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Related Concept Videos

Navier–Stokes Equations01:28

Navier–Stokes Equations

For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting...
Partial Differential Equations01:21

Partial Differential Equations

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Euler's Equations of Motion01:28

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In fluid mechanics, shear stresses arise from viscosity, which represents a fluid's internal resistance to deformation. For low-viscosity fluids, like water, these stresses are minimal, simplifying flow analysis by allowing the fluid to be treated as inviscid, or frictionless. In an inviscid fluid, shear stresses are absent, leaving only normal stresses, which act perpendicularly to fluid elements. Notably, pressure — defined as the negative of the normal stress — remains uniform across...
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Differential Form of Maxwell's Equations01:17

Differential Form of Maxwell's Equations

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Related Experiment Video

Updated: Jun 12, 2026

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180&#176; Curved Artery Test Section
11:00

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

Published on: July 19, 2016

Quantum algorithms for viscosity solutions to nonlinear Hamilton-Jacobi equations based on an entropy penalization

Shi Jin1,2,3, Nana Liu1,2,3,4

  • 1Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China.

Proceedings of the National Academy of Sciences of the United States of America
|June 10, 2026
PubMed
Summary

This study introduces a quantum framework for efficiently solving nonlinear Hamilton-Jacobi equations. The method approximates complex dynamics with linear ones, enabling quantum simulations for applications in optimal control and machine learning.

Keywords:
entropy penalizationnonlinear partial differential equationsquantum algorithmsviscosity solutions

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Last Updated: Jun 12, 2026

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180&#176; Curved Artery Test Section
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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Area of Science:

  • Applied Mathematics
  • Quantum Computing
  • Computational Science

Background:

  • Nonlinear Hamilton-Jacobi equations are crucial in diverse fields like optimal control, mean-field games, and machine learning.
  • Efficiently solving these equations, especially for convex Hamiltonians, presents significant computational challenges.

Purpose of the Study:

  • To develop a novel framework for the efficient extraction of viscosity solutions for nonlinear Hamilton-Jacobi equations with convex Hamiltonians.
  • To enable quantum simulations of these complex dynamics for broader applicability.

Main Methods:

  • An entropy penalization method is employed, generalizing the Cole-Hopf transform to convex Hamiltonians.
  • Viscous Hamilton-Jacobi dynamics are approximated by discrete-time linear dynamics, which in turn approximate a linear heat-like parabolic equation.
  • This approach extends to continuous-time dynamics and is suitable for quantum simulation.

Main Results:

  • The framework provides a method for approximating viscosity solutions of nonlinear Hamilton-Jacobi equations using linear dynamics.
  • The method's validity extends to arbitrary nonlinearities with convex Hamiltonians and for arbitrarily long times.
  • Quantum algorithms (analog and digital) are presented for extracting key properties of the viscosity solution without full state reconstruction.

Conclusions:

  • The developed framework offers an efficient and robust approach for solving nonlinear Hamilton-Jacobi equations using quantum computation.
  • This work overcomes a key obstacle in quantum algorithms for nonlinear partial differential equations, paving the way for advanced simulations.
  • The method's applicability spans various scientific and engineering domains requiring solutions to these types of equations.