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Grzegorz Karch

Publications and source records attributed to Grzegorz Karch.

49 records · Page 3Linked to original sources
arXiv2009

Infinite energy solutions to the homogeneous Boltzmann equation

The goal of this work is to present an approach to the homogeneous Boltzmann equation for Maxwellian molecules with a physical collision kernel which allows us to construct unique solutions to the initial value problem in a space of probability measures defined via the Fourier transform. In that space, the second moment of a measure is not assumed to be finite, so infinite energy solutions are not {\it a priori} excluded from our considerations. Moreover, we study the large time asymptotics of solutions and, in a particular case, we give an elementary proof of the asymptotic stability of self-similar solutions obtained by A.V. Bobylev and C. Cercignani [J. Stat. Phys. {\bf 106} (2002), 1039--1071].

math.AP↗
arXiv2008

Decay of mass for nonlinear equation with fractional Laplacian

The large time behavior of nonnegative solutions to the reaction-diffusion equation $\partial_t u=-(-Δ)^{α/2}u - u^p,$ $(α\in(0,2], p>1)$ posed on $\mathbb{R}^N$ and supplemented with an integrable initial condition is studied. We show that the anomalous diffusion term determines the large time asymptotics for $p>1+α/{N},$ while nonlinear effects win if $p\leq1+α/{N}.$

math.AP↗
arXiv2008

Blow up of solutions to generalized Keller--Segel model

The existence and nonexistence of global in time solutions is studied for a class of equations generalizing the chemotaxis model of Keller and Segel. These equations involve Lévy diffusion operators and general potential type nonlinear terms.

math.AP↗
arXiv2008

Far field asymptotics of solutions to convection equation with anomalous diffusion

The initial value problem for the conservation law $\partial_t u+(-Δ)^{α/2}u+\nabla \cdot f(u)=0$ is studied for $α\in (1,2)$ and under natural polynomial growth conditions imposed on the nonlinearity. We find the asymptotic expansion as $|x|\to \infty$ of solutions to this equation corresponding to initial conditions, decaying sufficiently fast at infinity.

math.AP↗
arXiv2007

Asymptotic profiles of solutions to convection-diffusion equations

The large time behavior of zero mass solutions to the Cauchy problem for a convection-diffusion equation. We provide conditions on the size and shape of the initial datum such that the large time asymptotics of solutions is given either by the derivative of the Guass-Weierstrass kernel or by a self-similar solution or by a hyperbolic N-wave

math.AP↗
arXiv2007

On convergence of solutions of fractal Burgers equation toward rarefaction waves

In the paper, the large time behavior of solutions of the Cauchy problem for the one dimensional fractal Burgers equation $u_t+(-\partial^2_x)^{α/2} u+uu_x=0$ with $α\in (1,2)$ is studied. It is shown that if the nondecreasing initial datum approaches the constant states $u_\pm$ ($u_-<u_+$) as $x\to \pm\infty$, respectively, then the corresponding solution converges toward the rarefaction wave, {\it i.e.} the unique entropy solution of the Riemann problem for the nonviscous Burgers equation.

math.AP↗
arXiv2006

Fractal Hamilton-Jacobi-KPZ equations

Nonlinear and nonlinear evolution equations of the form $u_t=Łu \pm|\nabla u|^q$, where $Ł$ is a pseudodifferential operator representing the infinitesimal generator of a Lévy stochastic process, have been derived as models for growing interfaces in the case when the continuous Brownian diffusion surface transport is augmented by a random hopping mechanism. The goal of this paper is to study properties of solutions to this equation resulting from the interplay between the strengths of the "diffusive" linear and "hyperbolic" nonlinear terms, posed in the whole space $\bbfR^N$, and supplemented with nonnegative, bounded, and sufficiently regular initial conditions.

math.AP↗
arXiv2003

About the regularized Navier--Stokes equations

The first goal of this paper is to study the large time behavior of solutions to the Cauchy problem for the 3-dimensional incompressible Navier-Stokes system. The Marcinkiewicz space $L^{3,\infty}$ is used to prove some asymptotic stability results for solutions with infinite energy. Next, this approach is applied to the analysis of two classical ``regularized'' Navier-Stokes systems. The first one was introduced by J. Leray and consists in ``mollifying'' the nonlinearity. The second one was proposed by J.L. Lions, who added the artificial hyper-viscosity $(-Δ)^{\ell/2}$, $\ell>2$, to the model. It is shown in the present paper that, in the whole space, solutions to those modified models converge as $t\to\infty$ toward solutions of the original Navier-Stokes system.

math.AP↗
arXiv2002

Smooth or singular solutions to the Navier--Stokes system ?

The existence of singular solutions of the incompressible Navier-Stokes system with singular external forces, the existence of regular solutions for more regular forces as well as the asymptotic stability of small solutions (including stationary ones), and a pointwise loss of smoothness for solutions are proved in the same function space of pseudomeasure type.

math.AP↗
arXiv2002

On zero mass solutions of viscous conservation laws

In the paper, we consider the large time behavior of solutions to the convection-diffusion equation u_t - Delta u + nabla cdot f(u) = 0 in R^n times [0,infinity), where f(u) ~ u^q as u --> 0. Under the assumption that q >= 1+1/(n+beta) and the initial condition u_0 satisfies: u_0 in L^1(R^n), integral_{R^n} u_0(x) dx = 0, and |e^{t Delta}u_0|_{L^1(R^n)} <= Ct^{-beta/2} for fixed beta in (0,1), all t>0, and a constant C, we show that the L^1-norm of the solution to the convection-diffusion equation decays with the rate t^{-beta/2} as t --> infinity. Moreover, we prove that, for small initial conditions, the exponent q^* = 1+1/(n+beta) is critical in the following sense. For q > q^* the large time behavior in L^p(R^n), 1 <= p <= infinity, of solutions is described by self-similar solutions to the linear heat equation. For q = q^*, we prove that the convection-diffusion equation with f(u) = u|u|^{q^*-1} has a family of self-similar solutions which play an important role in the large time asymptotics of general solutions.

math.AP↗