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Pointwise semigroup methods and stability of viscous shock waves
 Indiana Univ. Math. J
, 1998
"... Abstract. Considered as rest points of ODE on L p, stationary viscous shock waves present a critical case for which standard semigroup methods do not su ce to determine stability. More precisely, there is no spectral gap between stationary modes and essential spectrum of the linearized operator abou ..."
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Cited by 63 (32 self)
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Abstract. Considered as rest points of ODE on L p, stationary viscous shock waves present a critical case for which standard semigroup methods do not su ce to determine stability. More precisely, there is no spectral gap between stationary modes and essential spectrum of the linearized operator about the wave, a fact which precludes the usual analysis by decomposition into invariant subspaces. For this reason, there have been until recently no results on shock stability from the semigroup perspective except in the scalar or totally compressive case ([Sat], [K.2], resp.), each of which can be reduced to the standard semigroup setting by Sattinger's method of weighted norms. We overcome this di culty in the general case by the introduction of new, pointwise semigroup techniques, generalizing earlier work of Howard [H.1], Kapitula [K.12], and Zeng [Ze,LZe]. These techniques allow us to do \hard " analysis in PDE within the dynamical systems/semigroup framework: in particular, to obtain sharp, global pointwise bounds on the Green's function of the linearized operator around the wave, su cient for the analysis of linear and nonlinear stability. The method is general, and should nd applications
On the relation between elliptic and parabolic Harnack inequalities
, 2001
"... We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in que ..."
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Cited by 27 (3 self)
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We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1
Heat Equation And Reflected Brownian Motion In Time Dependent Domains
 J. Funct. Anal
"... This article is mostly devoted to questions of analytic nature; somewhat paradoxically in view of the original motivation, probability mostly plays here the role of a tool and not an end. The analytic literature on the heat equation and related problems is enormous and we would rather let the reader ..."
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Cited by 10 (4 self)
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This article is mostly devoted to questions of analytic nature; somewhat paradoxically in view of the original motivation, probability mostly plays here the role of a tool and not an end. The analytic literature on the heat equation and related problems is enormous and we would rather let the reader search the library than provide an exceedingly imperfect review. Crank (1984) provides an excellent review of various problems related to free and 1. Research partially supported by NSF grant DMS9700721. 2. Research partially supported by NSA grant MDA9049910104. 3. Research partially supported by NSF grant DMS{9801068 and ONR grants N0001493 {0295 and N000149810675. 1 moving bondaries. Although one can see obvious general similarities between our problem and the classical Stefan's problem, it remains to be seen if there exist any connections at the technical level. We will be much more specic on the probabilistic side as we feel that we can provide an eective guide to an uninitiated reader who wants to learn more about the reected Brownian motion. Brownian motion in time dependent domains belongs to \classical" subjects in probability. The model appears in the context of a problem often referred to as \boundary crossing." The literature on the problem is huge; we suggest Anderson and Pitt (1997) and Durbin (1992) as starting points. The boundary crossing problem was mainly motivated by statistical questions but the estimates derived in this area have been also applied to study Brownian path properties, see, e.g., Bass and Burdzy (1996) or Greenwood and Perkins (1983). In the context of our article, this classical model may be described as a Brownian motion killed on the boundary of a timedependent domain. The corresponding analytic problem may be called the heat...
Graphs between the elliptic and parabolic Harnack inequalities.
"... We present graphs that satisfy the uniform elliptic Harnack inequality, for harmonic functions, but not the stronger parabolic one, for solutions of the discrete heat equation. It is known from [9] that the parabolic Harnack inequality is equivalent to the conjunction of a volume regularity and a L ..."
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Cited by 7 (0 self)
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We present graphs that satisfy the uniform elliptic Harnack inequality, for harmonic functions, but not the stronger parabolic one, for solutions of the discrete heat equation. It is known from [9] that the parabolic Harnack inequality is equivalent to the conjunction of a volume regularity and a L 2 PoincarĂ© inequality. The first example of graph satisfying the elliptic but not the parabolic Harnack inequality is due to M. Barlow and R. Bass, see [2]. It satisfies the volume regularity and not the PoincarĂ© inequality. We construct another example that does not satisfy the volume regularity.