Do you want to publish a course? Click here

Uniqueness of very weak solutions for a fractional filtration equation

130   0   0.0 ( 0 )
 Added by Gabriele Grillo
 Publication date 2019
  fields
and research's language is English




Ask ChatGPT about the research

We prove existence and uniqueness of distributional, bounded, nonnegative solutions to a fractional filtration equation in ${mathbb R}^d$. With regards to uniqueness, it was shown even for more general equations in [19] that if two bounded solutions $u,w$ of (1.1) satisfy $u-win L^1({mathbb R}^dtimes(0,T))$, then $u=w$. We obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions, provided they are nonnegative. Indeed, we show that a minimal solution exists and that any other solution must coincide with it. As a consequence, distributional solutions have locally-finite energy.



rate research

Read More

109 - Mark Allen 2017
We prove uniqueness for weak solutions to abstract parabolic equations with the fractional Marchaud or Caputo time derivative. We consider weak solutions in time for divergence form equations when the fractional derivative is transferred to the test function.
In this paper we study the Cauchy problem for the Landau Hamiltonian wave equation, with time dependent irregular (distributional) electromagnetic field and similarly irregular velocity. For such equations, we describe the notion of a `very weak solution adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifier of the coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or distributional type solutions under conditions when such solutions also exist.
In this paper we prove the existence of weak solutions for a thermodynamically consistent phase-field model introduced in [26] in two and three dimensions of space. We use a notion of solution inspired by [18], where the pointwise internal energy balance is replaced by the total energy inequality complemented with a weak form of the entropy inequality. Moreover, we prove existence of local-in-time strong solutions and, finally, we show weak-strong uniqueness of solutions, meaning that every weak solution coincides with a local strong solution emanating from the same initial data, as long as the latter exists.
We show that the Hunter-Saxton equation $u_t+uu_x=frac14big(int_{-infty}^x dmu(t,z)- int^{infty}_x dmu(t,z)big)$ and $mu_t+(umu)_x=0$ has a unique, global, weak, and conservative solution $(u,mu)$ of the Cauchy problem on the line.
127 - Xiaoyu Zeng , Yimin Zhang 2017
For a class of Kirchhoff functional, we first give a complete classification with respect to the exponent $p$ for its $L^2$-normalized critical points, and show that the minimizer of the functional, if exists, is unique up to translations. Secondly, we search for the mountain pass type critical point for the functional on the $L^2$-normalized manifold, and also prove that this type critical point is unique up to translations. Our proof relies only on some simple energy estimates and avoids using the concentration-compactness principles. These conclusions extend some known results in previous papers.
comments
Fetching comments Fetching comments
Sign in to be able to follow your search criteria
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا