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user175203
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What does it mean that the function space $L^q_tL^p_x$ is invariant under the (3D) Navier-Stokes scaling $u(x,t) \mapsto \lambda u(\lambda x, \lambda^2 t)$ if $2/q + 3/p = 1$?

Does it mean that you compute the integral $$\left(\int_0^t\left( \left(\int_{\mathbb R^3} (\lambda u(\lambda x, \lambda^2 t))^p\right)^{1/p}\right)^qdt\right)^{1/q}$$$$\left(\int_0^t\left( \left(\int_{\mathbb R^3} (\lambda u(\lambda x, \lambda^2 t))^pdx\right)^{1/p}\right)^qdt\right)^{1/q}$$ and find that it is equal to the one with $\lambda = 1$ only for $2/q + 3/p = 1$?

What does it mean that the function space $L^q_tL^p_x$ is invariant under the (3D) Navier-Stokes scaling $u(x,t) \mapsto \lambda u(\lambda x, \lambda^2 t)$ if $2/q + 3/p = 1$?

Does it mean that you compute the integral $$\left(\int_0^t\left( \left(\int_{\mathbb R^3} (\lambda u(\lambda x, \lambda^2 t))^p\right)^{1/p}\right)^qdt\right)^{1/q}$$ and find that it is equal to the one with $\lambda = 1$ only for $2/q + 3/p = 1$?

What does it mean that the function space $L^q_tL^p_x$ is invariant under the (3D) Navier-Stokes scaling $u(x,t) \mapsto \lambda u(\lambda x, \lambda^2 t)$ if $2/q + 3/p = 1$?

Does it mean that you compute the integral $$\left(\int_0^t\left( \left(\int_{\mathbb R^3} (\lambda u(\lambda x, \lambda^2 t))^pdx\right)^{1/p}\right)^qdt\right)^{1/q}$$ and find that it is equal to the one with $\lambda = 1$ only for $2/q + 3/p = 1$?

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user175203
user175203
  • 61
  • 4

What does the "scaling invariant" Serrin condition mean?

What does it mean that the function space $L^q_tL^p_x$ is invariant under the (3D) Navier-Stokes scaling $u(x,t) \mapsto \lambda u(\lambda x, \lambda^2 t)$ if $2/q + 3/p = 1$?

Does it mean that you compute the integral $$\left(\int_0^t\left( \left(\int_{\mathbb R^3} (\lambda u(\lambda x, \lambda^2 t))^p\right)^{1/p}\right)^qdt\right)^{1/q}$$ and find that it is equal to the one with $\lambda = 1$ only for $2/q + 3/p = 1$?