SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5892-5993, January 2020. We study the long-time asymptotic behavior of the solution $q(x,t) $, $x\in\mathbb{R}$, $t\in\mathbb{R}^+$, of the modified Korteweg--de Vries equation (MKdV) $q_t+6q^2q_x+q_{xxx}=0$ with step-like initial datum $\scriptsize q(x,0)\to \Big\{\begin{array}{@{}l@{}l@{}} c_-\quad& {for $x\to-\infty$},\\ c_+\quad& {for

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5865-5891, January 2020. We study the system $ (*) \Big\{\!\!\begin{array}{c} u_t = D_1 \Delta u - \chi_1 \nabla \cdot (u \nabla v) + u(\lambda_1 - \mu_1 u + a_1 v) \\ v_t = D_2 \Delta v + \chi_2 \nabla \cdot (v \nabla u) + v(\lambda_2 - \mu_2 v - a_2 u) \end{array} $ (inter alia) for $D_1, D_2, \chi_1, \chi_2, \lambda_1, \lambda_2, \mu_1

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5840-5864, January 2020. We consider a parabolic-parabolic Keller--Segel system in a ball of $ \mathbb{R}^N $ under the Neumann boundary condition. This was introduced as a model of aggregation of bacteria. The aggregation is mathematically defined as finite-time blowup. When $ N = 2 $, an optimal criterion for finite-time blowup was obtained

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5792-5839, January 2020. We introduce a new class of models for emergent dynamics. It is based on a new communication protocol which incorporates two main features: short-range kernels which restrict the communication to local metric balls, and anisotropic communication kernels, adapted to the local density in these balls, which form topological

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5775-5791, January 2020. In the present paper we extend the $L^2$ Korn interpolation and second inequalities in thin domains, proven in [D. Harutyunyan, SIAM J. Math. Anal., 50 (2018), pp. 4964--4982], to the space $L^p$ for any $1

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5748-5774, January 2020. We investigate the large-time behavior of strong solutions to a two-phase fluid model in the whole space $\mathbb R^3$. This model was first derived by Choi [SIAM J. Math. Anal., 48 (2016), pp. 3090--3122] by taking the hydrodynamic limit from the Vlasov--Fokker--Planck/isentropic Navier--Stokes equations with

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5703-5747, January 2020. We present a rigorous framework for the local analysis of canards and slow passages through bifurcations in a wide class of infinite-dimensional dynamical systems with time-scale separation. The framework is applicable to models where an infinite-dimensional dynamical system for the fast variables is coupled to

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5675-5702, January 2020. On simple geodesic disks of constant curvature, we derive new functional relations for the geodesic X-ray transform, involving a certain class of elliptic differential operators whose ellipticity degenerates normally at the boundary. We then use these relations to derive sharp mapping properties for the X-ray transform

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5658-5674, January 2020. This paper studies a nonstrictly hyperbolic system of conservation laws in one space dimension which has been proposed in the literature as mathematically natural and as a model of magnetohydrodynamic (MHD) turbulence. It is shown that the associated Riemann problem possesses a one-parameter family of different

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5627-5657, January 2020. We study the inverse problem, or inverse design problem, for a time-evolution Hamilton--Jacobi equation. More precisely, given a target function $u_T$ and a time horizon $T>0$, we aim to construct all the initial conditions for which the viscosity solution coincides with $u_T$ at time $T$. As is common in this

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5598-5626, January 2020. We consider a Caratheodory differential equation with a state-dependent convex constraint that changes BV-continuously in time (a perturbed BV-continuous state-dependent sweeping process). By setting up an appropriate catching-up algorithm we prove solvability of the initial value problem. Then, for sweeping processes

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5554-5597, January 2020. We study sampling of functions $f$ and their images $Af$ under Fourier integral operators $A$ at rates $sh$ with $s$ fixed and $h$ a small parameter. We show that the Nyquist sampling limit of $Af$ and $f$ are related by the canonical relation of $A$ using semiclassical analysis. We apply this analysis to the Radon

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5533-5553, January 2020. We study the semiclassical limit from the time-dependent Hartree equation with Coulomb or gravitational potential to the Vlasov--Poisson equation. We prove convergence in trace norm for mixed states under a technical assumption on the solution of the Vlasov--Poisson equation. We exhibit a special class of mixed

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5500-5532, January 2020. In this contribution, we study the existence and uniqueness of nonlocal transport equations. The term “nonlocal” refers to the fact that the flux function's derivative will be integrated over a neighborhood of the corresponding space-time coordinate. We will demonstrate existence and uniqueness of weak solutions

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5467-5499, January 2020. We consider the fixed angle inverse scattering problem and show that a compactly supported potential is uniquely determined by its scattering amplitude for two opposite fixed angles. We also show that almost symmetric or horizontally controlled potentials are uniquely determined by their fixed angle scattering

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5441-5466, January 2020. The ability to manipulate the propagation of waves on subwavelength scales is important for many different physical applications. In this paper, we consider a honeycomb-lattice of subwavelength resonators and prove, for the first time, the existence of a Dirac dispersion cone at subwavelength scales. As shown in

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5422-5440, January 2020. We study the solution to a nonlinear stochastic heat equation in $d\geq 3$. The equation is driven by a Gaussian multiplicative noise that is white in time and smooth in space. For a small coupling constant, we prove (i) that the solution converges to the stationary distribution in large time and (ii) that the

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5389-5421, January 2020. We are interested in the description of small modulations in time and space of wave-train solutions to the complex Ginzburg--Landau equation $\partial_T \Psi = (1+ i \alpha) \partial_X^2 \Psi + \Psi - (1+i \beta ) \Psi |\Psi|^2$ near the Eckhaus boundary, that is, when the wave train is near the threshold of its

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5363-5388, January 2020. In this paper, we consider steady Euler flows in a planar bounded domain in which the vorticity is sharply concentrated in a finite number of disjoint regions of small diameter. Such flows are closely related to the point vortex model and can be regarded as desingularization of point vortices. By an adaption of

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5339-5362, January 2020. We consider an interacting particle system modeled as a system of $N$ stochastic differential equations driven by Brownian motions. We prove that the (mollified) empirical process converges, uniformly in time and space variables, to the solution of the two-dimensional Navier--Stokes equation written in vorticity

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 5287-5337, January 2020. Transonic shocks play a pivotal role in design of supersonic nozzles in gas dynamics. Previous studies have shown that for stationary compressible Euler flows in rectilinear ducts with constant cross-sections, they are not stable under perturbations of upcoming supersonic flows and back pressures at the exits of

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 5257-5286, January 2020. In this article we study the limit when $\alpha \to 0$ of solutions to the $\alpha$-Euler system in the half-plane, with no-slip boundary conditions. We establish the existence of subsequences converging to a weak solution of the 2D incompressible Euler equations, assuming nonnegative initial vorticities in the

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 5232-5256, January 2020. We study increasing stability in the inverse scattering source problem for the Helmholtz equation and the classical Lamé system in the three dimensional space from boundary data at multiple wave numbers. As additional data for source identification we use pressure or displacement at the boundary of the reference

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 5213-5231, January 2020. This paper concerns time-harmonic inverse source problems with a single far-field pattern in two dimensions, where the source term is compactly supported in an a priori given inhomogeneous background medium. For convex-polygonal source terms, we prove that the source support together with the zeroth and first-order

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 5196-5212, January 2020. The authors consider the problem of recovering an observable from certain measurements containing random errors. The observable is given by a pseudodifferential operator while the random errors are generated by a Gaussian white noise. The authors show how wave packets can be used to partially recover the observable

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 5155-5195, January 2020. We consider a class of sixth-order Cahn--Hilliard-type equations with logarithmic potential. This system is closely connected to some important phase-field models relevant in different applications, for instance, the functionalized Cahn--Hilliard equation that describes phase separation in mixtures of amphiphilic

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 5121-5154, January 2020. Supersonic flows for the two-dimensional (2D) steady full Euler system are studied. We construct a global nonisentropic rotational supersonic flow in a semi-infinite divergent duct. The flow satisfies the slip condition on the walls of the duct, and the state of the flow is given at the inlet of the duct. The solution

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 5090-5120, January 2020. This paper deals with the combined incompressible quasineutral limit of the weak martingale solution of the compressible Navier--Stokes--Poisson system perturbed by a stochastic forcing term in the whole space. In the framework of ill-prepared initial data, we show the convergence in law to a weak martingale solution

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 5066-5089, January 2020. Analytic global bifurcation theory is used to construct a large variety of families of steady periodic two-dimensional gravity water waves with real-analytic vorticity distributions, propagating in an incompressible fluid. The waves that are constructed can possess an arbitrary number of interior stagnation points

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 5036-5065, January 2020. In this paper, we shall study the parabolic-elliptic Keller--Segel system on the Poincaré disk model of the two-dimensional-hyperbolic space. We shall investigate how the negative curvature of this Riemannian manifold influences the solutions of this system. As in the two-dimensional Euclidean case, under the subcritical

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 5001-5035, January 2020. This paper rigorously establishes the stabilization effect of a background magnetic field on electrically conducting fluids, a phenomenon that has been widely observed in physical experiments and numerical simulations. This study is based on a 2 dimensional (2D) magnetohydrodynamic (MHD) system in which the velocity

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4971-5000, January 2020. We prove that the time-harmonic solutions to Maxwell's equations in a three-dimensional exterior domain converge to a certain static solution as the frequency tends to zero. We work in weighted Sobolev spaces and construct new compactly supported replacements for Dirichlet--Neumann fields. Moreover, we even show

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4937-4970, January 2020. We provide a rigorous mathematical study of an asymptotic model describing Darcy flow with free boundary in a small amplitude/large wavelength approximation. In particular, we prove several well-posedness results in critical spaces. Furthermore, we also study how the solution decays towards the flat equilibrium

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4900-4936, January 2020. We consider the spinless Pauli--Fierz Hamiltonian which describes a quantum system of nonrelativistic identical particles coupled to the quantized electromagnetic field. We study its time evolution in a mean-field limit where the number $N$ of charged particles gets large while the coupling to the radiation field

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4861-4899, January 2020. We study the two-dimensional water waves problem with surface tension in the case when there is a nonzero contact angle between the free surface and the bottom. In the presence of surface tension, dissipations take place at the contact point. Moreover, when the contact angle is less than $\frac{\pi}{6}$, no singularity

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4806-4860, January 2020. We prove an integral representation theorem for the $\mathrm{L}^1$-relaxation of the functional $\mathcal{F}\colon u\mapsto\int_\Omega f(x,u(x),\nabla u(x))\;\mathrm{d} x,\quad u\in\mathrm{W}^{1,1}(\Omega;\mathbb{R}^m)$, where $\Omega\subset\mathbb{R}^d$ ($d \geq 2$) is a bounded Lipschitz domain, to the space

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4783-4805, January 2020. It is well known that the optimal transport problem on the real line for the classical distance cost may not have a unique solution. In this paper we recover uniqueness by considering the transport problems where the costs are a power smaller than one of the distance and letting this parameter tend to one. A complete

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4751-4782, January 2020. We study the uncertainty principles related to the generalized Logan problem in $\mathbb{R}^{d}$. We define $\lambda(f)=\sup\{|x|\colon f(x)>0\}$ and $\tau(f)=\sup \{|x|\colon x\in \mathrm{supp}\,\widehat{f}\,\}$. One of our main results provides the complete solution of the following problem: for a fixed $m=0

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4705-4750, January 2020. In its original formulation the Krein matrix was used to locate the spectrum of first-order star-even polynomial operators where both operator coefficients are nonsingular. Such operators naturally arise when considering first-order-in-time Hamiltonian PDEs. Herein the matrix is reformulated to allow for operator

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4638-4704, January 2020. We construct global solutions on a full measure set with respect to the Gibbs measure for the one-dimensional cubic fractional nonlinear Schrödinger (FNLS) equation with weak dispersion $(-\partial_x^2)^{\alpha/2}$, $\alpha<2$, by quite different methods, depending on the value of $\alpha$. We show that if $\alpha>\frac{6}{5}$

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4616-4637, January 2020. In this paper, we establish the short time inviscid limit of the incompressible Navier--Stokes equations with critical Navier-slip boundary conditions for analytic data on half-space, a boundary condition that is physically derived from the hydrodynamic limit of the Boltzmann equations with the Maxwell boundary

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4565-4615, January 2020. This paper investigates the Keller--Segel model with quadratic cellular diffusion over a disk in $\mathbb R^2$ with a focus on the formation of its nontrivial patterns. We obtain explicit formulas of radially symmetric stationary solutions and such configurations give rise to the ring patterns and concentric airy

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4524-4564, January 2020. The long time behavior of solutions to stochastic porous media equations with nonlinear multiplicative noise on bounded domains with Dirichlet boundary data is studied. Based on weighted $L^{1}$-estimates, the existence and uniqueness of invariant measures with optimal bounds on the rate of mixing are proved. Along

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4506-4523, January 2020. We study stability of reconstruction in current density impedance imaging (CDII), that is, the inverse problem of recovering the conductivity of a body from the measurement of the magnitude of the current density vector field in the interior of the object. Our results show that CDII is stable with respect to errors

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4469-4505, January 2020. In this paper we prove Krylov's boundary gradient type estimates (and regularity) for solutions to fully nonlinear differential inequalities with unbounded coefficients and quadratic growth on the gradient with $C^{1,dini}$ boundary data. This means that drift coefficients and the right-hand side (RHS) are in $L^{q}$

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4421-4468, January 2020. In this paper, we rigorously justify the connection between the Qian--Sheng inertial $Q$-tensor model and the full Ericksen--Leslie model for the liquid crystal flow. By using the Hilbert expansion method, we prove that when the elastic coefficients tend to zero (also called the uniaxial limit), the solution to

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4391-4420, January 2020. We investigate how damping the lowest Fourier mode modifies the dynamics of the cubic Szegö equation. We show that there is a nonempty open subset of initial data generating trajectories with high Sobolev norms tending to infinity. In addition, we give a complete picture of this phenomenon on a reduced phase space

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4362-4390, January 2020. We study two quasi-static frictional contact problems on an infinite time interval for a viscoelastic material with long memory and a viscoplastic material, respectively, and with the constitutive laws described by functions which have a nonpolynomial growth. The contact condition is modeled by a subdifferential

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4314-4361, January 2020. In this work we study global well-posedness and large time behavior for a typical reaction-diffusion system, which include degenerate diffusion, and whose nonlinearities arise from chemical reactions. We show that there is an indirect diffusion effect, i.e., an effective diffusion for the nondiffusive species which

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4284-4313, January 2020. In this work we investigate some asymptotic properties of an age-structured Lotka--Volterra model, where a specific choice of the functional parameters allows us to formulate it as a delayed problem, for which we prove the existence of a unique coexistence equilibrium and characterize the existence of a periodic

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4238-4283, January 2020. The two-dimensional Benney--Luke equation is an isotropic model which describes long water waves of small amplitude in 3 dimensions whereas the KP-II equation is a unidirectional model for long waves with slow variation in the transverse direction. In the case where the surface tension is weak or negligible, linear

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4185-4237, January 2020. In this article, we prove global well-posedness and scattering for the defocusing quintic nonlinear Schrödinger equation on the cylinder $\mathbb{R} \times \mathbb{T}$ in $H^1$. We establish an infinite vector-valued linear profile decomposition in $L^2_x h^\alpha$, $0 < \alpha \le 1$, motivated by the linear profile

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4161-4184, January 2020. We study state-constraint static Hamilton--Jacobi equations in a sequence of domains $\{\Omega_k\}_{k \in \Bbb{N}}$ in $\Bbb{R}^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k\in \Bbb{N}$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4140-4160, January 2020. We discuss a Steklov-type problem for Maxwell's equations which is related to an interior Calderón operator and an appropriate Dirichlet-to-Neumann map. The corresponding Neumann-to-Dirichlet map turns out to be compact, and this provides a Fourier basis of Steklov eigenfunctions for the associated energy spaces

SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4101-4139, January 2020. In recent years there has been an emerging interest in PDE-like flows defined on finite graphs, with applications in clustering and image segmentation. In particular for image segmentation and semisupervised learning Bertozzi and Flenner [Multiscale Model. Simul., 10 (2012), pp. 1090--1118] developed an algorithm

SIAM Journal on Mathematical Analysis, Volume 52, Issue 4, Page 4068-4100, January 2020. We prove the sharp regularizing estimates for the gain term of the Boltzmann collision operator, including hard sphere, hard potential, and Maxwell molecule models. Our new estimates characterize both the regularization and the convolution properties of the gain term and have the following features. The regularizing

SIAM Journal on Mathematical Analysis, Volume 52, Issue 4, Page 4022-4067, January 2020. Following the global method for relaxation we prove an integral representation result for a large class of variational functionals naturally defined on the space of functions with bounded deformation. Mild additional continuity assumptions are required on the functionals.

SIAM Journal on Mathematical Analysis, Volume 52, Issue 4, Page 3990-4021, January 2020. A number of practically important imaging problems involve inverting the generalized Radon transform (GRT) ${\mathcal R}$ of a function $f$ in $\mathbb{R}^3$. On the other hand, not much is known about the spatial resolution of the reconstruction from discretized data. In this paper we study how accurately and

SIAM Journal on Mathematical Analysis, Volume 52, Issue 4, Page 3962-3989, January 2020. The long time behavior of solutions to the autonomous PDE system describing fluid diffusion in a viscoelastic porous medium with capillary hysteresis is studied with homogeneous Dirichlet conditions for the displacement of the solid and homogeneous Neumann boundary conditions for the capillary pressure. Although

SIAM Journal on Mathematical Analysis, Volume 52, Issue 4, Page 3881-3961, January 2020. The vortex method is a common numerical and theoretical approach used to implement the motion of an ideal flow, in which the vorticity is approximated by a sum of point vortices, so that the Euler equations read as a system of ordinary differential equations. Such a method is well justified in the full plane, thanks