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Results tagged with ap.analysis-of-pdes
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user 146998
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
3
votes
1
answer
166
views
Strichartz estimates and Schrödinger equation with derivative
Let
$$ iu_t + \Delta u = \Phi \cdot \nabla u$$
with initial data $u_0 \in H^s(\mathbb{R}^d)$ and $\Phi\colon \mathbb{R}^d \rightarrow \mathbb{R}^d$ sufficiently regular. Suppose I want to use Strichar …
- 469
1
vote
0
answers
118
views
Estimate of a product of functions where $1/p + 1/q >1$
I have the following problem. I need to estimate the following quantity:
$$|\nabla|^{-1}A u \nabla u \label{1}\tag{$*$}$$
I know that $A\in L^6(B_R)$, $u \in L^{14/5}\cap L^2(B_R)$ and $\nabla u \in L …
- 469
1
vote
0
answers
50
views
Strong convergence from elliptic equation
Consider the following equation
$$ \Delta A_n = -2\Im(\bar{u}_n(\nabla -iA_n)u_n) =: -J(u_n,A_n)$$
and suppose that $u_n \in L^{\infty}(I,H^1(\Omega))$ for some bounded $I\subset \mathbb{R}$, $\Omega\ …
- 469
1
vote
0
answers
87
views
Reference for an application of J. Simon "Compact sets in the space $L^p(0,T;B)$"
Theorem 5 p.84 of J. Simons paper "Compact sets in the space $L^p(0,T;B)$" states a generalization of the Aubin-Lions lemma which relaxes the required regularity in time to the existence of a modulus …
- 469
1
vote
1
answer
86
views
Why is this Hamiltonian flow of the Vlasov equation well defined?
Lions and Paul claim in their 1993 paper "Sur les mesures de Wigner" that the Hamiltonian flow
$$\dot{x} = \xi, \quad \dot{\xi} = - \nabla V(x) $$
of the Vlasov equation
$$\partial_t f + \xi \cdot \na …
- 469
2
votes
Accepted
Why is this Hamiltonian flow of the Vlasov equation well defined?
I have found an answer.
The condition $V\in C^{1,1}$ ensures that $\nabla V$ is Lipschitz which implies the existence of a global solution in the neighborhood of every point for the Cauchy problem of …
- 469
2
votes
0
answers
112
views
Behavior at infinity of an $L^2$ function with $L^2$ mixed second derivatives
If $f$, $\nabla_x \cdot \nabla_y f \in L^2(\mathbb{R}^d_x\times \mathbb{R}^d_y)$, what can be said about decay at infinity of $\nabla_x f$, $\nabla_y f$?
It is clear that $(\nabla_x^2 + \nabla_y^2) f …
- 469
1
vote
0
answers
88
views
Reference request: PDE of the form $(\Delta - |u|^2)f = F(u)$
I am interested in equations of the form
$$(\Delta -|u|^2)f = F(u)$$
where $F$ depends on $u$ and preferably on its derivative, too. $u$ is supposed to be given and $f$ the unknown. More precisely I a …
- 469
2
votes
2
answers
239
views
Why is this estimate about Besov norms true
For reference, I am reading the paper "Uniqueness of Finite Energy Solutions for Maxwell-Dirac and Maxwell-Klein-Gordon Equations" by Masmoudi and Nakanishi.
Let $A_0$ be a scalar function satisfying …
- 469
4
votes
1
answer
102
views
Gauge fixing for a semi-relativistic model involving electromagnetism
When studying non-relativistic charged particles in an electromagnetic field with self-interaction on usually relies on the Schrödinger-Maxwell system
\begin{align}
i\partial_t u = -(\nabla-iA)^2 u + …
- 469
2
votes
1
answer
203
views
Estimates for an elliptic PDE
Let's say I have an equation of the form $\Delta A = J$ where $J=u\nabla u + A|u|^2$ (Clarification: We are on $\mathbb{R}^3$ and $u$ is assumed to be in $H^1(\mathbb{R}^3)$). Then I could simply infe …
- 469