 # fa.practical evaluation – Does \$int_0^t Vert u_x(s,cdot) Vert_{L^2} ds le C\$ indicate \$Vert u_x (t,cdot) Vert_{L^2(mathbb R)} le C\$ within the warmth equation? Answer

Hello pricey customer to our community We will proffer you an answer to this query fa.practical evaluation – Does \$int_0^t Vert u_x(s,cdot) Vert_{L^2} ds le C\$ indicate \$Vert u_x (t,cdot) Vert_{L^2(mathbb R)} le C\$ within the warmth equation? ,and the respond will breathe typical via documented data sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning concerning the respond to this query.

fa.practical evaluation – Does \$int_0^t Vert u_x(s,cdot) Vert_{L^2} ds le C\$ indicate \$Vert u_x (t,cdot) Vert_{L^2(mathbb R)} le C\$ within the warmth equation?

For the parabolic equation
$$u_t + f(u)_x – u_{xx} = 0$$
one has
$$Vert u(t,cdot) Vert_{L^2(mathbb R)} + 2int_0^t Vert u_x(s,cdot) Vert_{L^2} ds le Vert u(0,cdot) Vert_{L^2(mathbb R)}.$$
If $$t le T$$ (for a hard and fast $$T>0$$), does the appraise above indicate too $$Vert u_x (t,cdot) Vert_{L^2(mathbb R)} le C$$
(with out time integration)?

In different phrases, one can too question: if $$int_0^T f(s) ds le C$$ is it undoubted that $$f(t) le C$$ for all $$t in [0,T]$$?

we’ll proffer you the answer to fa.practical evaluation – Does \$int_0^t Vert u_x(s,cdot) Vert_{L^2} ds le C\$ indicate \$Vert u_x (t,cdot) Vert_{L^2(mathbb R)} le C\$ within the warmth equation? query through our community which brings all of the solutions from a number of dependable sources.