ON A PARABOLIC FREE BOUNDARY EQUATION MODELING PRICE FORMATION

2009 ◽  
Vol 19 (10) ◽  
pp. 1929-1957 ◽  
Author(s):  
P. A. MARKOWICH ◽  
N. MATEVOSYAN ◽  
J.-F. PIETSCHMANN ◽  
M.-T. WOLFRAM
Keyword(s):  
Free Boundary ◽  
Existence And Uniqueness ◽  
Price Formation ◽  
Equation Modeling ◽  
Boundary Equation ◽  
Uniqueness Of Solutions ◽  
One Dimensional ◽  
Time Extension ◽  
Short Time ◽  
Time Existence

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.

scholarly journals Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

2006 ◽  
Vol 181 (3) ◽  
pp. 449-504 ◽  
Author(s):  
Olivier Alvarez ◽  
Philippe Hoch ◽  
Yann Le Bouar ◽  
Régis Monneau
Keyword(s):  
Existence And Uniqueness ◽  
Dislocation Dynamics ◽  
Uniqueness Of The Solution ◽  
Short Time ◽  
Time Existence

Long-time existence of classical solutions to a one-dimensional swelling gel

2014 ◽  
Vol 25 (01) ◽  
pp. 165-194 ◽  
Author(s):  
M. Carme Calderer ◽  
Robin Ming Chen
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Lagrangian Formulation ◽  
Classical Solutions ◽  
Governing Equations ◽  
Fixed Boundary ◽  
One Dimensional ◽  
Long Time Existence ◽  
Long Time ◽  
Time Existence

In this paper, we derived a model which describes the swelling dynamics of a gel and study the system in one-dimensional geometry with a free boundary. The governing equations are hyperbolic with a weakly dissipative source. Using a mass-Lagrangian formulation, the free boundary is transformed into a fixed boundary. We prove the existence of long-time C1-solutions to the transformed fixed boundary problem.

scholarly journals On the Solution of a Hyperbolic One-Dimensional Free Boundary Problem for a Maxwell Fluid

2011 ◽  
Vol 2011 ◽  
pp. 1-26
Author(s):  
Lorenzo Fusi ◽  
Angiolo Farina
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Boundary Data ◽  
Inner Core ◽  
Representation Formula ◽  
Maxwell Fluid ◽  
Explicit Representation ◽  
Boundary Equation ◽  
One Dimensional

We study a hyperbolic (telegrapher's equation) free boundary problem describing the pressure-driven channel flow of a Bingham-type fluid whose constitutive model was derived in the work of Fusi and Farina (2011). The free boundary is the surface that separates the inner core (where the velocity is uniform) from the external layer where the fluid behaves as an upper convected Maxwell fluid. We present a procedure to obtain an explicit representation formula for the solution. We then exploit such a representation to write the free boundary equation in terms of the initial and boundary data only. We also perform an asymptotic expansion in terms of a parameter tied to the rheological properties of the Maxwell fluid. Explicit formulas of the solutions for the various order of approximation are provided.

LOCAL EXISTENCE AND UNIQUENESS THEORY FOR THE SECOND SOUND EQUATION IN ONE SPACE DIMENSION

2012 ◽  
Vol 09 (01) ◽  
pp. 177-193 ◽  
Author(s):  
KEIICHI KATO ◽  
YUUSUKE SUGIYAMA
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Space Dimension ◽  
Local Existence ◽  
Second Sound ◽  
Uniqueness Of Solutions ◽  
Local Existence And Uniqueness ◽  
The Cauchy Problem ◽  
Time Existence ◽  
Uniqueness Theory

We study the local-in-time existence and uniqueness of the Cauchy problem for the nonlinear wave equation [Formula: see text], which is called the second sound equation. Assuming that u(0, x) = φ ≥ A > 0, φ ∈ C1, and ∂xφ ∈ Hs, we establish the uniqueness of solutions without restriction on their amplitude.

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

2017 ◽  
Vol 27 (05) ◽  
pp. 807-843 ◽  
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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