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Computer-assisted proofs in PDE: a survey (Q2321033)

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scientific article; zbMATH DE number 7098970

Language	Label	Description	Also known as
English	Computer-assisted proofs in PDE: a survey	scientific article; zbMATH DE number 7098970

Statements

scholarly article

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Computer-assisted proofs in PDE: a survey (English)

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Javier Gómez-Serrano

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S\(\vec{\text{e}}\)MA Journal

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publication date

28 August 2019

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full work available at URL

https://arxiv.org/abs/1810.00745

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zbMATH Keywords

PDE

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computer-assisted

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singularity

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incompressible

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describes a project that uses

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MaRDI profile type

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Well-posedness of two-phase Hele–Shaw flow without surface tension

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Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation

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Integration of Dissipative Partial Differential Equations: A Case Study

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Families of Periodic Solutions for Some Hamiltonian PDEs

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Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flow

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Performance of the Taylor series method for ODEs/DAEs

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Duchon-Robert solutions for the Rayleigh-Taylor and Muskat problems

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Local solvability and turning for the inhomogeneous Muskat problem

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New methods for high-dimensional verified quadrature

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Nonuniqueness of Weak Solutions to the SQG Equation

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Motions of vortex patches

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Global well-posedness for the two-dimensional Muskat problem with slope less than 1

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Rigorous numerics for ill-posed PDEs: periodic orbits in the Boussinesq equation

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Infinite energy solutions of the surface quasi-geostrophic equation

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Breakdown of smoothness for the Muskat problem

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Splash singularities for the one-phase Muskat problem in stable regimes

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Structural stability for the splash singularities of the water waves problem

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Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves

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Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations

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Uniformly rotating analytic global patch solutions for active scalars

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Global Smooth Solutions for the Inviscid SQG Equation

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Uniformly rotating smooth solutions for the incompressible 2D Euler equations

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Remarks on geometric properties of SQG sharp fronts and \(\alpha\)-patches

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The quasi-geostrophic equation in the Triebel Lizorkin spaces

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Well-posedness of the Muskat problem with \(H^2\) initial data

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On the Muskat problem: Global in time results in 2D and 3D

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On the global existence for the Muskat problem

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Global regularity for 2D Muskat equations with finite slope

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New numerical results for the surface quasi-geostrophic equation

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Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar

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Uniqueness for SQG patch solutions

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Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation

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Growth of solutions for QG and 2D Euler equations

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Almost sharp fronts for the surface quasi-geostrophic equation

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Evidence of singularities for a family of contour dynamics equations

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Contour dynamics of incompressible 3-D fluids in a porous medium with different densities

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A maximum principle for the Muskat problem for fluids with different densities

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Absence of squirt singularities for the multi-phase Muskat problem

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A note on stability shifting for the Muskat problem

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A note on stability shifting for the Muskat problem. II: From stable to unstable and back to stable

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The confined Muskat problem: differences with the deep water regime

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Global well-posedness for the 2D stable Muskat problem in $H^{3/2}$

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Non-splat singularity for the one-phase Muskat problem

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On the splash and splat singularities for the one-phase inhomogeneous Muskat problem

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A quasi-solution approach to nonlinear problems—the case of the Blasius similarity solution

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Analytical Approximation of the Blasius Similarity Solution with Rigorous Error Bounds

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Bifurcation from simple eigenvalues

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Rigorous Numerics for Global Dynamics: A Study of the Swift--Hohenberg Equation

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Doubly connected V-states for the generalized surface quasi-geostrophic equations

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An analytical and numerical study of steady patches in the disc

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Doubly connected \(V\)-states for the planar Euler equations

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An a posteriori KAM theorem for whiskered tori in Hamiltonian partial differential equations with applications to some ill-posed equations

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On the Two‐Dimensional Muskat Problem with Monotone Large Initial Data

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Imperfect bifurcation for the quasi-geostrophic shallow-water equations

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Stability of vortices in equilibrium with a cylinder

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On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results

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Relativistic stability of matter. I

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The spine of an SQG almost-sharp front

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Almost sharp fronts for SQG: the limit equations

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Aperiodicity of the Hamiltonian flow in the Thomas-Fermi potential

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Numerical Computations and Computer Assisted Proofs of Periodic Orbits of the Kuramoto--Sivashinsky Equation

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A Framework for the Numerical Computation and A Posteriori Verification of Invariant Objects of Evolution Equations

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A computer-assisted proof of the existence of traveling wave solutions to the scalar Euler equations with artificial viscosity

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A computer-assisted proof of the existence of solutions to a boundary value problem with an integral boundary condition

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Nonlinear stability of the Muskat problem with capillary pressure at the free boundary

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Homotopy hyperbolic 3-manifolds are hyperbolic

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A Posteriori Verification of Invariant Objects of Evolution Equations: Periodic Orbits in the Kuramoto--Sivashinsky PDE

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Existence for the \(\alpha \)-patch model and the QG sharp front in Sobolev spaces

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A survey for the Muskat problem and a new estimate

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On the Muskat problem with viscosity jump: global in time results

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Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem

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Non uniform rotating vortices and periodic orbits for the two-dimensional Euler equations

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On the existence of stationary patches

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On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof

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Global Existence for the Confined Muskat Problem

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Smooth Travelling-Wave Solutions to the Inviscid Surface Quasi-Geostrophic Equation

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A proof of the Kepler conjecture

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An algorithm to generate canonical transformations: Application to normal forms

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The parameterization method for invariant manifolds. From rigorous results to effective computations

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A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: rigorous results

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A Parameterization Method for the Computation of Invariant Tori and Their Whiskers in Quasi‐Periodic Maps: Explorations and Mechanisms for the Breakdown of Hyperbolicity

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Double bubbles minimize

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On the V-states for the generalized quasi-geostrophic equations

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Bifurcation of rotating patches from Kirchhoff vortices

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Degenerate bifurcation of the rotating patches

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Existence of corotating and counter-rotating vortex pairs for active scalar equations

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Boundary regularity of rotating vortex patches

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On rotating doubly connected vortices

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Existence of small loops in a bifurcation diagram near degenerate eigenvalues

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Numerical Software with Result Verification

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Self-similar singularity of a 1D model for the 3D axisymmetric Euler equations

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Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic

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A Software Package for the Numerical Integration of ODEs by Means of High-Order Taylor Methods

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A Simple Energy Pump for the Surface Quasi-geostrophic Equation

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Two adaptive Gauss-Legendre type algorithms for the verified computation of definite integrals

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A computer-assisted proof of the Feigenbaum conjectures

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Existence theorems for the 2D quasi-geostrophic equation with plane wave initial conditions

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An efficient and general numerical method to compute steady uniform vortices

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Existence and regularity of weak solutions to the quasi-geostrophic equations in the spaces \(L^p\) or \(\dot{H}^{-1/2}\)

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The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results

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An Approach to The Numerical Verification of Solutions for Nonlinear Elliptic Problems With Local Uniqueness

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Global flows with invariant measures for the inviscid modified SQG equations

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NUMERICAL VERIFICATION METHODS FOR SOLUTIONS OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

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Taylor forms -- use and limits.

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Large time decay estimates for the Muskat equation

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Local-existence for the inhomogeneous Muskat problem

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Relative equilibria with holes for the surface quasi-geostrophic equations

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Motivations for an arbitrary precision interval arithmetic and the MPFI library

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On the evolution of sharp fronts for the quasi‐geostrophic equation

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Verification methods: Rigorous results using floating-point arithmetic

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Obtuse Triangular Billiards II: One Hundred Degrees Worth of Periodic Trajectories

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A scenario for finite-time singularity in the quasigeostrophic model

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Global existence, singular solutions, and ill-posedness for the Muskat problem

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Global dynamics and fast indicators

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A rigorous ODE solver and Smale's 14th problem

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Stationary Coexistence of Hexagons and Rolls via Rigorous Computations

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Rigorous numerics in dynamics

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Validation of the bifurcation diagram in the 2D Ohta–Kawasaki problem

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Steady-state solutions of the Euler equations in two dimensions: Rotating and translating V-states with limiting cases. I: Numerical algorithms and results

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Quasi-geostrophic-type equations with initial data in Morrey spaces

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Solutions of the 2D quasi-geostrophic equation in Hölder spaces

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Numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element

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Rigorous numerics for dissipative partial differential equations. II: Periodic orbit for the Kuramoto-Sivashinsky PDE -- a computer-assisted proof

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Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation

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Identifiers

Mathematics Subject Classification ID

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zbMATH DE Number

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10.1007/S40324-019-00186-X

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Sitelinks

Mathematics(1 entry)

mardi Publication:2321033

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