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Results tagged with ap.analysis-of-pdes
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user 75422
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
1
vote
On solutions of the continuity equation
If $j$ has compact support in $\mathbb R_t\times\mathbb R^3_x$, $\rho$ will have compact support iff $\int_{-\infty}^\infty\nabla\cdot j_s\ ds\equiv0$.
The set of approximable $L^2$ solutions should …
- 3,085
5
votes
1
answer
314
views
Obstruction to Navier-Stokes blowup with cylindrical symmetry
Is there a known obstruction to cylindrically symmetric solutions (with swirl) of incompressible 3D Navier-Stokes blowing up in finite time ?
EDIT: in the whole space $\mathbb R^3$, I forgot to say.
…
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1
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Finding Free surface elevation in semi-infinite channel
What I seem to understand is that the bottom is at $y=-h+v(x-ct)$ where $v$ has compact support. That corresponds to an obstacle moving at a constant speed towards the infinite end. But it might mean …
- 3,085
2
votes
1
answer
110
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Smoluchowski-Poisson dynamics with atomic measures
"Smoluchowski-Poisson dynamics" is just a tentative provisional name I give to the following transport equation:$$\partial_t m+\nabla_x\cdot(um)=0$$where $u(x,t)\in\mathbb R^n$ ($x\in\mathbb R^n$, $t\ …
- 3,085
5
votes
The Biharmonic Eigenvalue Problem on a Rectangle with Dirichlet Boundary Conditions
At least the existence of eigenvalues and eigenfunctions (question 1) is routine common knowledge. The first one is the minimizer of $\int (\Delta u)^2$ subject to $u\in H^2_0$ and $\int u^2\le1$ (whi …
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0
votes
Accepted
The monotone operator in $BV$ space
$\varphi:=|.|_{TV}$ is a convex and (I think) lower semicontinuous function from $L^2$ to $[0,\infty]$. Its subdifferential $\partial\varphi$ is what you need to write the Euler-Lagrange equation, in …
- 3,085
2
votes
A $W^{1,2}_{loc}$ function with uniformly bounded integrals on compact subsets $W^{1,2}$?
Yes. There is an increasing sequence of compacts $K_n\subset\Omega$ whose union is $\Omega$, then $\int_{\Omega}(|f|^2+|\nabla f|^2)=\lim_{n\to\infty}\int_{K_n}(|f|^2+|\nabla f|^2)\le C$.
- 3,085
2
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An acting condition for a superposition operator from $H^1(\Omega)$ to $H^1(\Omega)$
The obvious sufficient condition for $v\mapsto f(v)$ to map $H^1(\Omega)$ to itself ($\Omega$ bounded) is $f$ globally Lipschitz, i.e. $f'\in L^\infty(\mathbb R)$. Add $f(0)=0$ for general $\Omega$.
…
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2
votes
Are Pointwise conditions studied?, do they make sense?, do they have any applications?
Such interpolatory conditions are better treated by solving your PDE $Af=\sum_ic_i\delta_{p_i}$ as distributions in the whole domain, with the $c_i$s to be determined so that $f(p_i)=d_i$. For ellipti …
- 3,085
1
vote
Accepted
Is this set of function belongs to $L^\infty$?
Let $\Omega=[-1,1]$ and $u(x)=u(-x)=0$ for $x\in[2^{-2k+1},2^{-2k+2})$ and $u(x)=u(-x)=2^{-k}$ for $x\in[2^{-2k},2^{-2k+1})$, $k=1,\ldots,n,\ldots$.
Doesn't this $u$ belong to your space $SBV\cap L^\ …
- 3,085
1
vote
Accepted
Does this time-dependent trace space have a name?
If you insist on "for all $t$ " as distinct from "for almost every $t$ ", one possible space is $C^1([0,\infty);H^{-1/2})\cap C^0([0,\infty);H^{1/2})$. A possible extension is the unique harmonic func …
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2
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Vorticity equation for generalized Naiver Stokes equations
Taking the curl of the original equation in $\mathbb R^3$ gives$$\partial_t\omega+\nu(-\Delta)^\alpha\omega+(u\cdot\nabla)\omega-(\omega\cdot\nabla) u=\nabla\times f$$ $$\nabla\times u=\omega$$ $$\nab …
- 3,085
2
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Stationary Navier-Stokes solutions
Tai-Peng Tsai's book Lectures on Navier-Stokes Equations (2018) cites as Theorem 8.3 (p.149) a theorem of V. Sverak (2011) that excludes the existence of minus one homogeneous solutions on $\mathbb R^ …
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8
votes
1
answer
328
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Continuous right inverse to the Laplacian operator on $C^\infty$
For each $f\in C^\infty(\mathbb R^n)$, there exists $u\in C^\infty(\mathbb R^n)$ such that $\Delta u=f$. This, I guess, has been known well before the more general Malgrange-Ehrenpreis theorem that sa …
- 3,085
6
votes
1
answer
423
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Stationary Navier-Stokes solutions
Are there known nontrivial ($u\neq0$) stationary solutions to Navier-Stokes equations in $\mathbb R^3$ ? Not square integrable of course (that's impossible), but with self-similar amplitudes of Fourie …
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