Related papers: On the one-dimensional continuity equation with a …

We establish existence and uniqueness results for initial-boundary value problems for transport equations in one space dimension with nearly incompressible velocity fields, under the sole assumption that the fields are bounded. In the case…

We consider the Cauchy problem for incompressible viscoelastic fluids in the whole space $\mathbb{R}^d$ ($d=2,3$). By introducing a new decomposition via Helmholtz's projections, we first provide an alternative proof on the existence of…

A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our…

Here we investigate the Cauchy problem for the inhomogeneous Navier-Stokes equations in the whole $n$-dimensional space. Under some smallness assumption on the data, we show the existence of global-in-time unique solutions in a critical…

We study the Cauchy problem for a multidimensional scalar conservation law with merely continuous flux vector in the class of Besicovitch almost periodic functions. The existence and uniqueness of entropy solutions are established. We…

This work addresses the question of uniqueness and regularity of the minimizers of a convex but not strictly convex integral functional with linear growth in a two-dimensional setting. The integrand -- whose precise form derives directly…

We study the Cauchy problem for a multidimensional scalar conservation law on the Bohr compactification of $\R^n$. The existence and uniqueness of entropy solutions are established in the general case of merely continuous flux vector. We…

We consider the equations governing incompressible, viscous fluids in three space dimensions, rotating around an inhomogeneous vector B(x): this is a generalization of the usual rotating fluid model (where B is constant). We prove the weak…

Here we investigate the Cauchy problem for the barotropic Navier-Stokes equations in R^n, in the critical Besov spaces setting. We improve recent results as regards the uniqueness condition: initial velocities in critical Besov spaces with…

This paper investigates the well posedness of ordinary differential equations and more precisely the existence (or uniqueness) of a flow through explicit compactness estimates. Instead of assuming a bounded divergence condition on the…

A linear stochastic continuity equation with non-regular coefficients is considered. We prove existence and uniqueness of strong solution, in the probabilistic sense, to the Cauchy problem when the vector field has low regularity, in which…

We face the well-posedness of linear transport Cauchy problems $$\begin{cases}\dfrac{\partial u}{\partial t} + b\cdot\nabla u + c\,u = f&(0,T)\times{\mathbb R}^n\\u(0,\cdot)=u_0\in L^\infty&{\mathbb R}^n\end{cases}$$ under borderline…

We establish existence and uniqueness results for initial boundary value problems with nearly incompressible vector fields. We then apply our results to establish well-posedness of the initial-boundary value problem for the Keyfitz and…

We consider the Cauchy problem for a $n\times n$ strictly hyperbolic system of balance laws $$ \{{array}{c} u_t+f(u)_x=g(x,u), x \in \mathbb{R}, t>0 u(0,.)=u_o \in L^1 \cap BV(\mathbb{R}; \mathbb{R}^n), | \lambda_i(u)| \geq c > 0 {for all}…

The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions…

This paper is concerned with the existence of compactly supported admissible solutions to the Cauchy problem for the isentropic compressible Euler equations. In more than one space dimension, convex integration techniques developed by De…

In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations is considered. When viscosity coefficients are given as a constant multiple of the density's power ($\rho^\delta$ with…

We consider the Cauchy problem for the continuity equation in space dimension ${d \geq 2}$. We construct a divergence-free velocity field uniformly bounded in all Sobolev spaces $W^{1,p}$, for $1 \leq p<\infty$, and a smooth compactly…

An evidence of temporal dis-continuity of the solution in $F^s_{1, \infty}(\mathbb{R}^d)$ is presented, which implies the ill-posedness of the Cauchy problem for the Euler equations. Continuity and weak-type continuity of the solutions in…

The vortex-wave system is a model for the evolution of 2D incompressible fluids in which the vorticity is split into a finite sum of Dirac masses plus an Lp part. Existence of a weak solution for this system was recently proved by Lopes…