Related Experiment Video
Updated: Aug 9, 2026
11:53
Spatial Multiobjective Optimization of Agricultural Conservation Practices using a SWAT Model and an Evolutionary Algorithm
Published on: December 9, 2012
Systems of conservation equations with a convex extension
1Courant Institute of Mathematical Sciences, New York University, New York, N.Y. 10012.
Summary
This study examines nonlinear conservation laws, revealing that an additional convex conserved quantity allows systems to be written in symmetric hyperbolic form. An entropy inequality is derived for discontinuous solutions.
Area of Science:
- Mathematics
- Applied Mathematics
- Physics
Background:
- First-order systems of nonlinear conservation laws are fundamental in fluid dynamics and other physical sciences.
- The existence of additional conservation laws can simplify the analysis of these systems.
- Understanding the properties of conserved quantities is crucial for developing accurate models.
Purpose of the Study:
- To investigate the implications of an additional conservation law in nonlinear systems.
- To determine conditions under which these systems can be transformed into a symmetric hyperbolic form.
- To derive an entropy inequality for discontinuous solutions.
Main Methods:
- Analysis of first-order nonlinear conservation laws.
- Mathematical transformation to a symmetric hyperbolic form.
- Derivation of an entropy inequality using established principles.
Main Results:
- Demonstration that an additional convex conserved quantity enables the system to be expressed in symmetric hyperbolic form.
- Successful derivation of an entropy inequality for discontinuous solutions.
- Provides a theoretical framework for analyzing systems with extra conserved quantities.
Conclusions:
- The convexity of an additional conserved quantity is key to achieving a symmetric hyperbolic form.
- The derived entropy inequality offers a tool for studying complex solutions.
- This work advances the theoretical understanding of nonlinear conservation laws.
Related Concept Videos
01:14Conservation of Energy in Control Volume
Consider a turbine operating under steady-flow conditions. The control volume is drawn around the turbine, with fluid entering at one point and exiting at another. The turbine extracts energy from the fluid, which performs mechanical work (shaft work).
For steady flow systems, the time derivative of the stored energy becomes zero since there is no energy accumulation within the control volume. This simplifies the energy equation to:
For steady flow systems, the time derivative of the stored energy becomes zero since there is no energy accumulation within the control volume. This simplifies the energy equation to:
01:30Equations of Equilibrium in Three Dimensions
When analyzing structures or systems at rest, it is necessary to ensure they are in equilibrium. This is where the vector and scalar equations of equilibrium come into play. These equations are crucial in ensuring a structure is stable and will not collapse or fall apart. The vector and scalar equations of equilibrium provide a framework for analyzing the forces acting on a body.
According to the vector equations of equilibrium, the vector sum of all the external forces acting on a body must...
According to the vector equations of equilibrium, the vector sum of all the external forces acting on a body must...
01:07Conservation of Mass in Fixed, Nondeforming Control Volume
The principle of conservation of mass is fundamental in fluid dynamics and is crucial for analyzing flow within fixed control volumes, such as pipes or ducts. This principle states that the total mass within a control volume remains constant unless altered by the inflow or outflow of mass through the control surfaces. This results in a vital relationship for steady, incompressible flow where the mass entering a system equals the mass leaving it.
In the case of a sewer pipe, which can be modeled...
In the case of a sewer pipe, which can be modeled...
01:20Separable Differential Equations
A separable differential equation is a type of first-order differential equation where the derivative dy/dx can be expressed as a product of two functions: one that depends only on x and another that depends only on y. This allows for the rearrangement of the equation so that all terms involving y are on one side, and all terms involving x are on the other. This process, known as the separation of variables, simplifies the process of solving the equation by enabling the integration of both...
01:14Conservation of Mass in Moving, Nondeforming Control Volume
Stormwater detention basins are essential in managing runoff during heavy rainfall, particularly in urban areas where impervious surfaces increase the risk of flooding. Understanding the conservation of mass in these systems allows engineers to optimize basin performance, balancing inflow, outflow, and water storage.
In the context of a detention basin, the conservation of mass states that the total mass of water entering the basin must equal the mass leaving the basin plus any accumulation of...
In the context of a detention basin, the conservation of mass states that the total mass of water entering the basin must equal the mass leaving the basin plus any accumulation of...
01:21Partial Differential Equations
A stone dropped into a still pond generates waves that propagate outward in circular patterns, creating a dynamic surface whose elevation depends on both position and time. At any given location, the water level oscillates as the wave passes, while at any fixed moment, the surface exhibits smooth, curved structures extending across space. This dual dependence requires a mathematical description that accounts for variation in multiple variables simultaneously.At a fixed point on the water...