Short time existence and uniqueness in Hölder spaces for the 2D dynamics of dislocation densities

We discuss existence and uniqueness of solutions for a one-dimensional parabolic evolution equation with a free boundary. This problem was introduced by Lasry and Lions as description of the dynamical formation of the price of a trading good. Short time existence and uniqueness is established by a contraction argument. Then we discuss the issue of global-in-time-extension of the local solution which is closely related to the regularity of the free boundary. We also present numerical results.

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A New Flow on Open Manifolds: Short Time Existence and Uniqueness

Bulletin of the Iranian Mathematical Society ◽

10.1007/s41980-020-00421-6 ◽

2020 ◽

Author(s):

Jurgen Eichhorn ◽

Hajar Ghahremani-Gol ◽

Asadollah Razavi

Keyword(s):

Existence And Uniqueness ◽

Open Manifolds ◽

Short Time ◽

Time Existence

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Existence and uniqueness theorems for the full three-dimensional Ericksen–Leslie system

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202517500178 ◽

2017 ◽

Vol 27 (05) ◽

pp. 807-843 ◽

Cited By ~ 2

Author(s):

Gregory A. Chechkin ◽

Tudor S. Ratiu ◽

Maxim S. Romanov ◽

Vyacheslav N. Samokhin

Keyword(s):

Existence And Uniqueness ◽

Three Dimensional ◽

Strong Solutions ◽

Periodic Case ◽

Initial Value ◽

Neumann Type ◽

Type Boundary ◽

Short Time ◽

Time Existence ◽

Leslie System

In this paper, we study the three-dimensional Ericksen–Leslie equations for the nematodynamics of liquid crystals. We prove short time existence and uniqueness of strong solutions for the initial value problem for the periodic case and in bounded domains with both Dirichlet- and Neumann-type boundary conditions.

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A Simple Proof of the Short-time Existence and Uniqueness for Ricci Flow

10.7546/crabs.2019.05.01 ◽

2019 ◽

Author(s):

Kaveh Eftekharinasab

Keyword(s):

Simple Proof ◽

Ricci Flow ◽

Existence And Uniqueness ◽

Short Time ◽

Time Existence

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MIXED FRACTIONAL DIFFERENTIATION OPERATORS IN HÖLDER SPACES DEFINED BY USUAL HÖLDER CONDITION

Chronos Journal ◽

10.31618/2658-7556-2019-38-11-1 ◽

2019 ◽

Author(s):

T. Mamatov ◽

R. Sabirova ◽

D. Barakaev

Keyword(s):

Fractional Derivative ◽

Fractional Differentiation ◽

Hölder Condition ◽

Hölder Spaces ◽

Main Interest ◽

Hölder Class ◽

Holder Class ◽

Function Of Two Variables ◽

Holder Spaces

We study mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The main interest being in the evaluation of the latter for the mixed fractional derivative in the cases Hölder class defined by usual Hölder condition

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Lorentz spaces in action on pressureless systems arising from models of collective behavior

Journal of Evolution Equations ◽

10.1007/s00028-021-00668-4 ◽

2021 ◽

Author(s):

Raphaël Danchin ◽

Piotr Bogusław Mucha ◽

Patrick Tolksdorf

Keyword(s):

Initial Data ◽

Collective Behavior ◽

Existence And Uniqueness ◽

Maximal Regularity ◽

Parabolic Systems ◽

Lorentz Spaces ◽

Regularity Estimate ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Time Existence

AbstractWe are concerned with global-in-time existence and uniqueness results for models of pressureless gases that come up in the description of phenomena in astrophysics or collective behavior. The initial data are rough: in particular, the density is only bounded. Our results are based on interpolation and parabolic maximal regularity, where Lorentz spaces play a key role. We establish a novel maximal regularity estimate for parabolic systems in $$L_{q,r}(0,T;L_p(\Omega ))$$ L q , r ( 0 , T ; L p ( Ω ) ) spaces.

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