Papers

The Immersed Boundary Method has long served as a robust computational framework for fluid-structure interactions, yet the rigorous analysis of 1D Peskin filaments anchored to rigid boundaries remains sparse. In this paper, we generalize the classical Peskin problem to the half-plane by considering an elastic filament whose endpoints are anchored to a no-slip wall. Moving beyond the algebraic complexity of the traditional Blake image system, we utilize the boundary-symmetric formulation of Gimbutas, Greengard, and Veerapaneni. This representation allows for a transparent decomposition of the hydrodynamic interactions into a free space principal part and a regularizing reflected component without resorting to hypersingular integral operators. Through this framework, we prove that the leading-order evolution of the anchored filament is governed by a fractional Laplacian equipped with homogeneous Dirichlet boundary conditions. We characterize the stationary states of the system, proving that all equilibria are circular arcs connecting the anchor points, a result that holds for a broad class of elastic energy densities. By framing the non-local dynamics in weighted little Hölder spaces, we establish local well posedness and prove that the filament exhibits instantaneous $C^\infty$ regularization in both space and time. This work provides a rigorous analytical foundation for anchored filaments in bounded domains and suggests a spectrally accurate numerical path for simulating tethered biological structures.

We study stochastic differential games with $N$ players, where interactions are determined by sequences of graphs in which the number of neighbours of each node remains bounded as $N$ grows, such as chain graphs or lattices. Our main goal is to quantify the phenomenon of the "unimportance of distant players" in such a large population, sparse regime: we show that, in order to determine the optimal trajectory in open-loop strategies of a given player with an arbitrarily small error, it suffices to consider a reduced game involving only the players at a certain distance in the graph, assigning arbitrary trajectories to the farther ones. Our main result provides an explicit non-asymptotic estimate in terms of the graph distance, valid independently of the time horizon $T$, under suitable convexity and monotonicity assumptions on the costs. Similar results are obtained for games in distributed strategies.

This thesis is devoted to the study of hyperbolic differential operators on globally hyperbolic manifolds, linear gauge theories and their quantisation. In the first part, we treat the Cauchy problem for symmetric hyperbolic systems and normally hyperbolic operators on globally hyperbolic manifolds from first principles. Although hyperbolic equations are usually studied with local interactions, there are strong motivations from several areas of mathematical physics to consider also nonlocal interactions. As an intermezzo, we therefore take a small deviation from the classical local theory and prove well-posedness of the Cauchy problem for symmetric hyperbolic systems coupled to a broad class of nonlocal potentials. The second part presents a detailed exposition of linear gauge theories in globally hyperbolic spacetimes. Linear gauge theories are yet another deviation from the concept of hyperbolicity: the corresponding equations of motion are generically non-hyperbolic; however, can always be reduced to a constrained hyperbolic dynamics once an appropriate gauge fixing procedure has been applied. We give a thorough analysis of their Cauchy problem and classical phase space, complemented by a detailed discussion of many examples of physical interest, and discuss their quantisation following the algebraic approach to quantum field theory. The final chapter is devoted to the quantisation of Maxwell's theory on globally hyperbolic spacetimes, with the goal of proving the existence of Hadamard states. The novelty of our approach lies in a new gauge-fixing procedure at the level of initial data, which allows us to suppress the unphysical degrees of freedom. This gauge is achieved by means of a new Hodge decomposition for differential k-forms in Sobolev spaces on complete Riemannian manifolds, while states are constructed using tools from pseudodifferential calculus.

Cell invasion and spatial pattern formation are two distinct manifestations of cellular self-organisation in development, regeneration, and disease. Here, we develop and analyse a unified theoretical framework that links these two seemingly different behaviours within a single mechanistic model for adhesion-mediated self-organisation in growing cell populations. Using a multiscale analysis, we show that the balance between cell-cell adhesion, self-diffusion, and proliferation controls the emergence of distinct collective dynamics. We find that for weak adhesion, tissues invade through stable monotone fronts. As adhesion increases, invasion slows, fronts become unstable, leading to aggregates and spatial patterns emerging behind the advancing edge. In two spatial dimensions, these instabilities generate fingering morphologies reminiscent of dysregulated invasion in cancer. Crucially, we show that density-dependent regulation of adhesion suppresses these instabilities and restores cohesive tissue expansion. Together, our results identify adhesion strength and its regulation as key determinants of whether tissues invade cohesively or fragment into patterns, and provide a unified framework for understanding collective migration, morphogenesis, and dysregulated growth.

This paper is a continuation of the work in \cite{kimzhang2024} concerning Hele-Shaw flow with both drift and source terms. We prove that, in a local neighborhood, if the free boundary is Lipschitz continuous with a sufficiently small Lipschitz constant, then the free boundary is $C^{1}$. As a corollary, we also consider the 2D vertical Hele-Shaw (or one-phase Muskat) problem with an advection term. We show that, provided the initial data and the advection term are small and the propagation speed is large, the free boundary becomes uniformly $C^1$ after a finite time.

This paper considers the homogeneous Neumann initial-boundary value problem for Alopecia Areata chemotaxis model with weakly singular sensitivity. For any appropriately regular initial conditions,it is shown that the problem admits a global boundedness of classical solutions in two spatial dimensions. Moreover, through the explicit construction of Lyapunov functions, we establish that the globally bounded solution converges exponentially to a constant steady state. The paper concludes with numerical experiments that serve to visually illustrate and corroborate some of the theoretically derived findings.

We study the entropy solution for a class of systems of nonlocal conservation laws in which the convective flux is convoluted with a kernel in both spatial and temporal variables. This formulation models the flux dependence on the solution within its spatial neighbourhood (nonlocal in space) as well as on prior states in time (nonlocal in time), thereby incorporating memory effects. In addition, employing a convergent finite volume approximation, the existence of the entropy solution is discussed. The uniqueness of such entropy solutions is also established. In addition, we analyze the asymptotic behaviour of the solutions as the support of the temporal convolution kernel shrinks, demonstrating the "memory-to-memoryless" effect and convergence to the entropy solution of the corresponding nonlocal conservation law without memory (i.e., nonlocal only in space). Convergence rate estimates are derived. In addition, the proposed numerical approximations are shown to be asymptotically compatible with this passage to the memoryless limit by deriving the corresponding asymptotic convergence rate estimates. The analysis is carried out in a very general setting, without imposing any geometric restrictions such as the convexity of the spatial and temporal convolution kernels, unlike the existing literature on the asymptotic analysis of nonlocal-in-space only conservation laws. To the best of our knowledge, this provides the first convergence and asymptotic analysis for finite volume schemes applied to nonlocal conservation laws with memory. Numerical experiments are included to illustrate the theory.

We consider the free boundary problem for relativistic plasma–vacuum interfaces in two and three spatial dimensions. The plasma flow is governed by the equations of ideal relativistic magnetohydrodynamics, while the vacuum magnetic and electric fields satisfy Maxwell's equations. The plasma and vacuum magnetic fields are tangential to the interface, which moves with the plasma flow. This yields a nonlinear, multidimensional hyperbolic problem with a free boundary that is characteristic of variable multiplicity. We identify a quantitative stability condition and establish the linear stability of three-dimensional relativistic plasma–vacuum interfaces in the sense that the variable-coefficient linearized problem satisfies energy estimates in anisotropic Sobolev spaces. In estimating tangential derivatives, we exploit an intrinsic cancellation effect to convert the boundary term into an instant integral. We then separate the estimate involving spatial derivatives from that involving time derivatives, so that the instant integral can be mainly absorbed by the instant energy under the stability condition. Moreover, we prove the local-in-time existence and uniqueness of solutions to the nonlinear problem in two-dimensional space, provided that the plasma and vacuum magnetic fields do not vanish simultaneously at any point of the initial interface. The proof combines the solvability and tame estimate of the linearized problem with a suitable modified Nash–Moser iteration. In particular, to establish its solvability, the two-dimensional linearized problem is reduced to a transport equation for the interface function and a hyperbolic boundary problem with maximally nonnegative boundary conditions.

We consider a system of two coupled first-order linear hyperbolic partial differential equations modeling heat transport in a counter-flow heat exchanger: one equation describes the transport of a hot fluid, and the other the transport of a cold fluid in the opposite direction. For this system, we design a boundary observer that uses only the temperature of the cold fluid measured at one boundary. Our approach is spectral: by assigning the spectrum of the operator governing the observation error dynamics to a prescribed region within the open left-half complex plane, we can freely tune the convergence rate of the observation error to zero in the $L^2$ norm. The main technical contribution is the proof that spectral stability, that is, the location of the spectrum in the open left-half plane, is equivalent to $L^2$ exponential stability of the origin for the observation error dynamics. This equivalence is established by showing that the operator governing the observation error dynamics satisfies the so-called spectral mapping property.

Lutwak's affine quermassintegral theory is a foundational component of modern affine Brunn–Minkowski theory. Developed in the 1980s, it provides affine analogues of the classical quermassintegrals and has led to a rich family of sharp affine isoperimetric inequalities. A central question in this program, going back to Lutwak's 1988 work, is an Alexandrov–Fenchel-type monotonicity principle for the normalized $L^{-n}$-moment quermassintegrals $I_{k,-n}$. In one form, this principle predicts that \[ I_{m,-n}(K)^{1/m}\ge I_{k,-n}(K)^{1/k}, \qquad 1\le m<k\le n . \] The question was recorded in Gardner's 2006 book Geometric Tomography as part of its problem list, and the comparison with the top dimension, $k=n$, was established by Milman and Yehudayoff in their 2023 JAMS paper. We show that the proposed monotonicity does not persist in the full range. More precisely, for every triple of integers $m,k,n$ satisfying $1\le m<k\le n-1$ and $n>(m+2)(k+2)-2$, there exists an origin-symmetric $C^2_+$ convex body $K\subset\mathbb R^n$ such that \[ I_{m,-n}(K)^{1/m} < I_{k,-n}(K)^{1/k}. \] The example is obtained from the Euclidean ball by an arbitrarily small degree-four spherical harmonic perturbation. On the positive side, we prove that the endpoint chain is true in dimension three: for every convex body $K\subset\mathbb R^3$, \[ I_{1,-3}(K)\ge I_{2,-3}(K)^{1/2}\ge I_{3,-3}(K)^{1/3}=1. \] The equality cases in both non-trivial inequalities are exactly ellipsoids, up to translation and nonsingular affine transformations.

This work is concerned with the probabilistic representation of solutions to the $p$-Laplace evolution equation $\frac{\partial u}{\partial t}={\rm div}(|\nabla u|^{p-2}\nabla u)$ in $(0,\infty)\times\mathbb{R}^d$, $u(0,x)=u_0(x),$ $x\in\mathbb{R}^d$. One proves that, if $p\geq 4$, and if $u_0$ is a probability density with compact support and $u_0\in L^2$, $|\nabla u_0|\in L^\infty$, then $u$ can be represented as $u(t,x)dx=\mathscr L_{X(t)}(dx)$, where $\mathscr L_{X(t)}$ denotes the time marginal law of $X$ at time $t$ with $X$ being a probabilistically weak solution to a corresponding McKean-Vlasov stochastic differential equation. This result is based on a new second order global regularity result for the weak solutions to the parabolic $p$-Laplace equation.

We study a thermodynamically consistent diffuse-interface model that describes the motion of two macroscopically immiscible, incompressible, and viscous Newtonian fluids with unmatched densities. This model is compatible with continuum mixture theory. It adopts a mass-averaged (barycentric) velocity so that the two-phase flow is quasi-incompressible: the velocity is no longer divergence-free, and the pressure enters the equation of the chemical potential. For the initial-boundary value problem in $\mathbb{T}^3$ with a class of physically relevant singular free energy densities, we prove the existence of global-in-time weak solutions. The proof relies on a suitable reduction of the original system to a Korteweg-type fluid model combined with a two-layer approximation, together with delicate estimates for the mass density and the phase-field variable inspired by the celebrated Bresch-Desjardins entropy. A key observation is that capillarity at the free interface provides a damping effect on the density evolution. For the limiting procedure, we derive delicate tail estimates to exclude possible concentrations of the singular potential, since no integrability of the pressure is available a priori. This work appears to be the first existence result for the Navier-Stokes/Cahn-Hilliard type system with unmatched densities and mass-averaged velocity without spatial regularization.

In the previous work [Interfaces Free Bound., 19, 351–369, 2017], de Queiroz and Shahgholian established the optimal $C^{1,\log}_{\mathrm{loc}}$ regularity of solutions for the obstacle problem with singular logarithmic forcing term $$-\Delta u = \log u\,\chi_{\{u>0\}} \quad \text{in } \Omega,$$ where $\Omega\subset\mathbb{R}^d$ ($d\geq 2$) is a smooth bounded domain. In our earlier work [arXiv:2408.08104, 2024], we proved the $C^{1,\alpha}$ regularity of the free boundary $\Omega\cap\partial\{u>0\}$ near regular points. In this paper, we investigate the more delicate structure of the singular free boundary. Since the nonlinearity $-\log u$ is singular near the free boundary and destroys the scaling invariance, so that neither the classical blow-up arguments nor the standard epiperimetric inequality [Weiss, Invent.\ Math., 138, 23–50, 1999] apply directly; moreover, the Weiss type monotonicity formula requires a variable-parameter correction that introduces non-integrable remainder terms into the energy estimates. Motivated by Colombo–Spolaor–Velichkov [Geom.\ Funct.\ Anal., 28, 1029–1061, 2018], we develop a new log-epiperimetric inequality for the modified Weiss energy, also proved by the direct method. A key novelty is the introduction of an auxiliary correction term $T$ that absorbs the non-integrable errors. As consequences, we establish a logarithmic energy decay, uniqueness of blow-ups at singular points, and a $C^{1,\log}$-type geometric description of the singular strata. In dimension two, the logarithmic modulus improves to a Hölder modulus.

We address Calderón's problem of stably determining the anisotropic complex admittivity $\sigma$ in a domain $\Omega\subset\mathbb{R}^n$, with $n\geq3$, representing a conducting medium, in terms of a Dirichlet-to-Neumann map locally prescribed on a non-empty portion $\Sigma$ of the boundary of $\Omega$, $\partial\Omega$. $\sigma$ is assumed to be of type $\sigma(\cdot)=A(\cdot,a(\cdot))$ in $\Omega$, where the one-parameter family of complex-symmetric matrices $[\lambda^{-1},\:\lambda]\ni t\mapsto A(\cdot,\: t)$ is assumed to be a-priori known and the scalar function $a$ is unknown. We establish Lipschitz and Hölder stability estimates at the boundary for $\sigma$ and its derivatives of arbitrary order on $\Sigma$, respectively, in terms of the local map.

In this work, we develop a new biological transmission model for Chagas disease. This model, set in two juxtaposed habitats with skew Brownian motion conditions at the interface, is composed of two reaction–diffusion equations and takes into account the sylvatic transmission. We write it as an abstract perturbed Cauchy problem using operator theory. Then, we show that the main operator, which models the dispersal process, generates an analytic semigroup in an adequate Banach space.

Motivated by the elastic-viscous-plastic (EVP) sea-ice model [E. C. Hunke and J. K. Dukowicz, J. Phys. Oceanogr., 27, 9 (1997), 1849–1867], which is used in large-scale numerical climate simulations, we proposed in [D. W. Boutros, X. Liu, M. Thomas and E. S. Titi, arXiv:2505.03080 (2025)] the use of the inviscid Voigt regularisation for the constitutive (stress-tensor) relation and proved the global well-posedness of the resulting model. The EVP model treats sea ice as a non-Newtonian fluid. In turn, elastic-viscous-plastic solids often involve a Kelvin-Voigt viscosity in terms of the strain rate. Therefore, in the present work we formulate an elastic-viscous-plastic sea-ice model with a Kelvin-Voigt regularisation in terms of the strain rate. In other words, we introduce the Voigt regularisation in the momentum balance rather than in the constitutive relation (for the stress tensor). We then prove the local well-posedness for the Kelvin-Voigt EVP model with the advection term, in the momentum balance, and the global well-posedness in the absence of the advection term (following a very standard approximation in the latter case). A crucial component of the proof of these results, is a new $L^\infty$-estimate for the stress tensor which relies on the damping structure. Note that, both with and without the advection term, we are able to handle the case of viscosity coefficients without a cutoff from above, which remains a major open problem for the closely related Hibler sea-ice model. We are also able to prove the existence of solutions for much less regular initial data compared to our previous paper on the Voigt-EVP model.

For the boundary value problem $$\left\{ \begin{array}{rcll} -\Delta_p u+u^{p-1}&=&|x|^{\alpha}u^{q-1}&\mbox{in }\Omega,\\ \frac{\displaystyle\partial u}{\displaystyle\partial{\bf n}}&=&0&\mbox{on }\partial \Omega, \end{array}\right. $$ in the unit ball $\Omega$, we investigate the properties of the positive radial solution. It is known, that for $1<p<n$, $\frac{(n-1)p}{n-p}<q<\frac{np}{n-p}$ and sufficiently large $\alpha$ this solution does not provide global minimum to the corresponding energy functional, see [M. Gazzini, E. Serra, 2008] for $p=2$ and [A.P. Shcheglova, 2018] in general case. Nevertheless, it is shown in [M. Gazzini, E. Serra, 2008] that for $n\ge 4$, $p=2$, $2<q<\frac{2n}{n-2}$ and sufficiently large $\alpha$ the radial solution is at least a local minimizer of the energy functional. We partially generalize this result. Namely, let $n\ge4$ and let $p>2$ be sufficiently close to $2$. Then for all $p<q<\frac{np}{n-p}$, for sufficiently large $\alpha$ the second variation of the energy functional is positive. The same holds true for all $2<p<n$ if $q>p$ is sufficiently close to $p$.

We study the asymptotic behavior at infinity of solutions to the supercritical Lagrangian mean curvature equation \[ \sum_{i=1}^n \arctan \lambda_i(D^2u)=\theta+f(x) \] on exterior domains in \(\mathbb R^n\), \(n\ge 2\), where \(|\theta|>((n-2)\pi)/2\). The perturbation \(f\) is assumed to be locally Lipschitz near infinity and to satisfy a decay condition with rate \(\beta>0\). The main new ingredient is a scale-dependent difference quotient argument, combined with a nonlocal potential method, which avoids differentiating \(f\) twice and yields quantitative Hessian convergence under only Lipschitz regularity. We establish optimal asymptotic expansions in all dimensions and for all decay rates \(\beta>0\), including the critical logarithmic cases. This improves previous results requiring higher regularity of \(f\) and faster decay in \cite{BJ2026}.

Let $(M^4, g, f)$ be a four-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f= \frac{1}{2}g$. If its scalar curvature is $1$, Cheng-Zhou \cite{Cheng-Zhou} proved that it is a finite quotient of $\mathbb{R}^2\times \mathbb{S}^2$. In this note we present an alternative proof by analyzing the asymptotic geometry at infinity.

We consider the inverse scattering problem for inhomogeneous media of compact support governed by the fractional s-Helmholtz equation, with $0<s<1$, in dimensions $d=1,2,3$. In particular, we study the determination of the support of the inhomogeneity from the far-field pattern of the scattered field generated by plane waves for all incident directions at a fixed frequency. The far-field pattern is defined as the principal term in the asymptotic expansion of the scattered field at infinity. It is shown in \cite{zilberberg2026limiting} that, up to a multiplicative constant, this coincides with the far-field pattern corresponding to the classical Helmholtz equation with the same inhomogeneity. Our approach is based on the development of the factorization method, which not only leads to an efficient and easily implementable reconstruction algorithm, but also provides a uniqueness result for determining the support of an admissible set of inhomogeneities. A fundamental ingredient in the analysis is a new transmission eigenvalue problem, whose eigenvalues must be excluded. Therefore, we prove that they are discrete with no finite accumulation points. We present numerical examples in dimension $d=2$ for both the direct and inverse problems.

We studied the Benamou–Brenier formulation of the Schrödinger problem, focusing on a gap between theoretical results and applications, that often involve measures with unbounded support. While the existing proof in the literature relies on the compactness of the marginals' supports to ensure the necessary regularity of the Schrödinger potentials, we extend the validity of the Benamou–Brenier formula to the larger class of sub-Gaussian probability measures. Exploiting fine estimates on the Hessian of the potentials and the entropic interpolation, we provide an almost self-contained proof that establishes the existence of a velocity field with the appropriate polynomial growth that ensures the right integrability. This result justifies the use of the dynamic formulation in more general settings, such as Gaussian and mixture-of-Gaussians models, important also for the applications.

We derive a general scheme to construct infinitely many probabilistic counterparts for solutions to nonlinear PDEs by recasting the latter as different nonlinear Fokker–Planck equations and by constructing, for each of these equations, a solution to the associated McKean–Vlasov SDE with one-dimensional time marginal densities given by the PDE solution. We utilize this scheme to prove that nonlinear Markov processes in the sense of McKean as introduced by Rehmeier–Röckner (J.\,Theor.\,Probab. 38, 60 (2025)) are not uniquely determined by their one-dimensional time marginals. This is in sharp contrast to the case of classical Markov processes, which are uniquely determined by their one-dimensional time marginals. We demonstrate our results by constructing a continuum of nonlinear Markov processes with one-dimensional time marginal densities given by the Barenblatt solutions to the porous medium and $p$-Laplace equations, as well as by the fundamental solution to the heat equation. This includes a novel martingale representation for the $p$-Laplace Barenblatt solutions. We also prove that a nonlinear Markov process is uniquely determined by its two-dimensional time marginals. Moreover, for the porous medium equation, we show that the different McKean–Vlasov SDEs we investigate are consistent with corresponding gradient flow interpretations of the equation in the sense of Otto calculus.

It has long been of interest to understand when control and observer-design methodologies developed for linear dynamical systems can provide local guarantees when applied to nonlinear systems. However, beyond establishing local stability, it is also important to characterize the region of attraction and performance in terms of system parameters and design gains–something that linearization-based approaches cannot achieve. For instance, although increasing an observer gain improves the convergence rate in the linear setting (as in backstepping design), the relationship between this gain and the convergence rate in the nonlinear setting need not be monotonic; understanding these dependencies is therefore crucial for reliable operation. In light of this discussion, we consider the one-dimensional quasilinear heat equation with state-dependent diffusivity, and design a boundary-output observer based on the backstepping design for a linear heat equation with constant diffusivity. Viewing the quasilinear system as a perturbation of the linear one, we establish exponential stability of the origin for the estimation error dynamics in $H^1$, with an estimate of the region of attraction depending on the system parameters, observer gains, and the mismatch between the nonlinear diffusivity and the constant design diffusivity. Importantly, the estimation error converges to zero rather than merely to a neighborhood scaling with this mismatch, even though, in contrast to backstepping-based stabilization of nonlinear PDEs, the mismatch need not decay along trajectories and may remain bounded away from zero, acting as a persistent state-dependent multiplicative perturbation. A technical challenge was to perform a sufficiently-fine Lyapunov analysis that does not yield overly conservative results such as mere boundedness of the estimation error.

In this paper, we establish Liouville-type theorems for parabolic differential inequalities with $(p,q)-$Laplacian operator on Riemannian manifolds. By a test function argument, we establish nonexistence results under suitable weighted volume growth assumptions involving potential. In particular, we can obtain nonexistence results for a wider class of parabolic inequalities.

We show that finite-energy weak solutions to the incompressible Navier–Stokes equations on a three-dimensional bounded smooth domain are regular up to the boundary, provided that the $L^4_tL^4_x$-norm of the solution is smaller than a constant depending only on the domain. This answers a problem raised in [D. Albritton, T. Barker, and C. Prange, J. Math. Fluid Mech. 25 (2023), Paper No. 49]. Our proof relies on a new slicing construction near the boundary of the domain.

We prove the following estimate \[ \|{e^{it\partial_x^2}f}\|_{L_{(t,x)\in \mathbb{T}^2}^6}\leq C (\log N)^{{1/6}} \|f\|_{L^2_x(\mathbb{T})}, \] assuming $\mbox{supp} (\hat f)\subset [-N,N]$ for $N>1$. The bound $(\log N)^{{1/6}}$ is sharp in view of the lower bound by Bourgain \cite{Bourgain}.

In this paper we consider higher order Schrödinger operators $$\mathcal L u=Lu+Vu,$$ where $L$ denotes a fourth order operator and $V\geq 0$ a suitable potential. We initiate our analysis by considering the constant coefficients differential operator $L=\Delta^2$. Subsequently, we extend our results to more general operators $L$ featuring suitable variable coefficients. We are interested in domain characterization and generation properties of these operators in $L^p(\mathbb{R}^N)$ for $p \in (1, \infty)$. To address this problems we employ a noncommutative version of the Dore-Venni theorem due to Monniaux and Prüss and we prove that the $L^p$-realization of $\mathcal L$ is quasi sectorial and, consequently, generates an analytic semigroup. Furthermore, this approach allows for a sharp characterization of the operator's domain as the intersection of the domains of the bilaplacian and the multiplication operator. The required assumptions allow to treat potentials that grow at infinity like $|x|^r$ for some $r<4$.

This paper concerns an inverse problem for the initial boundary value problem of the two-dimensional Navier-Stokes system defined in a bounded simply connected domain with slip, vorticity boundary conditions, and a global vorticity invariant constraint. We establish conditional Lipschitz stability and a local recovery for this inverse problem, where the velocity field and space-independent boundary vorticity are locally recovered from the given initial velocity field and the global vorticity invariant. Our analysis is based on well-posedness estimates and energy methods for the vorticity transport equation.

This paper presents two one-dimensional mathematical models describing automobile traffic flow on straight road segments at a signalized intersection. When the traffic light is permissive, the flow density and velocity are obtained by solving an initial-boundary value problem for a first-order hyperbolic system. When the signal is prohibitive, the same quantities are governed by a mixed system comprising a second-order parabolic equation for the velocity and a first-order equation for the density.

We consider a path-dependent Hamilton–Jacobi equation with coinvariant derivatives over the space of continuous functions. We prove two uniqueness results for viscosity (generalized) solutions defined in terms of coinvariantly smooth test functionals and a dense family of compact subsets of the space of continuous functions. It is assumed that the Hamiltonian is continuous and satisfies a local Lipschitz condition in the functional variable with respect to the supremum norm. When the Lipschitz constant satisfies a sublinear growth condition in the gradient (impulse) variable, uniqueness is established in the class of continuous viscosity solutions. In the general case, without any such growth conditions, uniqueness is established in the class of continuous viscosity solutions that satisfy an additional local Lipschitz condition. The proofs are based on the standard method of doubling variables, but use a novel penalty functional for constructing coinvariantly smooth test functionals. The obtained results generalize previously known ones by relaxing the assumptions on the Hamiltonian and/or enlarging the class of functionals in which uniqueness is established.