SELF-SIMILAR SOLUTIONS OF TWO-DIMENSIONAL CONSERVATION LAWS

2004 ◽  
Vol 01 (03) ◽  
pp. 445-492 ◽  
Author(s):  
BARBARA LEE KEYFITZ
Keyword(s):  
Conservation Laws ◽  
Elliptic Equations ◽  
Free Boundary Problems ◽  
Two Dimensional ◽  
Similar Reduction ◽  
Boundary Problems ◽  
Degenerate Elliptic ◽  
Preliminary Version ◽  
Self Similar ◽  
Similar Solutions

Self-similar reduction of an important class of two-dimensional conservation laws leads to boundary value problems for equations which change type. We have established a method for solving free boundary problems for quasilinear degenerate elliptic equations which arise when shocks interact with the subsonic (nonhyperbolic) part of the solution. This paper summarizes the principal features of the method.A preliminary version of these notes formed the basis of a series of three lectures at the Newton Institute in April, 2003. They are a report of research carried out jointly with Sunčica Čanić, Eun Heui Kim and Gray Lieberman.

scholarly journals Free boundary problems in shock reflection/diffraction and related transonic flow problems

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140276 ◽  
Author(s):  
Gui-Qiang Chen ◽  
Mikhail Feldman
Keyword(s):  
Free Boundary ◽  
High Speed ◽  
Free Boundary Problems ◽  
Fluid Flows ◽  
Shock Reflection ◽  
Two Dimensional ◽  
Open Problems ◽  
Boundary Problems ◽  
Flow Problems ◽  
Concave Corner

Shock waves are steep wavefronts that are fundamental in nature, especially in high-speed fluid flows. When a shock hits an obstacle, or a flying body meets a shock, shock reflection/diffraction phenomena occur. In this paper, we show how several long-standing shock reflection/diffraction problems can be formulated as free boundary problems, discuss some recent progress in developing mathematical ideas, approaches and techniques for solving these problems, and present some further open problems in this direction. In particular, these shock problems include von Neumann's problem for shock reflection–diffraction by two-dimensional wedges with concave corner, Lighthill's problem for shock diffraction by two-dimensional wedges with convex corner, and Prandtl-Meyer's problem for supersonic flow impinging onto solid wedges, which are also fundamental in the mathematical theory of multidimensional conservation laws.

scholarly journals Analysis of the self-similar solutions of a generalized Burger's equation with nonlinear damping

2001 ◽  
Vol 7 (3) ◽  
pp. 253-282 ◽  
Author(s):  
Ch. Srinivasa Rao ◽  
P. L. Sachdev ◽  
Mythily Ramaswamy
Keyword(s):  
Initial Conditions ◽  
Nonlinear Damping ◽  
Nonlinear Ordinary Differential Equation ◽  
The Self ◽  
Similar Reduction ◽  
Generalized Burgers Equation ◽  
Different Types ◽  
Self Similar ◽  
Parameter Ranges ◽  
Similar Solutions

The nonlinear ordinary differential equation resulting from the self-similar reduction of a generalized Burgers equation with nonlinear damping is studied in some detail. Assuming initial conditions at the origin we observe a wide variety of solutions – (positive) single hump, unbounded or those with a finite zero. The existence and nonexistence of positive bounded solutions with different types of decay (exponential or algebraic) to zero at infinity for specific parameter ranges are proved.

scholarly journals On two phase free boundary problems governed by elliptic equations with distributed sources

2014 ◽  
Vol 7 (4) ◽  
pp. 673-693 ◽  
Author(s):  
Daniela De Silva ◽  
◽  
Fausto Ferrari ◽  
Sandro Salsa ◽  
◽  
...  
Keyword(s):  
Free Boundary ◽  
Elliptic Equations ◽  
Free Boundary Problems ◽  
Two Phase ◽  
Boundary Problems

Existence of self-similar solutions of the two-dimensional Navier–Stokes equation for non-Newtonian fluids

2019 ◽  
Vol 26 (1/2) ◽  
pp. 167-178 ◽  
Author(s):  
Dongming Wei ◽  
Samer Al-Ashhab
Keyword(s):  
Newtonian Fluids ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Infinite Domain ◽  
Navier Stokes Equation ◽  
Continuity Equations ◽  
Self Similar ◽  
Similar Solutions ◽  
Two Parameters

The reduced problem of the Navier–Stokes and the continuity equations, in two-dimensional Cartesian coordinates with Eulerian description, for incompressible non-Newtonian fluids, is considered. The Ladyzhenskaya model, with a non-linear velocity dependent stress tensor is adopted, and leads to the governing equation of interest. The reduction is based on a self-similar transformation as demonstrated in existing literature, for two spatial variables and one time variable, resulting in an ODE defined on a semi-infinite domain. In our search for classical solutions, existence and uniqueness will be determined depending on the signs of two parameters with physical interpretation in the equation. Illustrations are included to highlight some of the main results.

Some experiments with singularities in linear elliptic partial differential equations

1971 ◽  
Vol 323 (1553) ◽  
pp. 179-190 ◽  
Keyword(s):  
Finite Difference ◽  
Elliptic Equations ◽  
Good Accuracy ◽  
Free Boundary Problems ◽  
Finite Difference Methods ◽  
Polar Coordinates ◽  
Isolated Singularity ◽  
Boundary Problems ◽  
Finite Difference Equations ◽  
Cylindrical Polar Coordinates

Error analysis for finite-difference methods reveals that very small intervals will be needed for good accuracy in the neighbourhood of boundary singularities in elliptic equations. Methods are discussed for reducing the labour by combining finite-difference equations remote from singularities with special formulae taking implicit account of the singularities. The analysis of the latter is also outlined, and two practical examples are treated, one with an isolated singularity in a problem with Cartesian coordinates and one with two singularities in a problem with cylindrical polar coordinates. Free boundary problems are considered briefly. Comparison is made with previous methods and some suggestions for further research are included.

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