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We investigate the existence of weak solutions for matrix-valued two-phase harmonic map flows with optimal lifespan, which arises as the limiting system of the matrix-valued Rubinstein-Sternberg-Keller problem studied by ({\em Invent.…

In this paper, we study the moduli spaces of noncollapsed Ricci flow solutions with bounded energy and scalar curvature. We show a weak compactness theorem for such moduli spaces and apply it to study isoperimetric constant control,…

We consider the one-dimensional shallow water equations (SW) in a finite channel with variable bottom topography. We pose several initial-boundary-value problems for the SW system, including problems with transparent (characteristic)…

Four integrable symplectic maps approximating two Hamiltonian flows from the relativistic Toda hierarchy are introduced. They are demostrated to belong to the same hierarchy and to examplify the general scheme for symplectic maps on groups…

In this paper, we establish a general convergence theorem for solutions of multivariate stochastic differential equations with countably many singular terms expressed as integrals with respect to local times. The processes under…

Continuous normalizing flows are known to be highly expressive and flexible, which allows for easier incorporation of large symmetries and makes them a powerful computational tool for lattice field theories. Building on previous work, we…

We show that for a broad family of noncompact homogeneous Riemannian manifolds, the corresponding homogeneous Ricci flow solutions have finite extinction time, thereby confirming the dynamical Alekseevskii conjecture for these spaces. As an…

Given a lattice \Gamma in a locally compact group G and a closed subgroup H of G, one has a natural action of \Gamma on the homogeneous space V=H\G. For an increasing family of finite subsets {\Gamma_T: T>0}, a dense orbit v\Gamma, v\in V,…

$\Gamma$-convergence techniques are used to give a characterization of the behavior of a family of heterogeneous multiple scale integral functionals. Periodicity, standard growth conditions and nonconvexity are assumed whereas a stronger…

We consider the question of whether solutions of variants of Teichm\"uller harmonic map flow from surfaces $M$ to general targets can degenerate in finite time. For the original flow from closed surfaces of genus at least $2$, as well as…

We consider the numerical approximation of compressible flow in a pipe network. Appropriate coupling conditions are formulated that allow us to derive a variational characterization of solutions and to prove global balance laws for the…

We study a projection-type gradient flow for equality-constrained maximisation of a smooth bilinear control objective on $\mathcal{H}=L^2(0,T;\mathbb{R})$, eliminating Lagrange multipliers through an $(M{+}1)\times(M{+}1)$ moving Gram…

We study the singularities for minimum time control-affine problems in 4D with 2D controls. After regularization, the problem boils down to the study of a bifurcation around some nilpotent equilibrium in the singular locus. We show that the…

We consider magnetic geodesic flows of the normal metrics on a class of homogeneous spaces, in particular (co)adjoint orbits of compact Lie groups. We give the proof of the non-commutative integrability of flows and show, in addition, for…

We study a one dimensional model for two-phase flows in heterogeneous media, in which the capillary pressure functions can be discontinuous with respect to space. We first give a model, leading to a system of degenerated non-linear…

In this paper, we develop a theory of new classes of discrete convex functions, called L-extendable functions and alternating L-convex functions, defined on the product of trees. We establish basic properties for optimization: a…

We consider the Ricci flow equation for invariant metrics on compact and connected homogeneous spaces whose isotropy representation decomposes into two irreducible inequivalent summands. By studying the corresponding dynamical system, we…

The sequential nature of autoregressive next-token prediction imposes a fundamental speed limit on large language models. While continuous flow models offer a path to parallel generation, they traditionally demand expensive iterative…

Accelerated gradient descent iterations are widely used in optimization. It is known that, in the continuous-time limit, these iterations converge to a second-order differential equation which we refer to as the accelerated gradient flow.…

We consider Lagrangian systems in the limit of infinitely many particles. It is shown that the corresponding discrete action functionals Gamma-converge to a continuum action functional acting on probability measures of particle…