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Results tagged with ap.analysis-of-pdes
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user 5371
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
Accepted
Constant in Poincaré Inequality
This is a fairly standard stuff. Suppose that the Stokes operator $A=-\Delta$ is defined on smooth divergence-free vector fields $u$ which satisfy the standard no-slip boundary condition $$\qquad\qqua …
- 22.3k
9
votes
PDEs as a tool in other domains in mathematics
PDEs are massively used in the theory of harmonic maps.
My personal favourite is a nice theorem by Lemaire and Sacks-Uhlenbeck.
Theorem. Suppose $M$ is a compact Riemann surface, possibly with bo …
3
votes
Accepted
Invertibility of the Laplacian
A careful exposition can be found in "Analytic Semigroups and Semilinear Initial Boundary Value Problems" by Kazuaki Taira.
He considers the mixed boundary value problem
$$\begin{cases} A u=f & \mb …
- 22.3k
12
votes
Harmonic Functions
$f$ is harmonic under the weaker assumption that it is just continuous.
Multiplying the identity
$$f(x+\delta,y+\delta)+f(x-\delta,y+\delta)+u(x-\delta,y-\delta) + f(x+\delta,y-\delta)-4f(x,y)=0$$
wi …
- 22.3k
4
votes
Trace space and Neumann boundary condition
If you are interested in $L^p$-theory, you are probably looking for solutions belonging to a Sobolev class $H^{s,p}(\Omega)$ with some $s>0$ and $p>1$. In this case, the Besov space
$B^{s-1-1/p,p}(\p …
- 22.3k
46
votes
Counterexamples in PDE
Scheffer has shown that there is a nontrivial weak solution $u(x,t)\in L^2(\mathbb R^2\times\mathbb R)$ to the incompressible Euler equations in 2D
$$\begin{cases} \frac{\partial u}{\partial t}+\nabla …
4
votes
Can the "physical argument" for the existence of a solution to Dirichlet's problem be made i...
The electrostatic intuition does lead to a correct mathematical formulation of the Dirichlet problem.
Let's consider an electric charge distribution of two thin layers (one layer is positive and the …
- 22.3k
19
votes
Accepted
Is there a mathematically precise definition of turbulence for solutions of Navier-Stokes?
There is probably no universally accepted mathematical definition of turbulence. (By the way, is there a physical one?) Moreover, the prevailing definitions seem to be highly volatile and time-depend …
- 22.3k
13
votes
Where was/is Compensated Compactness used?
Compensated compactness helps when one needs to find the limit of $u_n \cdot v_n$, where the sequences of vector fields $u_n$ and $v_n$ converge weakly in $L^2$: $u_n\rightharpoonup u$, $v_n\righth …
- 22.3k
33
votes
5
answers
3k
views
How to define a differential form on a fractal?
It is well known how to construct a Laplacian on a fractal using the Dirichlet forms (see e.g.
the survey article by Strichartz). This implies, in particular, that a fractal can be "heated", i.e. one …
- 22.3k