Related papers: Beale-Kato-Majda type condition for Burgers equati…
We consider certain Boltzmann type equations on a bounded physical and a bounded velocity space under the presence of both, reflective as well as diffusive boundary conditions. We introduce conditions on the shape of the physical space and…
We prove that for any $0 < s < 1/2$, the Benjamin--Ono equation on the torus is globally in time $C^0-$well-posed on the Sobolev space $H^{-s}(\T, \R)$,in the sense that the solution map, which is known to be defined for smooth data,…
We consider the motion described by the Navier-Stokes equations in a box with periodic boundary conditions. First we prove the existence of global strong two-dimensional solutions. Next we show the existence of global strong…
Given a brane tiling on a torus, we provide a new way to prove and generalise the recent results of Szendroi, Mozgovoy and Reineke regarding the Donaldson-Thomas theory of the moduli space of framed cyclic representations of the associated…
This paper proves that the 3-D Navier-Stokes system has a unique global solution under an assumpution on the initial data. That allow the data to be arbitrarily large in the scale invariant space \dot{B}_{\infty,\infty}^{-1}, which contains…
We study uniqueness of mild solutions to the two--dimensional incompressible Navier-Stokes equations on the torus in borderline spatial classes. While Lorentz-space methods yield uniqueness in $C([0,T);L^{2,1}(\mathbb{T}^2))$ via real…
In this paper, we study spatially homogeneous solutions of the Boltzmann equation in special relativity and in Robertson-Walker spacetimes. We obtain an analogue of the Povzner inequality in the relativistic case and use it to prove global…
This is a simplification of our prior work on the existence theory for the Rosseland-type equations. Inspired by the Rosseland equation in the conduction-radiation coupled heat transfer, we use the locally arbitrary growth conditions…
Considering the stochastic Navier-Stokes system in $\mathbb{R}^d$ forced by a multiplicative white noise, we establish the local existence and uniqueness of the strong solution when the initial data take values in the critical space…
Some classes of the rational, periodic and solitary wave solutions for the Burgers hierarchy are presented. The solutions for this hierarchy are obtained by using the generalized Cole - Hopf transformation.
We study global-in-time well-posedness and the behaviour and of the solution to Cauchy problem in the classical Keller-Segel system with logistic term \begin{equation*} \left. \aligned \partial_tn-\Delta n=&-\chi\nabla\cdot(n\nabla c)+\la…
We study 1D NLS with non-gauge invariant quadratic nonlinearity on the torus. The Cauchy problem admits trivial global solutions which are constant with respect to space. The non-existence of global solutions also has been studied only by…
In this paper we prove that the Benjamin-Ono equation, when considered on the torus, is an integrable (pseudo)differential equation in the strongest possible sense: it admits global Birkhoff coordinates on the space $L^2(\T)$. These are…
Whereas in a coordinate-dependent setting the Euler-Lagrange equations establish necessary conditions for solving variational problems in which both the integrands of functionals and the resulting paths are assumed to be sufficiently…
In this paper we solve the Monge problem on infinite dimensional Hilbert space endowed with a suitable Gaussian measure, that satisfies the Lebesgue differentiation theorem.
The present work deals with the global solvability as well as absolute continuity of the law of the solution to stochastic generalized Burgers-Huxley (SGBH) equation driven by multiplicative space-time white noise in a bounded interval of…
A box-ball system with more than one kind of balls is obtained by the generalized periodic discrete Toda equation (pd Toda eq.). We study the pd Toda equation in view of algebraic geometry. The time evolution of pd Toda eq. is linearized on…
We show that the Cauchy problem for a class of dispersive perturbations of Burgers' equations containing the low dispersion Benjamin-Ono equation $\partial$\_t u -- D^$\alpha$\_x $\partial$\_x u = $\partial$\_x(u^2), 0 < $\alpha$ $\le$ 1,…
We produce a finite time blow-up solution for nonlinear fractional heat equation ($\partial_t u + (-\Delta)^{\beta/2}u=u^k$) in modulation and Fourier amalgam spaces on the torus $\mathbb T^d$ and the Euclidean space $\mathbb R^d.$ This…
We investigate uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of fractional parabolic and elliptic equations.