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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.
12
votes
Accepted
Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\D...
"The" completion is not always a space of functions, for $N=1$ or $2$ for example it is a quotient $D^{-2}L^2/P_1$ (equivalence classes of functions $u\in H^2_{loc}$ with $\partial_i \partial_j u\in L …
- 3,085
8
votes
1
answer
328
views
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
views
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 …
- 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 …
- 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.
…
- 3,085
5
votes
Gaussian distribution, maximum entropy and the heat equation
Besides the central limit theorem, there is the connection between diffusion and Wasserstein distance $W_2(p,q)$ (the minimum integral of squared distance from $x$ to $T(x)$ when $T$ maps $p$ to $q$) …
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4
votes
Accepted
Is it true that for dimension d=3, if $v\in H_0^1(\Omega)$ but $v\notin L_\infty(\Omega)$ th...
If it is true that $\int e^u<\infty$ whenever $u\in W^{1,3}$, for $d=3$, and $W^{1,3}$ is not a subset of $L^\infty$, then, since $W_0^{1,3}\subset H_0^1$, there certainly exist essentially unbounded …
- 3,085
3
votes
Real analyticity of solution of heat equation
In the case of $\mathbb R^n$, the analyticity-in-time of the solution, with $u_0\in\mathcal S'$ (tempered distributions), stems from that of $e^{-(t+i\tau)|\xi|^2}$ which implies that of $u(t+i\tau,x) …
- 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
votes
Reformulation of the classical Navier-Stokes equation as a semilinear evolution equation and...
You are right, $(4)$ has to be understood in the special sense that the semigroup $S(t)$ (and the (Leray?) $L^2$-projection operator to $H$, as well) extend to a wider space, that of "distributional d …
- 3,085
2
votes
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
2
votes
Vorticity equation for generalized Naiver Stokes equations
Taking the curl of the original equation in $\mathbb R^3$ gives$$\partial_t\omega+\nu(-\Delta)^\alpha\omega+(u\cdot\nabla)\omega-(\omega\cdot\nabla) u=\nabla\times f$$ $$\nabla\times u=\omega$$ $$\nab …
- 3,085
2
votes
Stationary Navier-Stokes solutions
Tai-Peng Tsai's book Lectures on Navier-Stokes Equations (2018) cites as Theorem 8.3 (p.149) a theorem of V. Sverak (2011) that excludes the existence of minus one homogeneous solutions on $\mathbb R^ …
- 3,085
2
votes
1
answer
110
views
Smoluchowski-Poisson dynamics with atomic measures
"Smoluchowski-Poisson dynamics" is just a tentative provisional name I give to the following transport equation:$$\partial_t m+\nabla_x\cdot(um)=0$$where $u(x,t)\in\mathbb R^n$ ($x\in\mathbb R^n$, $t\ …
- 3,085