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Register a new userLarge-time rescaling behaviors for large data to the Hele-Shaw problem
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Yulin Lin
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This paper addresses a rescaling behavior of some classes of global solutions to the zero surface tension Hele-Shaw problem with injection at the origin, ${Omega(t)}_{tgeq 0}$. Here $Omega(0)$ is a small perturbation of $f(B_{1}(0),0)$ if $f(xi,t)$ is a global strong polynomial solution to the Polubarinova-Galin equation with injection at the origin and we prove the solution $Omega(t)$ is global as well. We rescale the domain $Omega(t)$ so that the new domain $Omega^{}(t)$ always has area $pi$ and we consider $partialOmega^{}(t)$ as the radial perturbation of the unit circle centered at the origin for $t$ large enough. It is shown that the radial perturbation decays algebraically as $t^{-lambda}$. This decay also implies that the curvature of $partialOmega^{}(t)$ decays to 1 algebraically as $t^{-lambda}$. The decay is faster if the low Richardson moments vanish. We also explain this work as the generalization of Vondenhoffs work which deals with the case that $f(xi,t)=a_{1}(t)xi$.
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The main goal of this paper is to give a precise description of rescaling behaviors of rational type global strong solutions to the Polubarinova-Galin equation. The Polubarinova-Galin equation is the reformulation of the zero surface tension Hele-Shaw problem with a single source at the origin by considering the moving domain as the Riemann mapping of the unit disk centered at the origin. The coefficients ${a_{k}(t)}_{kgeq 2}$ of the polynomial strong solution $f_{k_{0}}(xi,t)=sum_{i=1}^{k_{0}}a_{i}(t)xi^{i}$ decay to zero algebraically as $t^{-lambda_{k}}$ ($lambda_{k}=k/2$) and the decay is even faster if the low Richardson moments vanish. The dynamics for global solutions are discussed as well.
This paper gives a new and short proof of existence and uniqueness of the Polubarinova-Galin equation. The existence proof is an application of the main theorem in Lins paper. Furthermore, we can conclude that every strong solution can be approximated by many strong polynomial solutions locally in time.
The Theory of (2+1) Systems based on 2D Schrodinger Operator was started by S.Manakov, B.Dubrovin, I.Krichever and S.Novikov in 1976. The Analog of Lax Pairs introduced by Manakov, has a form $L_t=[L,H]-fL$ (The $L,H,f$-triples) where $L=partial_xpartial_y+Gpartial_y+S$ and $H,f$-some linear PDEs. Their Algebro-Geometric Solutions and therefore the full higher order hierarchies were constructed by B.Dubrovin, I.Krichever and S.Novikov. The Theory of 2D Inverse Spectral Problems for the Elliptic Operator $L$ with $x,y$ replaced by $z,bar{z}$, was started by B.Dubrovin, I.Krichever and S.Novikov: The Inverse Spectral Problem Data are taken from the complex Fermi-Curve consisting of all Bloch-Floquet Eigenfunctions $Lpsi=const$. Many interesting systems were found later. However, specific properties of the very first system, offered by Manakov for the verification of new method only, were not studied more than 10 years until B.Konopelchenko found in 1988 analogs of Backund Transformations for it. He pointed out on the Burgers-Type Reduction. Indeed, the present authors quite recently found very interesting extensions, reductions and applications of that system both in the theory of nonlinear evolution systems (The Self-Adjoint and 2D Burgers Hierarhies were invented, and corresponding reductions of Inverse Problem Data found) and in the Spectral Theory of Important Physical Operators (The Purely Magnetic 2D Pauli Operators). We call this system GKMMN by the names of authors who studied it.
We develop the Riemann-Hilbert problem approach to inverse scattering for the two-dimensional Schrodinger equation at fixed energy. We obtain global or gener
In this paper we continue the formal analysis of the long-time asymptotics of the homoenergetic solutions for the Boltzmann equation that we began in [18]. They have the form $fleft( x,v,tright) =gleft(v-Lleft( tright) x,tright) $ where $Lleft( tright) =Aleft(I+tAright) ^{-1}$ where $A$ is a constant matrix. Homoenergetic solutions satisfy an integro-differential equation which contains, in addition to the classical Boltzmann collision operator, a linear hyperbolic term. Depending on the properties of the collision kernel the collision and the hyperbolic terms might be of the same order of magnitude as $ttoinfty$, or the collision term could be the dominant one for large times, or the hyperbolic term could be the largest. The first case has been rigorously studied in [17]. Formal asymptotic expansions in the second case have been obtained in [18]. All the solutions obtained in this case can be approximated by Maxwellian distributions with changing temperature. In this paper we focus in the case where the hyperbolic terms are much larger than the collision term for large times (hyperbolic-dominated behavior). In the hyperbolic-dominated case it does not seem to be possible to describe in a simple way all the long time asymptotics of the solutions, but we discuss several physical situations and formulate precise conjectures. We give explicit formulas for the relationship between density, temperature and entropy for these solutions. These formulas differ greatly from the ones at equilibrium.
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