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Results tagged with ap.analysis-of-pdes
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user 33135
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
2
votes
Solution formular for Laplace equation
Maybe the book "Handbook of Linear Partial Differential Equations for Engineers and Scientists" by Polyanin, A. D. Chapman & Hall/CRC, 2002, can help you.
By the way, if $f=0$ the solution (up to a …
- 843
5
votes
Blow up of solutions to parabolic PDEs
Certainly the size of the domain can play a role. For the classical case
$$
\partial_t u=\partial_x^2 u+u^2, \text{ $x$ in }[-L,L]
$$
with Dirichlet BC I recomend Evan's book (Chapter 9, I think). in …
- 843
0
votes
How to use Gronwall's inequality?
I would think that the authors want to say the following:
Define
$$
Y(t)=1+\int_0^t w(\tau)^{\alpha'}d\tau.
$$
Then your inequality reduces to
$$
Y'(t)\leq Y(t),
$$
so Gronwall inequality (differen …
- 843
0
votes
1
answer
146
views
Dispersive estimate for linear semigroup
Let's consider the propagator corresponding to the one-dimensional equation
$$
u_t=\Lambda^\alpha u_x,\; u(x,0)=f(x)
$$
where
$$
\widehat{\Lambda^\alpha u}=|\xi|^\alpha\hat{u}(\xi),
$$
and $-1< \alph …
- 843
0
votes
Maximum of the solution of a parabolic PDE
As the initial data is perfectly smooth, I think that it is easy to apply the standard energy method to obtain a local in time existence in $H^s$ for any $s>0$. Take $s=3$. Then $g(t)=u(x^*,t)$ is a L …
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0
votes
1
answer
148
views
Integrability of the Poisson integral
Maybe this is rather obvious, but I'm stuck. Let's consider the Laplace equation in the upper half plane with boundary condition $g$, $i.e.$
$$
\Delta u(x,y)=0, u(x,0)=g(x).
$$
Then the solution is gi …
- 843
0
votes
Heat equation with Neumann BC
A very naive answer:
Assume that the initial data is positive (the same should be true dealing with absolute values...) and take p>2 (p=2 follows the same idea). Multiply the equation by $pu^{p-1}$ a …
- 843
0
votes
I have this linear PDE...
This is an elliptic operator. I think that Chapter 6 of Evans's book "Partial differential equations" is appropriate if the domain is bounded. In the whole plane you can use Fourier techniques, isn't …
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1
vote
Does $u_{t}=g(t)u_{x}^{2}$ blow-up for bounded positive g? What about $u_{t}=u_{xx}+g(t)u_{x...
At least for initial data verifying
$$
\|u_0\|_{L^\infty}\leq 1/3
$$
there is no blow up. The reason is that the $L^\infty$ norm is uniformly bounded:
$$
\|u(t)\|_{L^\infty}\leq \|u_0\|_{L^\infty}
$ …
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3
votes
Using Galerkin method for PDE with Neumann boundary condition?
I'll try my best:
As $\partial \Omega$ is measure zero and Lp functions are defined a.e., you need that the trace operator (see http://en.wikipedia.org/wiki/Trace_operator) is well defined. For insta …
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13
votes
4
answers
837
views
Mathematical difference between entropy and energy
I have a rather soft question. Let's assume that we consider the heat equation posed in $S^1$:
$$
\partial_t u=\partial_x^2u.
$$
It is well known that if we define the functionals
$$
H(t)=\int_{-\pi}^ …
- 843
10
votes
Accepted
integration by parts for the fractional Laplacian
You can integrate by parts:
$$
\int_{\mathbb{R}^d} (-\Delta)^s f(x) g(x)dx=\int_{\mathbb{R}^d} (-\Delta)^s g(x) f(x)dx.
$$
Using Fourier and $L^2$ the equality is obvious. Let's do "by hand" in $d=1 …
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2
votes
1
answer
234
views
A bound in Sobolev spaces of negative order
Let's consider the domain $U=[-\pi,\pi]\times[-1,1]$. Assume that we have two functions $f\in H^2$ and $g\in H^{1/2}$.
I wonder if the following bound is true:
$$
\|f g_{x_1}\|_{H^{-0.5}(U)}\leq C( …
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1
vote
weak solution of viscous Burgers equation with non-homogeneous Dirichlet boundary conditions
Try to substract $g(x)=\alpha-x(\alpha-\beta)$. Now the dirichlet conditions of the new unknown $h=\rho-g$ are homogeneous. The new problem is
$$
\partial_t h=\partial_u^2h-E\partial_u f(h)+F(g).
$$ …
- 843
0
votes
Blow-up for the quasilinear heat equation $u_t= u \ u_{x x}$ or the related $w_t= \left(w_x ...
I don't know if this is useful, but anyway: Let's compute
$$
\partial_t \partial_x u=\partial_x u\partial_x^2 u+u\partial_x^3u,
$$
and
$$
\partial_t \partial_x^2 u=\partial_x^2 u\partial_x^2 u+\partia …
- 843