Second Order Equations of Elliptic and
                    Parabolic Type

Abstract. We consider a boundary value problem for a Lamé type operator, which corresponds to a linearisation of the Navier-Stokes' equations for compressible flow of Newtonian fluids in the case where pressure is known. It consists of recovering a vector function, satisfying the parabolic Lamé type system in a cylindrical domain, via its values and the values of the boundary stress tensor on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural spaces of smooth functions and in the corresponding Hölder spaces; besides, additional initial data do not turn the problem to a well-posed one. Using the integral representation's method we obtain a uniqueness theorem and solvability conditions for the problem. We also describe conditions, providing dense solvabilty of the problem.

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“…Finally, as the boundary operators I and σ form a Dirichlet system on ∂Ω (see, for instance, [15]) we conclude that for each pair ( (15) and (16)…”

Section: Corollarymentioning

confidence: 70%

On a mixed problem for the parabolic Lamé type operator

Puzyrev

Shlapunov

2014

Journal of Inverse and Ill-Posed Problems

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“…Applying the maximum principle [17], we see thatr ≤ 0 in Q τ \ K . So for t ≤ τ we have r (t, x) ≤ (R + 1) + tδC 1 (R).…”

Section: Proof Of Lemma 38 Step 1 Let Us Write U(t) = V(t) + Z(t)mentioning

confidence: 92%

Stochastic CGL equations without linear dispersion in any space dimension

Kuksin

Nersesyan

2013

Stoch PDE: Anal Comp

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We consider the stochastic CGL equatioṅwhere ν > 0 and a ≥ 0, in a cube (or in a smooth bounded domain) with Dirichlet boundary condition. The force η is white in time, regular in x and non-degenerate. We study this equation in the space of continuous complex functions u(x), and prove that for any n it defines there a unique mixing Markov process. So for a large class of functionals f (u(·)) and for any solution u(t, x), the averaged observable E f (u(t, ·)) converges to a quantity, independent from the initial data u(0, x), and equal to the integral of f (u) against the unique stationary measure of the equation.

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“…Hitting times for a diffusion Krylov and Safonov in order to prove Theorem 3.4 resembles analytic ideas used in [89] (unfortunately, the book was not translated until much later; see [90]). The key idea is to prove a version of what Landis calls "growth lemma" (DiBenedetto [27] refers to the same phenomenon as "expansion of positivity").…”

Section: Operators In Nondivergence Formmentioning

confidence: 99%

Harnack Inequalities: An Introduction

Kaßmann

2007

Boundary Value Problems

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The aim of this article is to give an introduction to certain inequalities named after Carl Gustav Axel von Harnack. These inequalities were originally defined for harmonic functions in the plane and much later became an important tool in the general theory of harmonic functions and partial differential equations. We restrict ourselves mainly to the analytic perspective but comment on the geometric and probabilistic significance of Harnack inequalities. Our focus is on classical results rather than latest developments. We give many references to this topic but emphasize that neither the mathematical story of Harnack inequalities nor the list of references given here is complete.

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Second Order Equations of Elliptic and Parabolic Type

Cited by 126 publications

References 0 publications

On a mixed problem for the parabolic Lamé type operator

On a mixed problem for the parabolic Lamé type operator

Stochastic CGL equations without linear dispersion in any space dimension

Harnack Inequalities: An Introduction

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