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These notes address two problems. First, we investigate the question of ``how many'' are (in Baire sense) vector fields in $L^1_t L^q_x$, $q \in [1, \infty)$, for which existence and/or uniqueness of local, distributional solutions to the…

An example of a solution branch of the three dimensional Euler equation Cauchy problem is constructed which develops a singular velocity component and a singular vorticity component after finite time for some data which have Hoelder…

We derive a linear model of navigation in a two-layer fluid with a variable velocity of the ship. A spectral version of the model including a Rayleigh damping term is analyzed. We prove that the Cauchy problem has a unique solution if the…

The time-discretized, spatially continuous generalized Euler equations are a prototype example of multi-marginal optimal transport, yet the question whether they exhibit mass-splitting (or equivalently, whether they have solutions that are…

In this paper, we prove the global well-posedness of the classical solution to the 2D Cauchy problem of the compressible Navier-Stokes equations with arbitrarily large initial data when the shear viscosity $\mu$ is a positive constant and…

We consider the Cauchy problem for a second-order evolution equation, in which the problem operator is the sum of two self-adjoint operators. The main feature of the problem is that one of the operators is represented in the form of the…

This article is concerned with the study of weak solutions of a linear transport equation on a bounded domain with coupled boundary data for general non smooth space and time dependent velocity fields. The existence of solutions, its…

For the natural initial conditions $L^1$ in the density field (more generally a positive bounded Radon measure) and $L^\infty$ in the velocity field we obtain global approximate solutions to the Cauchy problem for the 3-D systems of…

Motivated by applications in model-free finance and quantitative risk management, we consider Fr\'echet classes of multivariate distribution functions where additional information on the joint distribution is assumed, while uncertainty in…

This paper is concerned with the Cauchy problem for the modified two-dimensional (2D) nonhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity. By fully using the structure of the system, we can obtain the key…

We investigate under which assumptions the flow associated to autonomous planar vector fields inherits the Sobolev or BV regularity of the vector field. We consider nearly incompressible and divergence-free vector fields, taking advantage…

Under general assumptions on the velocity field, it is possible to construct a flow that is forward untangled. Once such a flow has been selected, the associated transport problem is well-posed.

Motivated by modern machine learning applications where we only have access to empirical measures constructed from finite samples, we relax the marginal constraints of the classical Schr\"odinger bridge problem by penalizing the transport…

We propose two new alternating direction methods to solve "fully" nonsmooth constrained convex problems. Our algorithms have the best known worst-case iteration-complexity guarantee under mild assumptions for both the objective residual and…

The problem of describing the behavior of the solutions to the Navier-Stokes equations in three space dimensions has always been borderline. From one side, due to the viscosity term, smooth data seem to produce solutions with an everlasting…

We study the Cauchy problem for the chemotaxis Navier-Stokes equations and the Keller-Segel-Navier-Stokes system. Local-in-time and global-in-time solutions satisfying fundamental properties such as mass conservation and nonnegativity…

It is shown that if the system of the Euler equations has a special global in time smooth solution with the linear profile of velocity, then another solutions with Cauchy data, close in the Sobolev norm to the initial data of the given…

Four-dimensional spacetimes foliated by a two-parameter family of homologous two-surfaces are considered in Einstein's theory of gravity. By combining a 1+(1+2) decomposition, the canonical form of the spacetime metric and a suitable…

The use of the sine-Gordon equation as a model of magnetic flux propagation in Josephson junctions motivates studying the initial-value problem for this equation in the semiclassical limit in which the dispersion parameter $\e$ tends to…

In this paper we study existence of solutions of the initial-boundary value problems of the Navier-Stokes equations with a periodic boundary value condition for initial data in the Sobolev spaces $\mathcal{H}^{s}(\mathbb{T}^N)$ with a…