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Register a new userExistence and uniqueness of the Hele-Shaw problem with injection
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Yulin Lin
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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.
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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 $f(xi,t)=a_{1}(t)xi$.
We use a new method to prove uniqueness theorem for a coefficient inverse scattering problem without the phase information for the 3-D Helmholtz equation. We consider the case when only the modulus of the scattered wave field is measured and the phase is not measured. The spatially distributed refractive index is the subject of the interest in this problem. Applications of this problem are in imaging of nanostructures and biological cells.
We aim this paper to develop the classical lattice models with unbounded spin to the case of non-quadratic polynomial interaction. We demonstrate that the distinct relation between the growths of potentials leads to the uniqueness and the fast decay of correlations for Gibbs measure.
Consider a non-relativistic quantum particle with wave function inside a region $Omegasubset mathbb{R}^3$, and suppose that detectors are placed along the boundary $partial Omega$. The question how to compute the probability distribution of the time at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface; a particularly convincing definition, called the absorbing boundary rule, involves a time evolution for the particles wave function $psi$ expressed by a Schrodinger equation in $Omega$ together with an absorbing boundary condition on $partial Omega$ first considered by Werner in 1987, viz., $partial psi/partial n=ikappapsi$ with $kappa>0$ and $partial/partial n$ the normal derivative. We provide here a discussion of the rigorous mathematical foundation of this rule. First, for the viability of the rule it plays a crucial role that these two equations together uniquely define the time evolution of $psi$; we point out here how the Hille-Yosida theorem implies that the time evolution is well defined and given by a contraction semigroup. Second, we show that the collapse required for the $N$-particle version of the problem is well defined. Finally, we also prove analogous results for the Dirac equation.
We show that Tsirelsons problem concerning the set of quantum correlations and Connes embedding problem on finite approximations in von Neumann algebras (known to be equivalent to Kirchbergs QWEP conjecture) are essentially equivalent. Specifically, Tsirelsons problem asks whether the set of bipartite quantum correlations generated between tensor product separated systems is the same as the set of correlations between commuting C*-algebras. Connes embedding problem asks whether any separable II$_1$ factor is a subfactor of the ultrapower of the hyperfinite II$_1$ factor. We show that an affirmative answer to Connes question implies a positive answer to Tsirelsons. Conversely, a positve answer to a matrix valued version of Tsirelsons problem implies a positive one to Connes problem.
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