetic equations with stiff relaxation. The objective is to provide efficient methods for solving the underlying system of conservation laws. The construction is based on several ingredients: (i) a high order implicit upwind Discontinuous Galerkin approximation of the kinetic equations with easy-to-solve triangular linear systems; (ii) a second order asymptotic-preserving time integration based on symmetry arguments; (iii) a palindromic composition of the second order method for achieving higher orders in time. The method is then tested at orders 2, 4 and 6. It is asymptotic-preserving with respect to the stiff relaxation and accepts high CFL numbers.

preprint2016arXiv

On the vortex filament conjecture for Euler flows

In this paper, we study the evolution of a vortex filament in an incompressible ideal fluid. Under the assumption that the vorticity is concentrated along a smooth curve in $\mathbb{R}^3$, we prove that the curve evolves to leading order by binormal curvature flow. Our approach combines new estimates on the distance of the corresponding Hamiltonian-Possion structures with stability estimates recently developed in Ref. 15.

preprint2016arXiv

A Positive and Entropy-Satisfying Finite Volume Scheme for the Baer-Nunziato Model

We present a relaxation scheme for approximating the entropy dissipating weak solutions of the Baer-Nunziato two-phase flow model. This relaxation scheme is straightforwardly obtained as an extension of the relaxation scheme designed in [16] for the isentropic Baer-Nunziato model and consequently inherits its main properties. To our knowledge, this is the only existing scheme for which the approximated phase fractions, phase densities and phase internal energies are proven to remain positive without any restrictive condition other than a classical fully computable CFL condition. For ideal gas and stiffened gas equations of state, real values of the phasic speeds of sound are also proven to be maintained by the numerical scheme. It is also the only scheme for which a discrete entropy inequality is proven, under a CFL condition derived from the natural sub-characteristic condition associated with the relaxation approximation. This last property, which ensures the non-linear stability of the numerical method, is satisfied for any admissible equation of state. We provide a numerical study for the convergence of the approximate solutions towards some exact Riemann solutions. The numeric

535 works