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Results tagged with ap.analysis-of-pdes
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user 3948
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
A question on theorem 1.1 of Fritz John ultrahyperbolic pde
I don't have John's paper available at the moment, but are you sure
$$ | \xi - \eta| = \sum (\xi_i - \eta_i)^2 $$
and not
$$ | \xi - \eta|^2 = \sum (\xi_i - \eta_i)^2 ? $$
For one, I don't think J …
- 39.1k
1
vote
Accepted
Estimates for Klein-Gordon-Equation follow directly from Wave equation Estimates
I don't think they mean that you can literally just plug in Theorem 1.3 of Sterbenz-Rodnianski to get Lemma 3.1. I think they mean that the proof is basically the same, with suitable adjustments.
A …
- 39.1k
1
vote
Accepted
Estimating sums with restrictions to different Frequencies
To answer your question 1 (and I expect the answer to question 2 is similar):
Note that $I^+(f,g)$ first take the product $fg$ and act on it as a Fourier multiplier. So the frequency support of $I^+( …
- 39.1k
2
votes
Accepted
IBVP with transformed boundary conditions
You can't use your theorem. Your boundaries are not straight. Furthermore, your boundary conditions are independent of $t$, which requires $M = 0$, and this puts strong requirements on the initial dat …
- 39.1k
2
votes
Energy methods for first order systems
The literature on first order hyperbolic PDEs is vast.
Some places where you can get some answers:
Courant and Hilbert, Methods of mathematical physics, Vol II.
K.O. Friedrichs, "Symmetric hyperb …
- 39.1k
3
votes
Proof of global (in time) existence of classical solutions for 2D Euler equation in bounded ...
When you say you cannot find a proof in English reference, I assume you know about the work of Yudovitch, that of Wolibner, and of Bardos.
Are you, however, aware of the work of Kato, which is in En …
- 39.1k
3
votes
Accepted
Conservation law for the Camassa-Holm equation
The following should work.
First, denote by $U(t,x) = \int_{-\infty}^x u(t,y) ~dy$.
If you write the equation as
$$ u_t-u_{txx}+2k u_x=-3uu_x+u_xu_{xx}+(uu_{xx})_x$$
and take the primitive in $x$, you …
- 39.1k
3
votes
Lens-shaped vs globally hyperbolic
Christodoulou actually doesn't give the complete proof (which is why I said a proof needs to be extracted). Here let me give a proof of the geometric fact (the connection with the analysis will have t …
- 39.1k
3
votes
Accepted
Methods for determining domains of influence
I highly doubt the result you actually asked for is true.
Consider the linear wave equation on $(1+3)$ Minkowski space. The analytic domain of influence of a point $x$ as Lax defined it, which moral …
- 39.1k
5
votes
the inverse for the trace theorem
I think your question is answered in
Article (JonWal1978) Jonsson, A. & Wallin, H. A Whitney extension theorem in $L_p$ and Besov spaces Ann. Inst. Fourier (Grenoble), 1978, 28, vi, 139-192
and
A …
- 39.1k
2
votes
A basic stability question
Let $\Omega$ be the unit ball. Let $f$ be any non-positive radially symmetric smooth function and let $V_1 = V_2 = r f\partial_r$, hence smooth vector fields on $\Omega$.
Any and all radially decrea …
- 39.1k
2
votes
Accepted
A basic stability question
Let $\Omega$ be the unit ball in $\mathbb{R}^2$.
Let $u_k(x,y) = \tan^{-1}(k^3 x)$.
Let $v_k(x,y)$ be a function that agrees with $u_k$ on $\partial\Omega$, and is constant on the level sets of $\ …
- 39.1k
3
votes
Accepted
Stability and symmetries
First, you seem to be misunderstanding what orbital stability means.
The notion of orbital stability should be considered as in contrast against asymptotic stability. The latter notion is the stabil …
- 39.1k
2
votes
Accepted
Wavefront set of a product
In general you can only say that
$$ WF(uv) \subseteq \Bigg[ WF(u) \cup WF(v) \cup \Big[WF(u)+WF(v)\Big]\Bigg] $$
where the set
$$ WF(u) + WF(v) = \left\lbrace (x,\xi +\eta)| (x,\xi)\in WF(u), (x,\eta) …
- 39.1k
4
votes
Accepted
Sobolev imbedding result; proof
This first part is an extended comment: I think you have your scaling wrong.
Let $u_k = \phi(\lambda_k(s-s_k), \mu_n(t-t_k))$, where $\phi$ has compact support in the ball of radius 1.
We assume that …
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