Related papers: An Eulerian-Lagrangian Form for the Euler Equation…
The aim of the paper is to study some dynamic aspects coming from a tangent form, i.e. a time dependent differential form on a tangent bundle. The action on curves of a tangent form is natural associated with that of a second order…
We present the Euler-Lagrange and Hamilton's equations for a system whose configuration space is a unified product Lie group $G=M\bowtie_{\gamma} H$, for some $\gamma:M\times M \to H$. By reduction, then, we obtain the Euler-Lagrange type…
We construct examples and provide a classification of self-similar solutions to the two-dimensional incompressible Euler equations whose pseudo-velocity fields possess more than one stagnation point. These solutions are also homogeneous…
In this work a static solution of Einstein-Cartan (EC) equations in 2+1 dimensional space-time is given by considering classical spin-1/2 field as external source for torsion of the space-time. Here, the torsion tensor is obtained from…
The Navier-Stokes equations are considered by the use of the method of Lagrangians with covariant derivatives (MLCD) over spaces with affine connections and metrics. It is shown that the Euler-Lagrange equations appear as sufficient…
Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving…
We consider uniformly rotating incompressible Euler and Navier-Stokes equations. We study the suppression of vertical gradients of Lagrangian displacement ("vertical" refers to the direction of the rotation axis). We employ a formalism that…
In this paper we consider the incompressible porous media equation in the Sobolev spaces $H^s(\R^2), s > 2$. We prove that for $T > 0$ the time $T$ solution map $\rho_0 \mapsto \rho(T)$ is nowhere locally uniformly continuous. On the other…
In this paper we study cosmological solutions to the Einstein--Euler equations. We first establish the future stability of nonlinear perturbations of a class of homogeneous solutions to the relativistic Euler equations on fixed linearly…
Flows of one-dimensional continuum in Lagrangian coordinates are studied in the paper. Equations describing these flows are reduced to a single Euler-Lagrange equation which contains two undefined functions. Particular choices of the…
We study existence and convergence properties of least-energy symmetric solutions (l.e.s.s.) to the pure critical problem \begin{equation*} (-\Delta)^su_s=|u_s|^{2^\star_s-2}u_s, \quad u_s\in D^s_0(\Omega),\quad 2^\star_s:=\frac{2N}{N-2s},…
In this paper we study the existence and linear stability of almost periodic solutions for a NLS equation on the circle with external parameters. Starting from the seminal result of Bourgain (2005) on the quintic NLS, we propose a novel…
In this note we discuss the diffusive, vector-valued Burgers equations in a three-dimensional domain with periodic boundary conditions. We prove that given initial data in $H^{1/2}$ these equations admit a unique global solution that…
Lagrangian multiform theory is a variational framework for integrable systems. In this article we introduce a new formulation which is based on symplectic geometry and which treats position, momentum and time coordinates of a…
We study the local H\"older regularity of strong solutions $u$ of second-order uniformly elliptic equations having a gradient term with superquadratic growth $\gamma > 2$, and right-hand side in a Lebesgue space $L^q$. When $q >…
We establish the existence of infinitely many global and stationary solutions in $C(\mathbb{R};C^{\vartheta})$ space for some $\vartheta>0$ to the three dimensional Euler equations driven by an additive noise. The result is based on a new…
We provide a complete local well-posedness theory in $H^s$ based Sobolev spaces for the free boundary incompressible Euler equations with zero surface tension on a connected fluid domain. Our well-posedness theory includes: (i) Local…
We investigate the existence of minimizers of variational models with Eulerian-Lagrangian formulations. We consider energy functionals depending on the deformation of a body, defined on its reference configuration, and an Eulerian map…
We prove the non-uniform continuity of the data-to-solution map of the incompressible Euler equations in Besov spaces $B_{p,q}^{s}$, where the parameters $p, q$ and $s$ considered here are such that the local existence and uniqueness result…
We prove local existence and uniqueness for the two-dimensional Prandtl system in weighted Sobolev spaces under the Oleinik's monotonicity assumption. In particular we do not use the Crocco transform. Our proof is based on a new nonlinear…