Ngô Quốc Anh

February 26, 2010

Heat equations: The first pointwise estimates

Filed under: Các Bài Tập Nhỏ, PDEs — Ngô Quốc Anh @ 17:08

I am going to discuss a series of results involving pointwise estimates of heat and wave equations. Let us start with the following  simple problem

Consider the Cauchy problem for the heat operator in \mathbb R

\displaystyle u_t = u_{xx} \quad x\in \mathbb R, t>0

with the initial condition

\displaystyle u(x,0)=f(x), \quad x \in \mathbb R.

We assume f is bounded, continuous, and satisfies

\displaystyle\int_{ - \infty }^{ + \infty } {{{\left| {f(x)} \right|}^2}dx} < \infty.

We will show that there exists a constant C such that

\displaystyle \left| {u(x,t)} \right| \leqslant \frac{C}{{{t^{\frac{1}{4}}}}}

for all x \in \mathbb R and t>0.

Proof. The solution to the problem is given by the Poisson formula

\displaystyle u(x,t) = \frac{1}{{\sqrt {2\pi t} }}\int_{ - \infty }^{ + \infty } {f(y){e^{ - \frac{{{{(x - y)}^2}}}{{4t}}}}dy} .

By the Cauchy inequality, we have

\displaystyle\left| {u(x,t)} \right| \leqslant \frac{1}{{\sqrt {2\pi t} }}{\left( {\int_{ - \infty }^{ + \infty } {{{\left| {f(y)} \right|}^2}dy} } \right)^{\frac{1}{2}}}{\left( {\int_{ - \infty }^{ + \infty } {{e^{ - \frac{{{{(x - y)}^2}}}{{2t}}}}dy} } \right)^{\frac{1}{2}}}.

Set y=x+\sqrt{2t}\eta in the second integral of the above inequality. Then we get immediately

\displaystyle\int_{ - \infty }^{ + \infty } {{e^{ - \frac{{{{(x - y)}^2}}}{{2t}}}}dy} = \int_{ - \infty }^{ + \infty } {{e^{ - \frac{{{{(\sqrt {2t} \eta )}^2}}}{{2t}}}}\sqrt {2t} d\eta } = \sqrt {2t} \int_{ - \infty }^{ + \infty } {{e^{ - {\eta ^2}}}d\eta } .

Thus

\displaystyle\left| {u(x,t)} \right| \leqslant \frac{1}{{\sqrt {2\pi } {t^{\frac{1}{2}}}}}{\left( {\int_{ - \infty }^{ + \infty } {{{\left| {f(y)} \right|}^2}dy} } \right)^{\frac{1}{2}}}{\left( {\int_{ - \infty }^{ + \infty } {{e^{ - {\eta ^2}}}d\eta } } \right)^{\frac{1}{2}}}{\left( {2t} \right)^{\frac{1}{4}}} = \frac{C}{{{t^{\frac{1}{4}}}}}

for some constant C.

Note that the above pointwise estimate shows us that the function u is decreasing with respect to time t. This is reasonable because, for example, the heat equation predicts that if a hot body is placed in a box of cold water, the temperature of the body will decrease, and eventually (after infinite time, and subject to no external heat sources) the temperature in the box will equalize.

Advertisements

Leave a Comment »

No comments yet.

RSS feed for comments on this post. TrackBack URI

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Create a free website or blog at WordPress.com.

%d bloggers like this: