SIAM Journal on Mathematical Analysis, Volume 52, Issue 4, Page 3131-3148, January 2020. The Cauchy problem for the Yang--Mills system in three space dimensions with data in Fourier--Lebesgue spaces $\widehat{H}^{s,r}$ , $1 < r \le 2$ , is shown to be locally well-posed, where we have to assume only almost optimal minimal regularity for the data with respect to scaling as $r \to 1$ . This is true despite

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 2198-2200, January 2020. We correct the proof of Proposition 2.1 in the paper [S. Hittmeir and A. Jüngel, SIAM J. Math. Anal., 43 (2011), pp. 997--1022]. This proposition is used to prove the existence of global weak solutions to a Keller--Segel model with additional cross diffusion.

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 2179-2197, January 2020. The convective transport in a multicomponent isothermal compressible fluid subject to the mass continuity equations is considered. The velocity is proportional to the negative pressure gradient, according to Darcy's law, and the pressure is defined by a state equation imposed by the volume extension of the mixture

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 2158-2178, January 2020. We investigate a system of nonlocal transport equations in one spatial dimension. The system can be regarded as a model for the three-dimensional Euler equations in the hyperbolic flow scenario. We construct blowup solutions with control over the vorticity up to the blowup time. For a wide class of initial data

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 2134-2157, January 2020. We discuss the well-posedness of the “transient eddy current" magneto-quasi-static approximation of Maxwell's initial value problem with bounded and measurable conductivity, with sources, on a domain. We prove the existence and uniqueness of weak solutions, and we provide global Hölder estimates for the magnetic

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 2098-2133, January 2020. We study a stochastic system of $N$ interacting particles which models bimolecular chemical reaction-diffusion. In this model, each particle $i$ carries two attributes: the spatial location $X_t^i\in \mathbb{T}^d$, and the type $\Xi_t^i\in \{1,\ldots,n\}$. While $X_t^i$ is a standard (independent) diffusion process

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 2081-2097, January 2020. We consider weak solutions of the spatially inhomogeneous Landau equation with hard potentials ($\gamma \in (0,1]$), under the assumption that mass, energy, and entropy densities are under control. In this regime, with arbitrary initial data, we show that solutions satisfy pointwise Gaussian upper and lower bounds

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 2041-2080, January 2020. The fully parabolic Keller--Segel system is coupled to the incompressible Navier--Stokes equations through transport and buoyancy. It is shown that when posed with no-flux/no-flux/Dirichlet boundary conditions in smoothly bounded planar domains and along with appropriate assumptions on regularity of the initial

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 2000-2040, January 2020. Synchronization phenomena are ubiquitous in strongly correlated oscillatory systems, and the Kuramoto model serves as a prototype synchronization model for phase-coupled oscillators. In this paper, we consider its general type, called the Kuramoto--Daido model, and provide local sensitivity analysis for the Kuramoto--Daido

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1980-1999, January 2020. We consider the liquid drop model for nuclei interacting with a neutralizing homogeneous background of electrons. The regime we are interested in is when the fraction between the electronic and the nuclear charge densities is small. We show that in this dilute limit the thermodynamic ground state energy is given

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1943-1979, January 2020. We introduce a new definition of viscosity solution to path-dependent partial differential equations, which is a slight modification of the definition introduced in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204--236]. With the new definition, we prove the two important results, until now missing in the literature

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1903-1942, January 2020. A two species interacting system motivated by the density functional theory for triblock copolymers contains a long range interaction that affects the two species differently. In a two species periodic assembly of discs, the two species appear alternately on a lattice. A minimal two species periodic assembly is

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 3114-3130, January 2020. We show that the Kružkov solution of the Cauchy problem for a scalar conservation law in one spatial dimension propagates regulated initial data into regulated solutions at later times. The proof is based on standard front-tracking and an extension, due to Fraňková, of Helly's selection principle.

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 3093-3113, January 2020. In this paper, we find that the linearized collision operator $L$ of the non-cutoff Boltzmann equation with soft potential generates a strongly continuous semigroup on $H^k_n$, with $k,n\in\mathbb{R}$. In the theory of the Boltzmann equation without angular cutoff, the weighted Sobolev space plays a fundamental

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 3073-3092, January 2020. We develop a method of obtaining a hierarchy of new higher-order entropies in the context of compressible models with local and nonlocal diffusion and isentropic pressure. The local viscosity is allowed to degenerate as the density approaches vacuum. The method provides a tool to propagate initial regularity of

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 3052-3072, January 2020. We study the properties of the Hopf--Lax formula restricted to convex functions and provide a characterization of the optimal transfer plan for weak transport problems on the real line. On $n$ dimensional real space, we also provide a sufficient condition on the potential function such that the optimal plan of

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 3039-3051, January 2020. Let $f: (0,1)^d \rightarrow \mathbb{R}$ be a continuous function with zero mean and interpret $f_{+} = \max(f, 0)$ and $f_{-} = -\min(f, 0)$ as the densities of two measures. We prove that if the cost of transport from $f_{+}$ to $f_{-}$ is small, in terms of the Wasserstein distance $W_1 (f_+ , f_-)$, then the

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 3004-3038, January 2020. In bounded domains, the regularity of the solutions to boundary value problems depends on the geometry, and on the coefficients that enter into the definition of the model. This is in particular the case for the time-harmonic Maxwell equations, whose solutions are the electromagnetic fields. In this paper, emphasis

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2969-3003, January 2020. We prove that the initial value problem for the Dirac equation $(-i\gamma^\mu \partial_\mu + m) \psi= ( \frac{e^{- |x|}}{|x|} \ast ( \overline \psi \psi)) \psi \quad \text{in } \ \mathbb{R}^{1+3}$ is globally well-posed and the solution scatters to free waves asymptotically as $t \rightarrow \pm \infty$ if we start

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2945-2968, January 2020. In a plane polygon $P$ with straight sides, we prove analytic regularity of the Leray--Hopf solution of the stationary, viscous, and incompressible Navier--Stokes equations. We assume small data, analytic volume force, and no-slip boundary conditions. Analytic regularity is quantified in so-called countably normed

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2930-2944, January 2020. The study of positivity of solutions to the Boltzmann equation goes back to [T. Carleman, Acta Math., 60 (1933), pp. 91--146], and the initial argument of Carleman was developed in [A. Pulvirenti and B. Wennberg, Comm. Math. Phys., 183 (1997), pp. 145--160; C. Mouhot, Comm. Partial Differential Equations, 30 (2005)

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2895-2929, January 2020. The time evolution of a collisionless plasma is modeled by the relativistic Vlasov--Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We consider the case that the plasma is located in a bounded container $\Omega\subset\mathbb R^3$, for example

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2874-2894, January 2020. We consider the nonlinear Schrödinger equation with focusing power-type nonlinearity on compact graphs with at least one terminal edge, i.e., an edge ending with a vertex of degree 1. On the one hand, we introduce the associated action functional, and we provide a profile description of positive low action solutions

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2849-2873, January 2020. We suggest a new way of defining optimal transport of positive-semidefinite matrix-valued measures. It is inspired by a recent rendering of the incompressible Euler equations and related conservative systems as concave maximization problems. The main object of our attention is the Kantorovich--Bures metric space

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2803-2848, January 2020. This paper is concerned with the one-dimensional version of a specific member of the (abcd) family of Boussinesq systems having the higher possible dispersion and eigenvalues with nontrivial zeros. We will establish two different long time existence results for the solutions of the Cauchy problem. The first result

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2759-2802, January 2020. We consider dynamical transport metrics for probability measures on discretizations of a bounded convex domain in $\mathbb R^d$. These metrics are natural discrete counterparts to the Kantorovich metric $\mathbb{W}_2$, defined using a Benamou--Brenier-type formula. Under mild assumptions we prove an asymptotic

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2734-2758, January 2020. We consider the inverse problem for the general transport equation with external field, source term, and absorption coefficient. We show that the source and the absorption coefficients can be uniquely reconstructed from the boundary measurement in a Lipschitz stable manner. Specifically, the uniqueness and stability

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2689-2733, January 2020. This work is concerned with model reduction of stochastic differential equations and builds on the idea of replacing drift and noise coefficients of preselected relevant, e.g., slow variables by their conditional expectations. We extend recent results by Legoll and Leli\` evre [Nonlinearity 23 (2010), pp. 2131--2165]

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2655-2688, January 2020. In this article we study an inverse problem for the space-time fractional parabolic operator $(\partial_t-\Delta)^s+Q$ with $0

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2624-2654, January 2020. In this paper, we study the generation of interface for the solution of the mass conserved Allen--Cahn equation involving a nonlocal integral term. We show that, for a rather general class of initial functions that are independent of $\varepsilon$, the solution generally develops a steep transition layer of thickness

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2592-2623, January 2020. We prove a blow-up criterion in terms of an $L_2$-bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve diffusion flow, which has free boundary points supported on a line. The evolving

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2561-2591, January 2020. We prove a local variant of Einstein's formula for the effective viscosity of dilute suspensions, that is, $\mu^\prime=\mu ({{1+\frac 5 2\phi+o(\phi)}})$, where $\phi$ is the volume fraction of the suspended particles. Up to now rigorous justifications have only been obtained for dissipation functionals of the

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2531-2560, January 2020. In this paper we study pattern formation for a physical local/nonlocal interaction functional where the local attractive term is given by the 1-perimeter and the nonlocal repulsive term is the Yukawa (or screened Coulomb) potential. This model is physically interesting as it is the $\Gamma$-limit of a double Yukawa

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2491-2530, January 2020. In this paper, we show uniqueness and stability for the piecewise-smooth solutions, with a single discontinuity, to the Burgers--Hilbert equation constructed in Bressan and Zhang [Commun. Math. Sci., 15 (2017), pp. 165--184]. The Burgers--Hilbert equation is $u_t+(\frac{u^2}{2})_x={H}[u]$, where ${H}$ is the Hilbert

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2463-2490, January 2020. This paper is concerned with the application of time-domain boundary integral methods to a nonstationary boundary value problem for the thermo-elasto-dynamic equations, based on the Lubich approach via the Laplace transform. Fundamental solutions of the transformed thermo-elasto-dynamic equations are constructed

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2411-2462, January 2020. This paper is concerned with the Cauchy problem $u_{t}=u_{xx}+f(t,u),x\in \mathbb{R} ,t>0$ with initial function $u(0,\cdot)=u_0 (\cdot)\in L^{\infty} (\mathbb{R})$, where $f$ is a rather general nonlinearity that is periodic in $t$, and satisfies $f(\cdot,0)\equiv 0$ and that the corresponding ODE has a positive

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2383-2410, January 2020. We study the Cauchy problem for nonlinear, nonlocal operators that may be degenerate. Our general framework includes cases where the jump intensity is allowed to depend on the values of the solution itself, e.g., the porous medium equation with the fractional Laplacian and the parabolic fractional $p$-Laplacian

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2353-2382, January 2020. We regard the Cauchy problem for a particular Whitham--Boussinesq system modeling surface waves of an inviscid incompressible fluid layer. We are interested in well-posedness at a very low level of regularity. We derive dispersive and Strichartz estimates and implement them together with a fixed point argument

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2313-2352, January 2020. In this paper, we consider the existence and asymptotic behavior of the fermionic quantum BGK model, which is a relaxation model of the quantum Boltzmann equation for fermions. More precisely, we establish the existence of unique classical solutions and their exponentially fast stabilization when the initial data

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2275-2312, January 2020. We study the asymptotic behavior of solutions of a class of linear nonlocal measure valued differential equations with time delay. Our main result states that the solutions asymptotically exhibit a parabolic-like behavior in the large time, that is precisely expressed in terms of the heat kernel. Our proof relies

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2237-2274, January 2020. In this paper, we prove the existence of martingale solutions to the stochastic heat equation taking values in a Riemannian manifold, which admits the Wiener (Brownian bridge) measure on the Riemannian path (loop) space as an invariant measure using a suitable Dirichlet form. Using the Andersson--Driver approximation

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2201-2236, January 2020. In order to study the asymptotic behavior of a fluid in a domain of small thickness $\varepsilon$, it is common to use that the norm of the pressure $p_\varepsilon$ in $L^q$, $q>1$, is smaller than $C\|\nabla p_\varepsilon\|_{W^{-1,q}}/\varepsilon$. Our purpose in the present paper is to improve this estimate by

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1884-1902, January 2020. In the paper we consider the linear inverse problem that consists in recovering the initial state in a first order evolution equation generated by a skew-adjoint operator. We studied the well-posedness of the inversion in terms of the observation operator and the spectra of the skew-adjoint generator. The stability

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1864-1883, January 2020. We consider a class of the first order hyperbolic systems of PDEs that can be written as moments of a kinetic equation. We construct smooth, local solutions to the Cauchy problem using a discrete time, transport-projection splitting algorithm for the dual kinetic density. The method represents an alternative to

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1844-1863, January 2020. In this paper, we discuss a general approach to finding periodic solutions bifurcating from equilibrium points of classical Vlasov systems. The main access to the problem is chosen through the Hamiltonian representation of any Vlasov system, first put forward in [J. Fröhlich, A. Knowles, and S. Schwarz, Comm. Math

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1806-1843, January 2020. This paper is devoted to the global large solutions to the 3 dimensional compressible Navier--Stokes equations in the critical Besov spaces with initial data satisfying a nonlinear smallness condition. Here the “large solutions" mean that the any component of the initial velocity could be arbitrarily large. Moreover

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1786-1805, January 2020. This paper is concerned with the analysis of surface polariton resonance for nanoparticles in linear elasticity. With the presence of nanoparticles, we first derive the perturbed displacement field associated to a given elastic source field. It is shown that the leading-order term of the perturbed elastic wave

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1761-1785, January 2020. We identify a large class of objects---dissipative measure-valued (DMV) solutions to the Navier--Stokes--Fourier system---in which the strong solutions are stable. More precisely, a DMV solution coincides with the strong solution emanating from the same initial data as long as the latter exists. The DMV solutions

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1729-1760, January 2020. The question of well- and ill-posedness of entropy admissible solutions to the multi-dimensional systems of conservation laws has been studied recently in the case of isentropic Euler equations. In this context special initial data were considered, namely the 1D Riemann problem which is extended trivially to a

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1690-1728, January 2020. In this paper, we give a rigorous justification of the point vortex approximation to the family of modified surface quasi-geostrophic (mSQG) equations globally in time in both the inviscid and vanishing dissipative cases. This result completes the justification for the remaining range of the mSQG family unaddressed

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1639-1689, January 2020. The paper is dedicated to the asymptotic investigation of textiles as an elasticity problem on beam structures. The structure is subjected to a simultaneous homogenization and dimension reduction with respect to the asymptotic behavior of the beams' thickness and periodicity. Important for the problem are the contact

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1591-1638, January 2020. In this paper, we study a macroscopic description on the ensemble of Kuramoto oscillators with finite inertia in a random media characterized by a white noise. In a mesoscopic regime, it is well known that the dynamics of a large Kuramoto ensemble in a random media is governed by the Kuramoto--Sakaguchi--Fokker--Planck

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1567-1590, January 2020. This paper considers a system of two conservation laws with a Lipschitz continuous flow and deals with solutions to the Cauchy problems at a node. The system considered is the two-phase traffic model, proposed in [R. M. Colombo, F. Marcellini, and M. Rascle, SIAM J. Appl. Math., 70 (2010), pp. 2652--2666]. We are

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1549-1566, January 2020. This paper is devoted to an analysis of singular perturbations of inequality problems. For a general variational-hemivariational inequality, it is shown rigorously that under appropriate conditions, as the singular perturbation parameter approaches zero, the solution of the singularly perturbed problem converges

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1526-1548, January 2020. We consider the recovery of a source term $f(x,t)=p(x)q(t)$ for the nonhomogeneous heat equation in $\Omega\times (0,\infty)$ where $\Omega$ is a bounded domain in $\mathbb{R}^2$ with smooth boundary $\partial\Omega$ from overposed lateral data on a sparse subset of $\partial\Omega\times(0,\infty)$. Specifically

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1489-1525, January 2020. In this work we give optimal, i.e., necessary and sufficient, conditions for integrals of the calculus of variations to guarantee the existence of solutions---both weak and variational solutions---to the associated $L^2$-gradient flow. The initial values are merely $L^2$ functions with possibly infinite energy

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1463-1488, January 2020. In this paper, we are concerned with the system of Euler equations with time-dependent damping like $-\frac{\mu}{(1+t)^\lambda}u$ for physical parameters $\lambda\ge 0$ and $\mu>0$. It is well known that, when $0\le \lambda<1$, the time-asymptotic-degenerate damping plays the key role which makes the damped Euler

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1427-1462, January 2020. The mean-field analysis of a multipopulation agent-based model is performed. The model couples a particle dynamics driven by a nonlocal velocity with a Markov-type jump process on the probability that each agent has of belonging to a given population. A general functional analytic framework for the well-posedness

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1386-1426, January 2020. We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable traveling wave solutions to the deterministic system retain their orbital stability if the amplitude of the noise is sufficiently small. By applying a stochastic phase-shift

SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1363-1385, January 2020. We study the energy balance for the weak solutions of the three-dimensional compressible Navier--Stokes equations in a bounded domain. By constructing a global mollification combined with an independent boundary cut-off and then taking a double limit to prove the convergence of the resolved energy, we establish