Related papers: Nonlocal conservation laws for the two-dimensional…
Symmetries and conservation laws are studied for a generalized Westervelt equation which is a nonlinear partial differential equation modelling the propagation of sound waves in a compressible medium. This nonlinear wave equation is widely…
Recent work giving a classification of kinematic and vorticity conservation laws of compressible fluid flow for barotropic equations of state (where pressure is a function only of the fluid density) in $n>1$ spatial dimensions is extended…
In his 1892 paper [L. Bianchi, Sulla trasformazione di B\"{a}cklund per le superfici pseudosferiche, Rend. Mat. Acc. Lincei, s. 5, v. 1 (1892) 2, 3--12], L. Bianchi noticed, among other things, that quite simple transformations of the…
A complete classification of all low-order conservation laws is carried out for a system of coupled semilinear wave equations which is a natural two-component generalization of the nonlinear Klein-Gordon equation. The conserved quantities…
A generalization of the Euler's elastic problem, i.e., finding a stationary configuration (planar elastica) of the Bernoulli's thin ideal elastic rod with boundary conditions defined through fixed endpoints and/or tangents at the endpoints,…
We study the long-time behavior of solutions to the compressible Euler equations with frictional damping in the whole space, where we prescribe direction-dependent values for the density at spatial infinity. To this end, we transform the…
We prove the conservation of energy for weak and statistical solutions of the two-dimensional Euler equations, generated as strong (in an appropriate topology) limits of the underlying Navier-Stokes equations and a Monte Carlo-Spectral…
In this article, we explore some of the main mathematical problems connected to multidimensional fractional conservation laws driven by L\'evy processes. Making use of an adapted entropy formulation, a result of existence and uniqueness of…
We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential equations. In this paper, we…
A simple characterization of the action of symmetries on conservation laws of partial differential equations is studied by using the general method of conservation law multipliers. This action is used to define symmetry-invariant and…
This paper extends, to a class of systems of semi-linear hyperbolic second order PDEs in three variables, the geometric study of a single nonlinear hyperbolic PDE in the plane as presented in [Anderson I.M., Kamran N., Duke Math. J. 87…
We introduce notions of equivalence of conservation laws with respect to Lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such…
We consider a broad class of systems of nonlinear integro-differential equations posed on the real line that arise as Euler-Lagrange equations to energies involving nonlinear nonlocal interactions. Although these equations are not readily…
We consider variational principles related to V. I. Arnold's stability criteria for steady-state solutions of the two-dimensional incompressible Euler equation. Our goal is to investigate under which conditions the quadratic forms defined…
Solutions to a class of conservation laws with discontinuous flux are constructed relying on the Crandall-Liggett theory of nonlinear contractive semigroups~\cite{CL}. In particular, the paper studies the existence of backward Euler…
We lay down a set of requirements for a field theory to produce a covariant conservation law out of Noether's second theorem, and show that neither local invariance implies a covariant conservation law, nor the existence of a covariant…
In this paper, we study desingularization of vortices for the two-dimensional incompressible Euler equations in the full plane. We construct a family of steady vortex pairs for the Euler equations with a general vorticity function, which…
In this paper we prove that solutions of the 2D Euler equations in vorticity formulation obtained via vanishing viscosity approximation are renormalized.
We analyze the relativistic Euler equations of conservation laws of baryon number and momentum with a general pressure law. The existence of global-in-time bounded entropy solutions for the system is established by developing a compensated…
In this paper, the nonlinear Rosenau-Hyman equation with time dependent variable coefficients is considered for investigating its invariant properties, exact solutions and conservation laws. Using Lie classical method, we derive symmetries…