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We consider the Cauchy problem and the source problem for normally hyperbolic operators on the Minkowski spacetime, and study the determination of solutions from their integrals along null geodesics. For the Cauchy problem, we give a new…
The aim of these notes is to present in a comprehensive and relatively self-contained way some recent developments in the mathematical analysis of two-dimensional viscous flows. We consider the incompressible Navier-Stokes equations in the…
This article deals with the design of saturated controls in the context of partial differential equations. It focuses on a Korteweg-de Vries equation, which is a nonlinear mathematical model of waves on shallow water surfaces. Two different…
We study stability of non-compact gradient Kaehler-Ricci flow solitons with positive holomorphic bisectional curvature. Our main result is that any compactly supported perturbation and appropriately decaying perturbations of the Kaehler…
In this paper, we study the asymptotic behaviors of solutions to the inhomogeneous Navier-Stokes-Vlasov system in $\mathbb{R}^{3}\times\mathbb{R}^{3}$, where the initial fluid density is allowed to vanish. We establish the uniform bound of…
Under mild regularity assumptions, the transport problem is stable in the following sense: if a sequence of optimal transport plans $\pi_1, \pi_2, \ldots$ converges weakly to a transport plan $\pi$, then $\pi$ is also optimal (between its…
We study the large-data Cauchy problem for two dimensional Oldroyd model of incompressible viscoelastic fluids. We prove the global-in-time existence of the Leray-Hopf type weak solutions in the physical energy space. Our method relies on a…
We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a C^1 dynamical stability theorem of the mean curvature flow for…
This paper is focused on some properties of paramonotone operators on Banach spaces and their application to certain feasibility problems for convex sets in a Hilbert space and convex systems in the Euclidean space. In particular, it shows…
We study properties of continuous semi-homogeneous operators of degree $k$ via various functions (e.g. measures of noncompactness) on all bounded subsets of a Banach space. We prove necessary and sufficient conditions for these functions to…
We introduce a comprehensive framework for subordinacy theory applicable to long-range operators on $\ell^2(\mathbb Z)$, bridging dynamical systems and spectral analysis. For finite-range operators, we establish a correspondence between the…
We study the Cauchy problem for the first order evolutive Hamilton-Jacobi equation with a Lipschitz initial condition. The Hamiltonian is not necessarily convex in the momentum variable and not a priori compactly supported. We build and…
We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional…
Partial differential equations endowed with a Hamiltonian structure, like the Korteweg--de Vries equation and many other more or less classical models, are known to admit rich families of periodic travelling waves. The stability theory for…
In this paper, we establish the well-posedness of Cauchy problems for weak solutions to second-order degenerate parabolic equations with a non-smooth, time-dependent degenerate elliptic part that includes both bounded and unbounded…
Since its elaboration by Whitham, almost fifty years ago, modulation theory has been known to be closely related to the stability of periodic traveling waves. However, it is only recently that this relationship has been elucidated, and that…
The monotonicity and stability of difference schemes for, in general, hyperbolic systems of conservation laws with source terms are studied. The basic approach is to investigate the stability and monotonicity of a non-linear scheme in terms…
We consider a size-structured aggregation and growth model of phytoplankton community proposed by Ackleh and Fitzpatrick [2]. The model accounts for basic biological phenomena in phytoplankton community such as growth, gravitational…
Extending the approach of Grillakis-Shatah-Strauss, Bronski-Johnson-Kapitula, and others for Hamiltonian systems, we explore relations between the constrained variational problem $\min_{X:C(X)=c_0} \mathcal{E}(X)$, $c_0\in \RM^r$, and…
In this paper, we investigate the problems of Cauchy theory and exponential stability for the inhomogeneous Boltzmann equation without angular cut-off. We only deal with the physical case of hard potentials type interactions (with a…