arXiv subjects

Steve Shkoller

Publications and source records attributed to Steve Shkoller.

At least 37 records · Page 2Linked to original sources

arXiv2019

Formation of shocks for 2D isentropic compressible Euler

We consider the 2D isentropic compressible Euler equations, with pressure law $p(ρ) = (\sfrac{1}γ) ρ^γ$, with $γ>1$. We provide an elementary constructive proof of shock formation from smooth initial datum of finite energy, with no vacuum regions, and with {nontrivial vorticity}. We prove that for initial data which has minimum slope $- {\sfrac{1}{ \eps}}$, for $ \eps>0$ taken sufficiently small relative to the $\OO(1)$ amplitude, there exist smooth solutions to the Euler equations which form a shock in time $\OO(\eps)$. The blowup time and location can be explicitly computed and solutions at the blowup time are of cusp-type, with Hölder $C^ {\sfrac{1}{3}}$ regularity. Our objective is the construction of solutions with inherent $\OO(1)$ vorticity at the shock. As such, rather than perturbing from an irrotational regime, we instead construct solutions with dynamics dominated by purely azimuthal wave motion. We consider homogenous solutions to the Euler equations and use Riemann-type variables to obtain a system of forced transport equations. Using a transformation to modulated self-similar variables and pointwise estimates for the ensuing system of transport equations, we show the global stability, in self-similar time, of a smooth blowup profile.

math.AP↗

arXiv2019

Global existence of near-affine solutions to the compressible Euler equations

We establish global existence of solutions to the compressible Euler equations, in the case that a finite volume of ideal gas expands into vacuum. Vacuum states can occur with either smooth or singular sound speed, the latter corresponding to the so-called physical vacuum singularity when the enthalpy vanishes on the vacuum wave front like the distance function. In this instance, the Euler equations lose hyperbolicity and form a degenerate system of conservation laws, for which a local existence theory has only recently been developed. Sideris found a class of expanding finite degree-of-freedom global-in-time affine solutions, obtained by solving nonlinear ODEs. In three space dimensions, the stability of these affine solutions, and hence global existence of solutions, was established by Hadžić \& Jang with the pressure-density relation $p = ρ^γ$ with the constraint that $1< γ\le {\frac{5}{3}} $. They asked if a different approach could go beyond the $γ> {\frac{5}{3}} $ threshold. We provide an affirmative answer to their question, and prove stability of affine flows and global existence for all $γ>1$, thus also establishing global existence for the shallow water equations when $γ=2$.

math.AP↗

arXiv2019

A space-time smooth artificial viscosity method with wavelet noise indicator and shock collision scheme, Part 2: the 2-D case

This is the second part to our companion paper [18]. Herein, we generalize to two space dimensions the C-method developed in [20,18] for adding localized, space-time smooth artificial viscosity to nonlinear systems of conservation laws that propagate shock waves, rarefaction waves, and contact discontinuities. For gas dynamics, the C-method couples the Euler equations to scalar reaction-diffusion equations, which we call C-equations, whose solutions serve as space-time smooth artificial viscosity indicators for shocks and contacts. We develop a high-order numerical algorithm for gas dynamics in 2-D which can accurately simulate the Rayleigh-Taylor (RT) instability with Kelvin-Helmholtz (KH) roll-up of the contact discontinuity, as well as shock collision and bounce-back. We implement both directionally isotropic and anisotropic artificial viscosity schemes, the latter adding diffusion only in directions tangential to the evolving front. We additionally produce a novel shock collision indicator function, which naturally activates during shock collision, and then smoothly deactivates. Moreover, we implement a high-frequency 2-D wavelet-based noise detector together with an efficient and localized noise removal algorithm. We provide numerical results for some classical 2-D test problems, including the RT problem, the Noh problem, a circular explosion problem from the Liska and Wendroff [13] review paper, the Sedov blast wave problem, the double Mach 10 reflection test, and a shock-wall collision problem. In particular, we show that our artificial viscosity method can eliminate the wall-heating phenomenon for the Noh problem, and thereby produce an accurate, non-oscillatory solution, even though our simplified WENO-type scheme fails to run for this problem.

physics.comp-ph↗

arXiv2019

A space-time smooth artificial viscosity method with wavelet noise indicator and shock collision scheme, Part 1: the 1-D case

In this first part of two papers, we extend the C-method developed in [40] for adding localized, space-time smooth artificial viscosity to nonlinear systems of conservation laws that propagate shock waves, rarefaction waves, and contact discontinuities in one space dimension. For gas dynamics, the C-method couples the Euler equations to a scalar reaction-diffusion equation, whose solution $C$ serves as a space-time smooth artificial viscosity indicator. The purpose of this paper is the development of a high-order numerical algorithm for shock-wall collision and bounce-back. Specifically, we generalize the original C-method by adding a new collision indicator, which naturally activates during shock-wall collision. Additionally, we implement a new high-frequency wavelet-based noise detector together with an efficient and localized noise removal algorithm. To test the methodology, we use a highly simplified WENO-based discretization scheme. We show that our scheme improves the order of accuracy of our WENO algorithm, handles extremely strong discontinuities (ranging up to nine orders of magnitude), allows for shock collision and bounce back, and removes high frequency noise. The causes of the well-known "wall heating" phenomenon are discussed, and we demonstrate that this particular pathology can be effectively treated in the framework of the C-method. This method is generalized to two space dimensions in the second part of this work [41].

physics.comp-ph↗

arXiv2018

Rigorous Asymptotic Models of Water Waves

We develop a rigorous asymptotic derivation for two mathematical models of water waves that capture the full nonlinearity of the Euler equations up to quadratic and cubic interactions, respectively. Specifically, letting epsilon denote an asymptotic parameter denoting the steepness of the water wave, we use a Stokes expansion in epsilon to derive a set of linear recursion relations for the tangential component of velocity, the stream function, and the water wave parameterization. The solution of the water waves system is obtained as an infinite sum of solutions to linear problems at each epsilon^k level, and truncation of this series leads to our two asymptotic models, that we call the quadratic and cubic h-models. Using the growth rate of the Catalan numbers (from number theory), we prove well-posedness of the h-models in spaces of analytic functions, and prove error bounds for solutions of the h-models compared against solutions of the water waves system. We also show that the Craig-Sulem models of water waves can be obtained from our asymptotic procedure and that their WW2 model is well-posed in our functional framework. We then develop a novel numerical algorithm to solve the quadratic and cubic h-models as well as the full water waves system. For three very different examples, we show that the agreement between the model equations and the water waves solution is excellent, even when the wave steepness is quite large. We also present a numerical example of corner formation for water waves.

math.AP↗

arXiv2016

Well-posedness and decay to equilibrium for the Muskat problem with discontinuous permeability

We first prove local-in-time well-posedness for the Muskat problem, modeling fluid flow in a two-dimensional inhomogeneous porous media. The permeability of the porous medium is described by a step function, with a jump discontinuity across the fixed-in-time curve $(x_1,-1+f(x_1))$, while the interface separating the fluid from the vacuum region is given by the time-dependent curve $(x_1,h(x_1,t))$. Our estimates are based on a new methodology that relies upon a careful study of the PDE system, coupling Darcy's law and incompressibility of the fluid, rather than the analysis of the singular integral contour equation for the interface function $h$. We are able to develop an existence theory for any initial interface given by $h_0 \in H^2$ and any permeability curve-of-discontinuity that is given by $f \in H^{2.5}$. In particular, our method allows for both curves to have (pointwise) unbounded curvature. In the case that the permeability discontinuity is the set $f=0$, we prove global existence and decay to equilibrium for small initial data. This decay is obtained using a new energy-energy dissipation inequality that couples tangential derivatives of the velocity in the bulk of the fluid with the curvature of the interface. To the best of our knowledge, this is the first global existence result for the Muskat problem with discontinuous permeability.

math.AP↗

arXiv2016

Nonuniqueness of weak solutions to the SQG equation

We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 in the survey arXiv:1111.2700 by De Lellis and Székelyhidi Jr. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian.

math.AP↗

arXiv2016

A model for Rayleigh-Taylor mixing and interface turn-over

We first develop a new mathematical model for two-fluid interface motion, subjected to the Rayleigh-Taylor (RT) instability in two-dimensional fluid flow, which in its simplest form, is given by $ h_{tt}(α,t) = A g\, Λh - \fracσ{ρ^++ρ^-} Λ^3 h - A \partial_α(H h_t h_t) $, where $Λ= H \partial_ α$ and $H$ denotes the Hilbert transform. In this so-called $h$-model, $A$ is the Atwood number, $g$ is the acceleration, $ σ$ is surface tension, and $ρ^\pm$ denotes the densities of the two fluids. Under a certain stability condition, we prove that this so-called $h$-model is both locally and globally well-posed. Numerical simulations of the $h$-model show that the interface can quickly grow due to nonlinearity, and then stabilize when the lighter fluid is on top of the heavier fluid and acceleration is directed downward. In the unstable case of a heavier fluid being supported by the lighter fluid, we find good agreement for the growth of the mixing layer with experimental data in the "rocket rig" experiment of Read of Youngs. We then derive an RT interface model with a general parameterization $z(α,t)$ such that $ z_{tt}= Λ\bigg{[}\frac{A}{|\partial_αz|^2}H\left(z_t\cdot (\partial_αz)^\perp H(z_t\cdot (\partial_αz)^\perp)\right) + A g z_2 \bigg{]} \frac{(\partial_αz)^\perp}{|\partial_αz|^2} +z_t\cdot (\partial_αz)^\perp\left(\frac{(\partial_αz_t)^\perp}{|\partial_αz|^2}-\frac{(\partial_αz)^\perp 2(\partial_αz\cdot \partial_αz_t)}{|\partial_αz|^4}\right)$. This more general RT $z$-model allows for interface turn-over. Numerical simulations of the $z$-model show an even better agreement with the predicted mixing layer growth for the "rocket rig" experiment.

math.AP↗

arXiv2016

Local well-posedness and Global stability of the Two-Phase Stefan problem

The two-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition such as ice melting to water. This is accomplished by solving the heat equation on a time-dependent domain, composed of two regions separated by an a priori unknown moving boundary which is transported by the difference (or jump) of the normal derivatives of the temperature in each phase. We establish local-in-time well-posedness and a global-in-time stability result for arbitrary sufficiently smooth domains and small initial temperatures. To this end, we develop a higher-order energy with natural weights adapted to the problem and combine it with Hopf-type inequalities. This extends the previous work by Hadzic and Shkoller [31,32] on the one-phase Stefan problem to the setting of two-phase problems, and simplifies the proof significantly.

math.AP↗

arXiv2016

Well-posedness for the classical Stefan problem and the zero surface tension limit

We develop a framework for a unified treatment of well-posedness for the Stefan problem with or without surface tension. In the absence of surface tension, we establish well-posedness in Sobolev spaces for the classical Stefan problem. We introduce a new velocity variable which extends the velocity of the moving free-boundary into the interior domain. The equation satisfied by this velocity is used for the analysis in place of the heat equation satisfied by the temperature. Solutions to the classical Stefan problem are then constructed as the limit of solutions to a carefully chosen sequence of approximations to the velocity equation, in which the moving free-boundary is regularized and the boundary condition is modified in a such a way as to preserve the basic nonlinear structure of the original problem. With our methodology, we simultaneously find the required stability condition for well-posedness and obtain new estimates for the regularity of the moving free-boundary. Finally, we prove that solutions of the Stefan problem with positive surface tension $σ$ converge to solutions of the classical Stefan problem as $σ\to 0$.

math.AP↗

arXiv2016

On the impossibility of finite-time splash singularities for vortex sheets

In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. By means of elementary arguments, we prove that such a singularity cannot occur in finite time for vortex sheet evolution, i.e. for the two-phase incompressible Euler equations. We prove this by contradiction; we assume that a splash singularity does indeed occur in finite time. Based on this assumption, we find precise blow-up rates for the components of the velocity gradient which, in turn, allow us to characterize the geometry of the evolving interface just prior to self-intersection. The constraints on the geometry then lead to an impossible outcome, showing that our assumption of a finite-time splash singularity was false.

math.AP↗

arXiv2015

A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary

We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has played an important role in prior work on the non-relativistic compressible Euler equations. Our main result is the derivation, relative to Lagrangian (also known as co-moving) coordinates, of local-in-time a priori estimates for the solution. The solution features a fluid-vacuum boundary, transported by the fluid four-velocity, along which the hyperbolicity of the equations degenerates. In this context, the relativistic Euler equations are equivalent to a degenerate quasilinear hyperbolic wave-map-like system that cannot be treated using standard energy methods.

math.AP↗

arXiv2015

Regularity of the velocity field for Euler vortex patch evolution

We consider the vortex patch problem for both the 2-D and 3-D incompressible Euler equations. In 2-D, we prove that for vortex patches with $H^{k-0.5}$ Sobolev-class contour regularity, $k \ge 4$, the velocity field on both sides of the vortex patch boundary has $H^k$ regularity for all time. In 3-D, we establish existence of solutions to the vortex patch problem on a finite-time interval $[0,T]$, and we simultaneously establish the $H^{k-0.5}$ regularity of the two-dimensional vortex patch boundary, as well as the $H^k$ regularity of the velocity fields on both sides of vortex patch boundary, for $k \ge 3$.

math.AP↗

arXiv2015

Solvability and regularity for an elliptic system prescribing the curl, divergence, and partial trace of a vector field on Sobolev-class domains

We provide a self-contained proof of the solvability and regularity of a Hodge-type elliptic system, wherein the divergence and curl of a vector field are prescribed in an open, bounded, Sobolev-class domain, and either the normal component or the tangential components of the vector field are prescribed on the boundary. The proof is based on a regularity theory for vector elliptic equations set on Sobolev-class domains and with Sobolev-class coefficients.

math.AP↗

arXiv2015

On the splash singularity for the free-surface of a Navier-Stokes fluid

In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. We prove that for $d$-dimensional flows, $d=2$ or $3$, the free-surface of a viscous water wave, modeled by the incompressible Navier-Stokes equations with moving free-boundary, has a finite-time splash singularity. In particular, we prove that given a sufficiently smooth initial boundary and divergence-free velocity field, the interface will self-intersect in finite time.

math.AP↗

arXiv2015

Global stability of steady states in the classical Stefan problem

The classical one-phase Stefan problem (without surface tension) allows for a continuum of steady state solutions, given by an arbitrary (but sufficiently smooth) domain together with zero temperature. We prove global-in-time stability of such steady states, assuming a sufficient degree of smoothness on the initial domain, but without any a priori restriction on the convexity properties of the initial shape. This is an extension of our previous result [28] in which we studied nearly spherical shapes.

math.AP↗

arXiv2014

Well-posedness of the Muskat problem with $H^2$ initial data

We study the dynamics of the interface between two incompressible fluids in a two-dimensional porous medium whose flow is modeled by the Muskat equations. For the two-phase Muskat problem, we establish global well-posedness and decay to equilibrium for small $H^2$ perturbations of the rest state. For the one-phase Muskat problem, we prove local well-posedness for $H^2$ initial data of arbitrary size. Finally, we show that solutions to the Muskat equations instantaneously become infinitely smooth.

math.AP↗

arXiv2013

Global stability and decay for the classical Stefan problem

The classical one-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition, such as ice melting to water. This is accomplished by solving the heat equation on a time-dependent domain whose boundary is transported by the normal derivative of the temperature along the evolving and a priori unknown free-boundary. We establish a global-in-time stability result for nearly spherical geometries and small temperatures, using a novel hybrid methodology, which combines energy estimates, decay estimates, and Hopf-type inequalities.

math.AP↗