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Graduate · Critical norm scaling

Critical norm scaling: An exponent just below critical

Critical norm scaling: investigate an exponent just below critical with final spatial factor λ = 3; lebesgue exponent p = 2.5.

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Watch the relationship

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Critical norm scaling: An exponent just below critical. Current scale λ: 1. Norm scaling exponent: -0.2. Lᵖ norm ratio: 1. Volume ratio: 1Amplitude and volume compete01.320.252.134norm ratioscale factor λ → · labeled axes rescale to this model
The base field is smooth, compactly supported, and nonzero on R³. This is a spatial norm calculation with finite p, not an evolved PDE solution or a statement settling any fluid regularity question. The λ convention is explicitly λu(λx).

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Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Current scale λ
1
Norm scaling exponent
-0.2
Lᵖ norm ratio
1
Volume ratio
1

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Understand what you are seeing

The idea behind the motion.

Changing amplitude and spatial concentration together affects different norms differently. In three dimensions, the change of variables contributes λ⁻³ to volume, while the amplitude contributes λᵖ to the pth-power integral. Their balance makes p=3 invariant under this particular spatial scaling. This investigation starts with Final spatial factor λ = 3; Lebesgue exponent p = 2.5. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

uλ(x)=λu(λx); ||uλ||ₚ/||u||ₚ=λ^(1−3/p)

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Fix a finite exponent p and a nonzero base field with finite Lᵖ norm. The starting case is “An exponent just below critical.”

  2. STEP 2

    Follow the changing quantity

    Move the spatial scale from one toward the selected λ. Compare the norm ratio with the amplitude and volume factors.

  3. STEP 3

    Explain and test the result

    Set p=3 and check that the norm ratio is one at every scale. Compare p below and above three without turning a scaling identity into a regularity theorem.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Final spatial factor λ = 3; Lebesgue exponent p = 2.5. Pause the timeline at 40%. Given current scale λ = 1.8, calculate norm scaling exponent, lᵖ norm ratio, volume ratio. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

The exponent is 1−3/2.5=-0.2. At λ=1.8, the norm ratio is (1.8)^(-0.2)=0.88909; the volume ratio is (1.8)^(−3)=0.171468. Results: Norm scaling exponent: -0.2; Lᵖ norm ratio: 0.889; Volume ratio: 0.171. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

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Graduate · Calculus

Find which velocity norm survives Navier–Stokes scaling

Compute norm scaling in three spatial dimensions.