scholarly journals Weak solution of parabolic complex Monge–Ampère equation II

2016 ◽  
Vol 27 (12) ◽  
pp. 1650098
Author(s):  
Do Hoang Son
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Pseudoconvex Domain ◽  
Plurisubharmonic Function ◽  
Initial Condition ◽  
Strictly Pseudoconvex Domain ◽  
Monge Ampere

We study the equation [Formula: see text] in [Formula: see text], where [Formula: see text] and [Formula: see text] is a bounded strictly pseudoconvex domain in [Formula: see text], with the boundary condition [Formula: see text] and the initial condition [Formula: see text]. In this paper, we consider the case, where [Formula: see text] is smooth and [Formula: see text] is an arbitrary plurisubharmonic function in a neighborhood of [Formula: see text] satisfying [Formula: see text].

Existence and Hölder continuity to solutions of the complex Monge–Ampère-type equations with measures vanishing on pluripolar subsets

2021 ◽  
Author(s):  
Le Mau Hai ◽  
Vu Van Quan
Keyword(s):  
Pseudoconvex Domain ◽  
Hölder Continuity ◽  
Type Equation ◽  
Holder Continuity ◽  
Continuous Solutions ◽  
Hölder Continuous ◽  
Strictly Pseudoconvex Domain ◽  
Monge Ampere

In this paper, we establish existence of Hölder continuous solutions to the complex Monge–Ampère-type equation with measures vanishing on pluripolar subsets of a bounded strictly pseudoconvex domain [Formula: see text] in [Formula: see text].

scholarly journals Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation

2020 ◽  
Vol 18 (1) ◽  
pp. 1302-1316
Author(s):  
Guobing Fan ◽  
Zhifeng Yang
Keyword(s):  
Boundary Condition ◽  
Optimal Control ◽  
Weak Solution ◽  
Existence And Uniqueness ◽  
Optimal Solution ◽  
Dispersive Equation ◽  
Weak Dissipation ◽  
Formula Type

Abstract In this paper, we investigate the problem for optimal control of a viscous generalized \theta -type dispersive equation (VG \theta -type DE) with weak dissipation. First, we prove the existence and uniqueness of weak solution to the equation. Then, we present the optimal control of a VG \theta -type DE with weak dissipation under boundary condition and prove the existence of optimal solution to the problem.

scholarly journals Сlassical solution of the mixed problem for the one-dimensional wave equation with the nonsmooth second initial condition

2020 ◽  
Vol 64 (6) ◽  
pp. 657-662
Author(s):  
V. I. Korzyuk ◽  
J. V. Rudzko
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Mixed Problem ◽  
Method Of Characteristics ◽  
Smooth Boundary ◽  
Analytical Form ◽  
Initial Condition ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave

In this article, we study the classical solution of the mixed problem in a quarter of a plane and a half-plane for a one-dimensional wave equation. On the bottom of the boundary, Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at one point. Smooth boundary condition is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. Uniqueness is proved and conditions are established under which a piecewise-smooth solution exists. The problem with linking conditions is considered.

Time, direction of

2018 ◽  
Author(s):  
Jos Uffink
Keyword(s):  
Boundary Condition ◽  
General Principle ◽  
Room Temperature ◽  
Hot Water ◽  
Initial Condition ◽  
Arrow Of Time ◽  
Time Asymmetry ◽  
Time Direction ◽  
James Clerk Maxwell ◽  
Ice Cube

You can pour a tumblerful of water into the sea, but you can never get that same tumblerful of water out again. James Clerk Maxwell gave this as an example of an irreversible process. There are many other homely examples: coffee and milk will mix if stirred, but white coffee does not unmix if stirred in reverse. An ice cube in a glass of hot water will melt, but we never see water at room temperature spontaneously separate into ice and hot water. Physical theories like thermodynamics or hydrodynamics, which codify this type of irreversible phenomenon, do not allow the same kind of behaviour in the forward and backward direction of time. There is thus a striking asymmetry in the two temporal directions. This is usually referred to as the ‘direction of time’ (or ‘time asymmetry’ or ‘anisotropy’ or the ‘arrow of time’). The source of this asymmetry has been sought in various theories of physics, both classical and quantum. Some explanations appeal to some sort of boundary condition, typically an initial condition, which the explanation admits to be, not a law of the theory, but a matter of happenstance. Other explanations advocate some additional general principle about, for example, temporally asymmetric notions of causality or randomness.

Hankel Operators on Pseudoconvex Domains of Finite Type in ℂ2

1998 ◽  
Vol 50 (3) ◽  
pp. 658-672 ◽  
Author(s):  
Frédéric Symesak
Keyword(s):  
Hardy Space ◽  
Finite Type ◽  
Pseudoconvex Domain ◽  
Bergman Spaces ◽  
Hankel Operators ◽  
Weighted Bergman Spaces ◽  
Pseudoconvex Domains ◽  
Sufficient Condition ◽  
Schatten Class ◽  
Strictly Pseudoconvex Domain

AbstractThe aimof this paper is to study small Hankel operators h on the Hardy space or on weighted Bergman spaces,where Ω is a finite type domain in ℂ2 or a strictly pseudoconvex domain in ℂn. We give a sufficient condition on the symbol ƒ so that h belongs to the Schatten class Sp, 1 ≤ p < +∞.

scholarly journals On the evolution of solutions of Burgers equation on the positive quarter-plane

2019 ◽  
Vol 52 (1) ◽  
pp. 237-248
Author(s):  
Esen Hanaç
Keyword(s):  
Boundary Condition ◽  
Analytic Solution ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Condition ◽  
Quarter Plane ◽  
Initial Boundary Value ◽  
Good Agreement

AbstractIn this paper we investigate an initial-boundary value problem for the Burgers equation on the positive quarter-plane; $\matrix{ {{v_t} + v{v_x} - {v_{xx}} = 0,\,\,\,x > 0,\,\,\,t > 0,} \cr {v\left( {x,0} \right) = {u_ + },\,\,\,x > 0,} \cr {v\left( {0,t} \right) = {u_b},\,\,t > 0,} \cr }$ where x and t represent distance and time, respectively, and u+ is an initial condition, ub is a boundary condition which are constants (u+ ≠ ub). Analytic solution of above problem is solved depending on parameters (u+ and ub) then compared with numerical solutions to show there is a good agreement with each solutions.

scholarly journals On the existence of Kobayashi and Bergman metrics for Model domains

2020 ◽  
Author(s):  
Nikolay Shcherbina
Keyword(s):  
Entire Function ◽  
Continuous Function ◽  
Pseudoconvex Domain ◽  
Plurisubharmonic Function ◽  
Holomorphic Curves ◽  
List Type ◽  
Bergman Metric ◽  
The Core ◽  
Bounded Smooth ◽  
Bergman Metrics

Abstract We prove that for a pseudoconvex domain of the form $${\mathfrak {A}} = \{(z, w) \in {\mathbb {C}}^2 : v > F(z, u)\}$$ A = { ( z , w ) ∈ C 2 : v > F ( z , u ) } , where $$w = u + iv$$ w = u + i v and F is a continuous function on $${\mathbb {C}}_z \times {\mathbb {R}}_u$$ C z × R u , the following conditions are equivalent: The domain $$\mathfrak {A}$$ A is Kobayashi hyperbolic. The domain $$\mathfrak {A}$$ A is Brody hyperbolic. The domain $$\mathfrak {A}$$ A possesses a Bergman metric. The domain $$\mathfrak {A}$$ A possesses a bounded smooth strictly plurisubharmonic function, i.e. the core $$\mathfrak {c}(\mathfrak {A})$$ c ( A ) of $$\mathfrak {A}$$ A is empty. The graph $$\Gamma (F)$$ Γ ( F ) of F can not be represented as a foliation by holomorphic curves of a very special form, namely, as a foliation by translations of the graph $$\Gamma ({\mathcal H})$$ Γ ( H ) of just one entire function $${\mathcal {H}} : {\mathbb {C}}_z \rightarrow {\mathbb {C}}_w$$ H : C z → C w .

scholarly journals The Sedov Blast Wave as a Radial Piston Verification Test

2016 ◽  
Vol 1 (3) ◽  
Author(s):  
Clark Pederson ◽  
Bart Brown ◽  
Nathaniel Morgan
Keyword(s):  
Energy Source ◽  
Boundary Condition ◽  
Point Source ◽  
Blast Wave ◽  
Time Varying ◽  
Initial Condition ◽  
Piston Problem ◽  
Great Utility ◽  
Verification Problem ◽  
Verification Test

The Sedov blast wave is of great utility as a verification problem for hydrodynamic methods. The typical implementation uses an energized cell of finite dimensions to represent the energy point source. This approximation can be avoided by directly finding the effects of the energy source as a boundary condition (BC). The proposed method transforms the Sedov problem into an outward moving radial piston problem with a time-varying velocity. A portion of the mesh adjacent to the origin is removed and the boundaries of this hole are forced with the velocities from the Sedov solution. This verification test is implemented on two types of meshes, and convergence is shown. The results from the typical initial condition (IC) method and the new BC method are compared.

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