Learn with Amar
Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Compute norm scaling in three spatial dimensions.
- Justify the conclusion "p=3 is scale-invariant; p=2 scales as λ⁻¹ᐟ²" using the stated assumptions.
Before you start
Changes of variables and Lp norms.
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 uλ(x,t)=λu(λx,λ²t), determine how the spatial Lp norm scales on R³.
Why this math matters
Compute norm scaling in three spatial dimensions. 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 domain is R³ and the scale factor λ is positive.
- For the finite-p formula, 1≤p<∞ and the norm is finite; the L∞ case is treated separately.
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.
Find which velocity norm survives Navier–Stokes scaling
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For uλ(x,t)=λu(λx,λ²t), determine how the spatial Lp norm scales on R³.
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
∫|uλ(x,t)|ᵖdx=λᵖλ⁻³∫|u(y,λ²t)|ᵖdy
Substitute y=λx, including the volume Jacobian.
Work through the mathematics
||uλ(t)||p=λ^(1−3/p)||u(λ²t)||p
Taking the p-th root identifies the scaling exponent.
Check the conclusion
p=3 is scale-invariant; p=2 scales as λ⁻¹ᐟ²
The energy norm alone becomes smaller under concentration scaling, illustrating why an L² bound is not automatically a critical regularity bound.
The result
p=3 is scale-invariant; p=2 scales as λ⁻¹ᐟ²
The energy norm alone becomes smaller under concentration scaling, illustrating why an L² bound is not automatically a critical regularity bound.
Common mistakes to catch
- Domain and boundary conditions may not be preserved by scaling.
- Scale invariance alone is not a regularity proof.
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
How does the L∞ norm scale?
Show a hint
Take the supremum after rescaling.
Reveal answer and explanation
By λ
A spatial coordinate change does not reduce the maximum amplitude.
Practice 2
What pressure scaling accompanies this velocity scaling?
Show a hint
Match the gradient term to the acceleration scale.
Reveal answer and explanation
pλ(x,t)=λ²p(λx,λ²t)
Its gradient then scales as λ³, matching the other momentum terms.
Take the idea with you
Use dimensional scaling to identify which proposed a priori estimate could control concentration.
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: Remove the longitudinal component of a Fourier velocity
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