scholarly journals Some Properties of Solutions to Weakly Hypoelliptic Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Christian Bär
Keyword(s):  
Parabolic Equations ◽  
Complex Analysis ◽  
Bounded Solution ◽  
Removable Singularity ◽  
Local Solution ◽  
Singularity Theorem ◽  
Hypoelliptic Equation ◽  
Constant Coefficients ◽  
Hypoelliptic Equations ◽  
Liouville’S Theorem

A linear different operatorLis called weakly hypoelliptic if any local solutionuofLu=0is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which coverall elliptic, overdetermined elliptic, subelliptic, and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients, we show that Liouville's theorem holds, any bounded solution must be constant, and anyLp-solution must vanish.

REGULARITY OF WEAK SOLUTIONS FOR THE NAVIER–STOKES EQUATIONS IN THE CLASS L∞(BMO-1)

2012 ◽  
Vol 14 (03) ◽  
pp. 1250020 ◽  
Author(s):  
WENDONG WANG ◽  
ZHIFEI ZHANG
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Stokes Equations ◽  
Removable Singularity ◽  
Navier Stokes ◽  
Singularity Theorem ◽  
Navier Stokes Equations ◽  
Regularity Of Weak Solutions ◽  
Mild Assumption

We study the regularity of weak solution for the Navier–Stokes equations in the class L∞( BMO-1). It is proved that the weak solution in L∞( BMO-1) is regular if it satisfies a mild assumption on the vorticity direction, or it is axisymmetric. A removable singularity theorem in ∈ L∞( VMO-1) is also proved.

scholarly journals ON THE CONSTRUCTION OF A FUNDAMENTAL SYSTEM OF SOLUTIONS OF A LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS OF AN ARBITRARY ORDER

2020 ◽  
Vol 70 (2) ◽  
pp. 53-58
Author(s):  
P.B. Beisebay ◽  
◽  
G.H. Mukhamediev ◽  
Keyword(s):  
Differential Equation ◽  
Complex Analysis ◽  
Arbitrary Order ◽  
Fundamental System ◽  
Homogeneous Differential Equation ◽  
Complex Roots ◽  
Linear Homogeneous Differential Equation ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Homogeneous Linear

The paper proposes a method of presentation topics «On the construction of a fundamental system of solutions of a linear homogeneous differential equation with constant coefficients of an arbitrary order». In the traditional presentation of this topic in the case when the characteristic equation has complex roots, the particular solutions of the equation corresponding to them are constructed by applying the elements of complex analysis. In consequence of that, for students in the field, whose training programs included the theory of linear differential equations with constant coefficients and at the same time does not include the study of the theory of complex analysis, types of private solving the equation in this case is given without substantiation, or as a known fact, only for this case, previously issued elements complex analysis. Offered in the presentation technique differs from the traditional presentation of the topic in that it partial solutions scheme for constructing fundamental system of homogeneous linear equation with constant coefficients of arbitrary order is based only on the basis of the properties of the differential form corresponding to the left side of the equation, without using the elements of the theory of complex analysis.

scholarly journals The cost of controlling strongly degenerate parabolic equations

2020 ◽  
Vol 26 ◽  
pp. 2
Author(s):  
P. Cannarsa ◽  
P. Martinez ◽  
J. Vancostenoble
Keyword(s):  
Parabolic Equations ◽  
Complex Analysis ◽  
Degenerate Parabolic Equations ◽  
Control Cost ◽  
Locally Distributed Control ◽  
Degenerate Parabolic ◽  
The Moment ◽  
The Cost ◽  
Strongly Degenerate ◽  
Degeneracy Parameter

We consider the typical one-dimensional strongly degenerate parabolic operator Pu = ut − (xαux)x with 0 < x < ℓ and α ∈ (0, 2), controlled either by a boundary control acting at x = ℓ, or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter α. We prove that the control cost blows up with an explicit exponential rate, as eC/((2−α)2T), when α → 2− and/or T → 0+. Our analysis builds on earlier results and methods (based on functional analysis and complex analysis techniques) developed by several authors such as Fattorini-Russel, Seidman, Güichal, Tenenbaum-Tucsnak and Lissy for the classical heat equation. In particular, we use the moment method and related constructions of suitable biorthogonal families, as well as new fine properties of the Bessel functions Jν of large order ν (obtained by ordinary differential equations techniques).

scholarly journals Construction of fundamental solutions of hypoelliptic equations by the use of a probabilistic method

1978 ◽  
Vol 72 ◽  
pp. 103-126 ◽  
Author(s):  
T. Matsuzawa
Keyword(s):  
Parabolic Equations ◽  
Probabilistic Method ◽  
Fundamental Solutions ◽  
Second Order ◽  
Degenerate Parabolic Equations ◽  
Degenerate Parabolic ◽  
Hypoelliptic Equations

A. Friedman, [5] has constructed the fundamental solutions for a class of degenerate parabolic equations of the second order by making use of a probabilistic method and obtained the estimates for the fundamental solutions, especially near the degenerating manifolds (obstacles) of given operators, where a probabilistic method plays an important role.

scholarly journals Iterative solution of quasilinear parabolic equations by parabolic equations with constant coefficients

1982 ◽  
Vol 24 (2) ◽  
pp. 203-222
Author(s):  
John E. Lavery
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Quasilinear Parabolic Equations ◽  
Constant Coefficients ◽  
Initial Boundary Value ◽  
Quasilinear Parabolic

AbstractA method for solving quasilinear parabolic equations of the typesthat differs radically from previously known methods is proposed. For each initial-boundary-value problem of one of these types that has boundary conditions of the first kind (second kind), a conjugate initial-boundary-value problem of the other type that has boundary conditions of the second kind (first kind) is defined. Based on the relations connecting the solutions of a pair of conjugate problems, a series of parabolic equations with constant coefficients that do not change step to step is constructed. The method proposed consists in calculating the solutions of the equations of this series. It is shown to have linear convergence. Results of a series of numerical experiments in a finite-difference setting show that one particular implementation of the proposed method has a smaller domain of convergence than Newton's method but that it sometimes converges faster within that domain.

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