Learn with Amar
Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Derive the energy balance and understand what lower-regularity limits can retain.
- Justify the conclusion "½d||u||₂²/dt+ν||∇u||₂²=0" using the stated assumptions.
Before you start
Weak solutions, integration by parts, and kinetic energy.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For a smooth unforced periodic incompressible velocity, derive the L² energy law.
Why this math matters
Derive the energy balance and understand what lower-regularity limits can retain. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The smooth calculation uses periodic boundaries or stationary homogeneous no-slip boundaries u=0.
- Viscosity is a positive constant.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Separate a smooth energy identity from a weak energy inequality
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For a smooth unforced periodic incompressible velocity, derive the L² energy law.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
Take the L² inner product of the equation with u
The time term becomes half the derivative of the squared L² norm.
Work through the mathematics
∫((u·∇)u)·u=0 and ∫∇p·u=0
Divergence-free transport and periodicity cancel the nonlinear and pressure contributions.
Check the conclusion
½d||u||₂²/dt+ν||∇u||₂²=0
For suitable finite-energy weak solutions, one generally works with the corresponding integrated inequality rather than assuming every smooth manipulation remains valid.
The result
½d||u||₂²/dt+ν||∇u||₂²=0
For suitable finite-energy weak solutions, one generally works with the corresponding integrated inequality rather than assuming every smooth manipulation remains valid.
Common mistakes to catch
- Do not upgrade an inequality to equality without regularity justification.
- Energy control and pointwise regularity are different estimates.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What term appears with body force f?
Show a hint
Retain its inner product with velocity.
Reveal answer and explanation
∫f·u
Force can inject or remove energy depending on its alignment with u.
Practice 2
Does finite kinetic energy alone bound the pointwise maximum speed?
Show a hint
Compare L² and L∞ control.
Reveal answer and explanation
No
A field can concentrate large values in a small region while its L² norm remains finite.
Take the idea with you
Identify the exact estimate preserved when approximating a PDE solution by smoother fields.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Identify the three-dimensional vorticity stretching term
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