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Search Results: 1 - 10 of 1222 matches for " Grzegorz Karch "
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Far field asymptotics of solutions to convection equation with anomalous diffusion
Lorenzo Brandolese,Grzegorz Karch
Mathematics , 2008,
Abstract: The initial value problem for the conservation law $\partial_t u+(-\Delta)^{\alpha/2}u+\nabla \cdot f(u)=0$ is studied for $\alpha\in (1,2)$ 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.
Blow up of solutions to generalized Keller--Segel model
Piotr Biler,Grzegorz Karch
Mathematics , 2008,
Abstract: 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 L\'evy diffusion operators and general potential type nonlinear terms.
Decay of mass for nonlinear equation with fractional Laplacian
Ahmad Fino,Grzegorz Karch
Mathematics , 2008,
Abstract: The large time behavior of nonnegative solutions to the reaction-diffusion equation $\partial_t u=-(-\Delta)^{\alpha/2}u - u^p,$ $(\alpha\in(0,2], p>1)$ 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 $p>1+{\alpha}/{N},$ while nonlinear effects win if $p\leq1+{\alpha}/{N}.$
Infinite energy solutions to the homogeneous Boltzmann equation
Marco Cannone,Grzegorz Karch
Mathematics , 2009,
Abstract: 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].
Blow-up versus global existence of solutions to aggregation equations
Grzegorz Karch,Kanako Suzuki
Mathematics , 2010,
Abstract: A class of nonlinear viscous transport equations describing aggregation phenomena in biology is considered. Optimal conditions on an interaction potential are obtained which lead either to the existence or to the nonexistence of global-in-time solutions.
Spikes and diffusion waves in one-dimensional model of chemotaxis
Grzegorz Karch,Kanako Suzuki
Mathematics , 2010, DOI: 10.1088/0951-7715/23/12/007
Abstract: We consider the one-dimensional initial value problem for the viscous transport equation with nonlocal velocity $u_t = u_{xx} - \left(u (K^\prime \ast u)\right)_{x}$ with a given kernel $K'\in L^1(\R)$. We show the existence of global-in-time nonnegative solutions and we study their large time asymptotics. Depending on $K'$, we obtain either linear diffusion waves ({\it i.e.}~the fundamental solution of the heat equation) or nonlinear diffusion waves (the fundamental solution of the viscous Burgers equation) in asymptotic expansions of solutions as $t\to\infty$. Moreover, for certain aggregation kernels, we show a concentration of solution on an initial time interval, which resemble a phenomenon of the spike creation, typical in chemotaxis models.
Smooth or singular solutions to the Navier--Stokes system ?
Marco Cannone,Grzegorz Karch
Mathematics , 2002,
Abstract: 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.
About the regularized Navier--Stokes equations
Marco Cannone,Grzegorz Karch
Mathematics , 2003,
Abstract: 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 $(-\Delta)^{\ell/2}$, $\ell>2$, 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.
Time-dependent singularities in the Navier-Stokes system
Grzegorz Karch,Xiaoxin Zheng
Mathematics , 2014,
Abstract: We show that, for a given H\"older continuous curve in $\{(\gamma(t),t)\,:\, t>0\} \subset R^3\times R^+$, there exists a solution to the Navier-Stokes system for an incompressible fluid in $R^3$ which is smooth outside this curve and singular on it. This is a pointwise solution of the system outside the curve, however, as a distributional solution on $R^3\times R^+$, it solves an analogous Navier-Stokes system with a singular force concentrated on the curve.
Asymptotic stability of Landau solutions to Navier-Stokes system
Grzegorz Karch,Dominika Pilarczyk
Mathematics , 2011, DOI: 10.1007/s00205-011-0415-1
Abstract: It is known that the three dimensional Navier-Stokes system for an incompressible fluid in the whole space has a one parameter family of explicit stationary solutions, which are axisymmetric and homogeneous of degree -1. We show that these solutions are asymptotically stable under any $L^2$-perturbation.
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